FDS_LAB_Practical_8_Probability.ipynb

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        "<a href=\"https://colab.research.google.com/github/ypushpa9420/FDS-Lab-PG1/blob/FDS_Practical_10/FDS_LAB_Practical_8_Probability.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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    {
      "cell_type": "markdown",
      "metadata": {
        "id": "r_qavwqb0r10"
      },
      "source": [
        "**FDS LAB Assignment:**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2BYW9GdE05jg"
      },
      "source": [
        "# **Probability**\r\n",
        "\r\n",
        "\r\n",
        "**Definition :** Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.The probability of all the events in a sample space adds up to 1.\r\n",
        "\r\n",
        "**Formula for Probability**\r\n",
        "\r\n",
        "The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.\r\n",
        "\r\n",
        "**Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "W5uVC8G2SeOF"
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      "source": [
        "**1. Types of Events in Probability.**\r\n",
        "\r\n",
        "Some of the important probability events are:\r\n",
        "\r\n",
        "*Impossible and Certain Events\r\n",
        "\r\n",
        "*Simple Events\r\n",
        "\r\n",
        "*Compound Events\r\n",
        "\r\n",
        "*Independent and Dependent Events\r\n",
        "\r\n",
        "*Mutually Exclusive Events\r\n",
        "\r\n",
        "*Complementary Events\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "L4w8US34Sp1R"
      },
      "source": [
        "1 #Impossible and Certain Events"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lxUuPXPpSz5h"
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      "source": [
        "If the probability of occurrence of an event is 0, such an event is called an impossible event and if the probability of occurrence of an event is 1, it is called a sure event. In other words, the empty set 1 is an impossible event and the sample space S is a sure event."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "43XkzRjLTK0h"
      },
      "source": [
        "eg. 1:\r\n",
        " A spinner has 4 equal sectors colored yellow, blue, green, and red. What is the probability of landing on purple after spinning the spinner?\r\n",
        "Probability: It is impossible to land on purple since the spinner does not contain this color.\r\n",
        "\r\n",
        "P(purple) =0/ = 0\r\n",
        "\r\n",
        "eg. 2:\r\n",
        " A teacher chooses a student at random from a class of 30 girls. What is the probability that the student chosen is a girl?\r\n",
        "Probability: Since all the students in the class are girls, the teacher is certain to choose a girl.\r\n",
        "\r\n",
        "P(girl) = 30/30 = 1\r\n",
        "\r\n",
        "eg.3:\r\n",
        " A single 6-sided die is rolled. What is the probability of rolling a number less than 7?\r\n",
        "Probability: Rolling a number less than 7 is a certain event since a single die has 6 sides, numbered 1 through 6.\r\n",
        "\r\n",
        "P(number < 7) = 6/6 = 1\r\n",
        "\r\n",
        "eg. 4:\r\n",
        " A total of five cards are chosen at random from a standard deck of 52 playing cards. What is the probability of choosing 5 aces?\r\n",
        "Probability: It is impossible to choose 5 aces since a standard deck of cards has only 4 of a kind. This is an impossible event.\r\n",
        "\r\n",
        "P(5 aces) = 0 /52 = 0\r\n",
        "\r\n",
        "eg. 5:\r\n",
        " A glass jar contains 15 red marbles. If a single marble is chosen at random from the jar, what is the probability that it is red?\r\n",
        "Probability: Choosing a red marble is certain to occur since all 15 marbles in the jar are red. This is a certain event.\r\n",
        "\r\n",
        "P(red) = 15/15 = 1"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Z0-lBMWMTyF7"
      },
      "source": [
        "2 #Simple Events and Compound Events"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9CQSwuigaGHi"
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      "source": [
        "Any event consisting of a single point of the sample space is known as a simple event in probability. For example, if S = {56 , 78 , 96 , 54 , 89} and E = {78} then E is a simple event.\r\n",
        "\r\n",
        "Contrary to the simple event, if any event consists of more than one single point of the sample space then such an event is called a compound event. Considering the same example again, if S = {56 ,78 ,96 ,54 ,89}, E1 = {56 ,54 }, E2 = {78 ,56 ,89 } then, E1 and E2 represent two compound events."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FajWU4P1T5oJ"
      },
      "source": [
        "def event_probability(n,s):\r\n",
        "  return n/s"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WjoO_zg8U7aa"
      },
      "source": [
        "eg.1"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cmSxWMdHUCwK",
        "outputId": "fc5ac188-90e0-4b58-fed3-bdb5f68839ce"
      },
      "source": [
        "# probability of getting 3 when a die is rolled\r\n",
        "ns = 6 #n(S) = {1,2,3,4,5,6}\r\n",
        "na = 1 #n(A) = {3}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of getting 3 is:\",pa)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of getting 3 is: 0.16666666666666666\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-sKyTDsPU_mq"
      },
      "source": [
        "eg.2"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ewHHw8sLU4Jy",
        "outputId": "ffddd26f-8dc7-4662-ee72-64b10946027e"
      },
      "source": [
        "# probability of getting Heads when a fair coin is tossed\r\n",
        "ns = 2 #n(S) = {H,T}\r\n",
        "na = 1 #n(A) = {H}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of getting head is:\",pa)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of getting head is: 0.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EzN0oGHfVFBS"
      },
      "source": [
        "eg.3"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ozJWY1BuVG_t",
        "outputId": "49a081b8-c9d5-419a-da7b-8c4684a71276"
      },
      "source": [
        "# probability of atleast getting one head when a coin is tossed thrice\r\n",
        "ns = 8 #n(S) = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}\r\n",
        "na = 7 #n(A) = {HHH, HHT, HTH, THH, TTH, THT, HTT}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of getting atleast one head is:\",pa)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of getting atleast one head is: 0.875\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Tlpgvg9uVP7p"
      },
      "source": [
        "eg.4"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "HZhQT2deVRb7",
        "outputId": "5753bfb2-f52d-4f91-fded-8c42c877c065"
      },
      "source": [
        "#probabiltiy of not getting 5 when a fair die is rolled\r\n",
        "ns = 6 #n(S) = {1,2,3,4,5,6}\r\n",
        "na = 5 #n(A) = {1,2,3,4,6}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of not getting 5 is:\",pa)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of not getting 5 is: 0.8333333333333334\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xyHiIqD4VflB"
      },
      "source": [
        "eg.5"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "lL9u2MhTVhHU",
        "outputId": "dbe075d6-fe61-4584-f065-93c3f9abdfde"
      },
      "source": [
        "# A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at random from the jar, what is the probability that it is not blue?\r\n",
        "ns = 10 #n(S) = {5red,3blue,2green}\r\n",
        "na = 7 #n(A) = {5red, 2green}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of getting not blue jellybean is:\",pa)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of getting not blue jellybean is: 0.7\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XfeWQW3LWKQZ",
        "outputId": "beac8f1c-897f-42c5-f7b6-544eec927496"
      },
      "source": [
        "# A number x is chosen at random from the numbers -2, -1, 0 , 1, 2. Then the probability that x2 < 2 is?\r\n",
        "ns = 5 #n(S) = {-2, -1, 0 , 1, 2}\r\n",
        "na = 3 #n(A) = {-2, -1, 0}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of x2 < 2:\",pa)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of x2 < 2: 0.6\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "hMdVlOuAWMFp",
        "outputId": "3bdd2d23-5f38-47bd-ee8c-1f11602654f0"
      },
      "source": [
        "#A number from 1 to 5 is chosen at random. What is the probability that the number chosen is not odd?\r\n",
        "ns = 5 #n(S) = {1,2,3,4,5}\r\n",
        "na = 2 #n(A) = {2,4}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probability of getting not odd number:\",pa)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of getting not odd number: 0.4\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "IT2BOPXSVmqS"
      },
      "source": [
        "3 #Independent and Dependent Events"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gyRi6NSKacr6"
      },
      "source": [
        "If the occurrence of any event is completely unaffected by the occurrence of any other event, such events are known as an independent event in probability and the events which are affected by other events are known as dependent events."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "M9GObVPtXBRL"
      },
      "source": [
        "eg.1"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "7DQKjSHjVscT",
        "outputId": "61481039-ac4b-4b93-9dbb-b49c2e940f6d"
      },
      "source": [
        "# If the probability that person A will be alive in 20 years is 0.7 and the probability that person B will be alive in\r\n",
        "# 20 years is 0.5, what is the probability that they will both be alive in 20 years?\r\n",
        "\r\n",
        "#These are independent events, so\r\n",
        "P = 0.7*0.5\r\n",
        "print(\"probability that they will be alive after 20 years is:\",P)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability that they will be alive after 20 years is: 0.35\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0bKT4l2VXDPi"
      },
      "source": [
        "eg.2"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3AhGloRSW5HK",
        "outputId": "4a83ca27-61ee-4f7e-fc9d-6cd22f2c9a96"
      },
      "source": [
        "#A fair die is tossed twice. Find the probability of getting a 4 or 5 on the first toss and a 1,2, or 3 in the second toss.\r\n",
        "pa = event_probability(2,6) # probability of getting a 4 or 5 on the first toss\r\n",
        "pb = event_probability(3,6) # probability of getting 1,2,3 in second toss\r\n",
        "P = pa*pb \r\n",
        "print(\"probability of getting a 4 or 5 on the first toss and a 1,2, or 3 in the second toss is:\",P)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability of getting a 4 or 5 on the first toss and a 1,2, or 3 in the second toss is: 0.16666666666666666\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kge_LnHFXGWB"
      },
      "source": [
        "eg.3"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "15IaYxReXKir",
        "outputId": "50bd0a52-43f1-4172-a1e6-7d6c4957ae02"
      },
      "source": [
        "# A bag contains 5 white marbles, 3 black marbles and 2 green marbles. In each draw, a marble is drawn from the bag\r\n",
        "# and not replaced. In three draws, find the probability of obtaining white, black and green in that order.\r\n",
        "pw = event_probability(5,10)\r\n",
        "pb = event_probability(3,9)\r\n",
        "pg = event_probability(2,8)\r\n",
        "print(\"the probability of obtaining white, black and green in that order is \",(pw*pb*pg))\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "the probability of obtaining white, black and green in that order is  0.041666666666666664\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DgcCEo5xXNOC"
      },
      "source": [
        "4 #Mutually Exclusive Events"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yVJGQgxSakep"
      },
      "source": [
        "If the occurrence of one event excludes the occurrence of another event, such events are mutually exclusive events i.e. two events don’t have any common point. For example, if S = {1 , 2 , 3 , 4 , 5 , 6} and E1, E2 are two events such that E1 consists of numbers less than 3 and E2 consists of numbers greater than 4.\r\n",
        "\r\n",
        "So, E1 = {1,2} and E2 = {5,6} .\r\n",
        "\r\n",
        "Then, E1 and E2 are mutually exclusive."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZnAL89H7XXLa"
      },
      "source": [
        "eg.1"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "1Z8zucryXYxp",
        "outputId": "cc70b6ce-8357-48b2-eacc-6bd053cba46e"
      },
      "source": [
        "#What is the probability of a die showing a number 3 or number 5?\r\n",
        "pa = 1/6 # pa is the probability of getting a number 3\r\n",
        "\r\n",
        "pb = 1/6 #pb is the probability of getting a number 5\r\n",
        "\r\n",
        "Pab = pa + pb #Pab is the probability of getting 3 or 5\r\n",
        "print(\"Therefore, the probability of a die showing 3 or 5 is\",Pab)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Therefore, the probability of a die showing 3 or 5 is 0.3333333333333333\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "flZbRbokXhsi"
      },
      "source": [
        "eg.2"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Xak93PM7XjvC",
        "outputId": "fa1a674d-17a9-4b25-e530-721b73233925"
      },
      "source": [
        "# Sample Space\r\n",
        "cards = 52\r\n",
        "\r\n",
        "# Calculate the probability of drawing a heart or a club\r\n",
        "hearts = 13\r\n",
        "clubs = 13\r\n",
        "heart_or_club = event_probability(hearts, cards) + event_probability(clubs, cards)\r\n",
        "print(heart_or_club )"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZuR0t0-iX_WC"
      },
      "source": [
        "eg.3"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JQha3E80YAsy",
        "outputId": "ad4cda0b-3e32-4b4e-c9b4-5f201a828028"
      },
      "source": [
        "# Calculate the probability of drawing an ace, king, or a queen\r\n",
        "aces = 4\r\n",
        "kings = 4\r\n",
        "queens = 4\r\n",
        "ace_king_or_queen = event_probability(aces, cards) + event_probability(kings, cards) + event_probability(queens, cards)\r\n",
        "\r\n",
        "print(heart_or_club)\r\n",
        "print(ace_king_or_queen)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.5\n",
            "0.23076923076923078\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oFsU8ro-YMUi"
      },
      "source": [
        "eg.4"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "PxiXmzYTYRv6",
        "outputId": "dee4b7ec-44f5-439d-f3e7-742446005ca8"
      },
      "source": [
        "# In a deck of 52 card, the probability of getting king or queen is equal to:\r\n",
        "queen = 4\r\n",
        "king = 4\r\n",
        "king_or_queen = event_probability(queen, cards) + event_probability(king, cards)\r\n",
        "print(king_or_queen)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.15384615384615385\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1FWWxZ-YYgTu"
      },
      "source": [
        "eg.5"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "aGgj-XHuYilY",
        "outputId": "e923ddf6-f0f6-416f-dcd5-a021ff7909cf"
      },
      "source": [
        "# Calculate the probability of drawing a heart or an ace\r\n",
        "hearts = 13\r\n",
        "aces = 4\r\n",
        "ace_of_hearts = 1\r\n",
        "heart_or_ace = event_probability(hearts, cards) + event_probability(aces, cards) - event_probability(ace_of_hearts, cards)\r\n",
        "print(round(heart_or_ace, 1))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.3\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "c--I-8ouY0Fq"
      },
      "source": [
        "eg.6"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "gxOvDdqWY1_T",
        "outputId": "ab1713f0-21dc-47eb-d7ee-b5fd3cd56e36"
      },
      "source": [
        "red_cards = 26\r\n",
        "face_cards = 12\r\n",
        "red_face_cards = 6\r\n",
        "red_or_face_cards = event_probability(red_cards, cards) + event_probability(face_cards, cards) - event_probability(red_face_cards, cards)\r\n",
        "\r\n",
        "print(round(heart_or_ace, 1))\r\n",
        "print(round(red_or_face_cards, 1))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.3\n",
            "0.6\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "iGu2mHWzY8M6"
      },
      "source": [
        "5 #Complementary Events"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "COxN4k8yauMK"
      },
      "source": [
        "For any event E1 there exists another event E1‘ which represents the remaining elements of the sample space S.\r\n",
        "\r\n",
        "E1 = S − E1‘\r\n",
        "\r\n",
        "If a dice is rolled then the sample space S is given as S = {1 , 2 , 3 , 4 , 5 , 6 }. If event E1 represents all the outcomes which is greater than 4, then E1 = {5, 6} and E1‘ = {1, 2, 3, 4}.\r\n",
        "\r\n",
        "Thus E1‘ is the complement of the event E1.\r\n",
        "\r\n",
        "Similarly, the complement of E1, E2, E3……….En will be represented as E1‘, E2‘, E3‘……….En‘"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "03WMhj85ZE6T"
      },
      "source": [
        "eg.1\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "J4_3XyLJZGgx",
        "outputId": "de71829e-799b-4fbf-d050-9e69d86f7c3e"
      },
      "source": [
        "#probabiltiy of not getting 5 when a fair die is rolled\r\n",
        "ns = 6 #n(S) = {1,2,3,4,5,6}\r\n",
        "na = 1 #n(A) = {5}\r\n",
        "pa = na/ns # P(A)\r\n",
        "print(\"probabilty of not getting 5 is:\",1-pa)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probabilty of not getting 5 is: 0.8333333333333334\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "J5LT4k_yZOcR"
      },
      "source": [
        "eg.2\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Ob2CJJ9kZQEq",
        "outputId": "ceaa16ab-c4d3-492a-ff09-fe3af1279470"
      },
      "source": [
        "\r\n",
        "# a probability of getting a white balls from a bag of balls is 1/4\r\n",
        "p = 1/4\r\n",
        "# What is the probability of not getting a white ball?\r\n",
        "print(\"Probability of not getting white ball is \",1-p)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Probability of not getting white ball is  0.75\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Bh9amROeZUsL"
      },
      "source": [
        "eg.3\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6YPC0Bx1ZWjx",
        "outputId": "281fc77e-7fe1-4b43-fbfc-03197c50a2df"
      },
      "source": [
        "# A number is chosen at random from a set of whole numbers from 1 to 50.\r\n",
        "# Calculate the probability that the chosen number is not a perfect square.\r\n",
        "n = 50 # Sample space\r\n",
        "a = 7 #{1,4,9,16,25,36,49}\r\n",
        "p = event_probability(a,n)\r\n",
        "print(\"the probability that the chosen number is not a perfect square is\",1-p)\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "the probability that the chosen number is not a perfect square is 0.86\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "f90wybtEu-4t"
      },
      "source": [
        "**2. Rules of Probability**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "InqGiXAeQavL"
      },
      "source": [
        "Basic Rules of Probability\r\n",
        "\r\n",
        "1.The chance that some event happens given all events that could happen in that category.\r\n",
        "\r\n",
        "Probabilities denote the chance that something happens, not the guarantee 0≤p(x)≤1 or 0%≤p(x)≤100%\r\n",
        "\r\n",
        "For any set of associated but separate events, ∑ip(xi)=1\r\n",
        "\r\n",
        "For related events A and B, the probablity that they happen together is p(A∩B)=p(A)⋅p(B)\r\n",
        "\r\n",
        "The probability that either one of them happens is p(A∪B)=p(A)+p(B)−p(A∩B)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rc-2usn2UEI7"
      },
      "source": [
        "Complementary rule for even"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vgLAAtgmorq5",
        "outputId": "81b96083-ca7f-42ad-9b00-1318496f4255"
      },
      "source": [
        "import math \r\n",
        "def findProb(l, r): \r\n",
        "      \r\n",
        "    # Count of perfect squares  \r\n",
        "    countOfPS = (math.floor(math.sqrt(r)) - \r\n",
        "                  math.ceil(math.sqrt(l)) + 1) \r\n",
        "      \r\n",
        "    # Total numbers in range l to r  \r\n",
        "    total = r - l + 1\r\n",
        "  \r\n",
        "    # Calculating probability  \r\n",
        "    prob = countOfPS / total \r\n",
        "      \r\n",
        "    return prob  \r\n",
        "      \r\n",
        "# Driver code  \r\n",
        "if __name__=='__main__': \r\n",
        "      \r\n",
        "    L = 16\r\n",
        "    R = 25\r\n",
        "      \r\n",
        "    print(findProb(L, R))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.2\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "j28S2qNsqUie"
      },
      "source": [
        "Complementary rule for odds"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "sqjxb1uSqUyV",
        "outputId": "fc0b7e9a-bd5b-4f44-f262-fc37ac1d3dd4"
      },
      "source": [
        "MAX=21\r\n",
        "\r\n",
        "fact=[0]*MAX\r\n",
        "\r\n",
        "# Returns probability of \r\n",
        "# getting at least k \r\n",
        "# heads in n tosses. \r\n",
        "def probability(k, n): \r\n",
        "\tans = 0\r\n",
        "\tfor i in range(k,n+1): \r\n",
        "\r\n",
        "\t\t# Probability of getting exactly i \r\n",
        "\t\t# heads out of n heads \r\n",
        "\t\tans += fact[n] / (fact[i] * fact[n - i]) \r\n",
        "\r\n",
        "\t# Note: 1 << n = pow(2, n) \r\n",
        "\tans = ans / (1 << n) \r\n",
        "\treturn ans \r\n",
        "\r\n",
        "def precompute(): \r\n",
        "\t\r\n",
        "\t# Preprocess all factorial \r\n",
        "\t# only upto 19, \r\n",
        "\t# as after that it \r\n",
        "\t# will overflow \r\n",
        "\tfact[0] = 1\r\n",
        "\tfact[1] = 1\r\n",
        "\r\n",
        "\tfor i in range(2,20): \r\n",
        "\t\tfact[i] = fact[i - 1] * i \r\n",
        "\r\n",
        "# Driver code \r\n",
        "if __name__=='__main__': \r\n",
        "\tprecompute() \r\n",
        "\r\n",
        "\t# Probability of getting 2 \r\n",
        "\t# head out of 3 coins \r\n",
        "\tprint(probability(2, 3)) \r\n",
        "\r\n",
        "\t# Probability of getting \r\n",
        "\t# 3 head out of 6 coins \r\n",
        "\tprint(probability(3, 6)) \r\n",
        "\r\n",
        "\t# Probability of getting \r\n",
        "\t# 12 head out of 18 coins \r\n",
        "\tprint(probability(12, 18)) \r\n",
        "\t\r\n",
        "\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.5\n",
            "0.65625\n",
            "0.1189422607421875\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "E1xeWs9gvCA7"
      },
      "source": [
        "Multiplication rule:\r\n",
        "\r\n",
        "P(A and B) = P(A) * P(B), "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "kcC6x7M_vDxe",
        "outputId": "a3b028cb-0cd0-4114-cc76-797b516ebbad"
      },
      "source": [
        "\r\n",
        "import numpy \r\n",
        "import numpy as np\r\n",
        " \r\n",
        "temps = [31, 32, 32, 31, 28, 29, 31, 38, 32, 31, 30, 29, 30, 31, 26] \r\n",
        " \r\n",
        "num_items = len(temps) \r\n",
        "product = 1. \r\n",
        " \r\n",
        "for temperature in temps: \r\n",
        "    product *= temperature \r\n",
        "     \r\n",
        "geometric_mean = product**(1./num_items) \r\n",
        " \r\n",
        "print(geometric_mean)   # == 30.634\r\n",
        "\r\n",
        "import numpy as np\r\n",
        "\r\n",
        "# create user preferences\r\n",
        "user_pref = np.array([5, 1, 3])\r\n",
        "\r\n",
        "# create a random movie matrix of 10,000 movies\r\n",
        "movies = np.random.randint(5,size=(3,1000))+1\r\n",
        "\r\n",
        "# Note that the randint will make random integers from 0-4\r\n",
        "# so I added a 1 at the end to increase the scale from 1-5\r\n",
        "\r\n",
        "print(user_pref.shape) # (1, 3)\r\n",
        "\r\n",
        "print(movies.shape)    # (3, 1000)\r\n",
        "\r\n",
        "# np.dot does both dot products and matrix multiplication\r\n",
        "recs = np.dot(user_pref, movies)\r\n",
        "\r\n",
        "import time\r\n",
        "\r\n",
        "for i in (10000, 100000, 1000000, 10000000, 100000000):\r\n",
        "    movies = np.random.randint(5,size=(3,i))+1\r\n",
        "    now = time.time()\r\n",
        "    np.dot(user_pref, movies)\r\n",
        "    print (time.time() - now), \"seconds to run\", i, \"movies\""
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "30.63473484374659\n",
            "(3,)\n",
            "(3, 1000)\n",
            "0.00021696090698242188\n",
            "0.0016713142395019531\n",
            "0.01735711097717285\n",
            "0.12940454483032227\n",
            "1.2655282020568848\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cdcxcTmsPMzq"
      },
      "source": [
        "Addition rule\r\n",
        "\r\n",
        "P\r\n",
        "(\r\n",
        "A\r\n",
        "∪\r\n",
        "B\r\n",
        ")\r\n",
        "=\r\n",
        "P\r\n",
        "(\r\n",
        "A\r\n",
        ")\r\n",
        "+\r\n",
        "P\r\n",
        "(\r\n",
        "B\r\n",
        ")\r\n",
        "−\r\n",
        "P\r\n",
        "(\r\n",
        "A\r\n",
        "∩\r\n",
        "B\r\n",
        ")"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6xbIyalDO-_c",
        "outputId": "a92058dd-f395-4775-967e-16b2f7e6a7b3"
      },
      "source": [
        "\r\n",
        "def event_probability(event_outcomes, sample_space):\r\n",
        "    probability = (event_outcomes / sample_space) * 100\r\n",
        "    return round(probability, 1)\r\n",
        "\r\n",
        "# Sample Space\r\n",
        "cards = 52\r\n",
        "\r\n",
        "# Determine the probability of drawing a heart\r\n",
        "hearts = 13\r\n",
        "heart_probability = event_probability(hearts, cards)\r\n",
        "\r\n",
        "# Determine the probability of drawing a face card\r\n",
        "face_cards = 12\r\n",
        "face_card_probability = event_probability(face_cards, cards)\r\n",
        "\r\n",
        "# Determine the probability of drawing the queen of hearts\r\n",
        "queen_of_hearts = 1\r\n",
        "queen_of_hearts_probability = event_probability(queen_of_hearts, cards)\r\n",
        "\r\n",
        "# Print each probability\r\n",
        "print(str(heart_probability) )\r\n",
        "print(str(face_card_probability))\r\n",
        "print(str(queen_of_hearts_probability))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "25.0\n",
            "23.1\n",
            "1.9\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "j9Tn-zIyP1Wm"
      },
      "source": [
        "Conditional Probability"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "jmdJeK6vRlp6",
        "outputId": "d85eff7c-8c4d-4f77-9c37-98d3137e4b0f"
      },
      "source": [
        "import numpy as np\r\n",
        "def coinFlip(p):    \r\n",
        "    #perform the binomial distribution (returns 0 or 1)    \r\n",
        "    result = np.random.binomial(1,p)\r\n",
        "#return flip to be added to numpy array    \r\n",
        "    return result\r\n",
        "'''Main Area'''\r\n",
        "#probability of heads vs. tails. This can be changed.\r\n",
        "probability = .5\r\n",
        "#num of flips required. This can be changed.\r\n",
        "n = 10\r\n",
        "#initiate array\r\n",
        "fullResults = np.arange(n)\r\n",
        "#perform desired numbered of flips at required probability set above\r\n",
        "for i in range(0, n):    \r\n",
        "    fullResults[i] = coinFlip(probability)    \r\n",
        "    i+=1\r\n",
        "#print results\r\n",
        "print(\"probability is set to \", probability)\r\n",
        "print(\"Tails = 0, Heads = 1: \", fullResults)\r\n",
        "#Total up heads and tails for easy user experience \r\n",
        "print(\"Head Count: \", np.count_nonzero(fullResults == 1))\r\n",
        "print(\"Tail Count: \", np.count_nonzero(fullResults == 0))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "probability is set to  0.5\n",
            "Tails = 0, Heads = 1:  [0 0 0 0 0 0 1 0 0 1]\n",
            "Head Count:  2\n",
            "Tail Count:  8\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6wSruof4URPA"
      },
      "source": [
        "P(A∪B)=P(A)+P(B)−P(A∩B)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Z-oRAuNFTOFn",
        "outputId": "05a86156-a045-4069-e5e1-083a350aaf31"
      },
      "source": [
        "# Sample Space\r\n",
        "cards = 52\r\n",
        "hole_cards = 2\r\n",
        "# Your opponent provided you information... use it!\r\n",
        "opponents_hole_cards = 2 \r\n",
        "\r\n",
        "turn_community_cards = 4\r\n",
        "cards = cards - (hole_cards + opponents_hole_cards + turn_community_cards)\r\n",
        "\r\n",
        "# Outcomes\r\n",
        "diamonds = 13\r\n",
        "diamonds_drawn = 4\r\n",
        "\r\n",
        "# You can't count the two diamonds that won't help you win\r\n",
        "diamond_non_outs = 2 \r\n",
        "\r\n",
        "outs = diamonds - diamonds_drawn - diamond_non_outs\r\n",
        "\r\n",
        "# Determine win probability\r\n",
        "win_probability = outs / cards\r\n",
        "\r\n",
        "# Determine expected value\r\n",
        "pot = 60\r\n",
        "ev = pot * win_probability\r\n",
        "\r\n",
        "# Print ev and appropriate decision\r\n",
        "call_amount = 20\r\n",
        "if ev >= 20:\r\n",
        "    print(round(ev, 2), 'Call')\r\n",
        "else:\r\n",
        "    print(round(ev, 2), 'Fold')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "9.55 Fold\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "rmQl8GkHwa07"
      },
      "source": [
        "**3. Permutations and Combinations**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BwKkrw8YCHe4",
        "outputId": "93e7a927-cc4b-4894-dec6-e503fc2acc4f"
      },
      "source": [
        "                                                          #Permutation and Combination\r\n",
        "\r\n",
        "#------------------------------------------------------------------ Permutation ------------------------------------------------------------------------------\r\n",
        "#What is Permutation?\r\n",
        "#Permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging (reordering) its elements.\r\n",
        "#The formula for Permutation (non-repetitive) is :\r\n",
        "#P (n, r) =n! /(n-r)!  where n is the total no of element in a set and r is the no of elements taken from a given set.\r\n",
        "#The formula for Permutation (repetitive) is:\r\n",
        "#P (n, r) =n^r\r\n",
        "\r\n",
        "#--------------------------------------------------NON- REPITATIVE FORMULA FOR PERMUTATION---------------------------------------------------------------------\r\n",
        "\r\n",
        "import math\r\n",
        "#n=int (input (\"Enter the size of set : \"))\r\n",
        "#r=int (input (\"Enter the size of elements taken from the set : \"))\r\n",
        "\r\n",
        "def per(n,r):\r\n",
        "  return (math.factorial(n)/(math.factorial(n-r)))\r\n",
        "\r\n",
        "#------------------------------------------------------ REPITATIVE FORMULA FOR PERMUTATION---------------------------------------------------------------------\r\n",
        "\r\n",
        "#a=int (input (\"Enter the size of set : \"))\r\n",
        "#b=int (input (\"Enter the size of elements taken from the set : \"))\r\n",
        "\r\n",
        "def per_with_rep(n,r):\r\n",
        "  return n**r\r\n",
        "\r\n",
        "\r\n",
        "#example 1 :\r\n",
        "\r\n",
        "#To find the number of permutations of pocket Aces from a pack of cards such that you only pick 2 cards\r\n",
        "\r\n",
        "#With no repitation.\r\n",
        "\r\n",
        "#There are four Aces in a deck of cards, and these are all the different combinations of pocket Aces\r\n",
        "\r\n",
        "#/1.Ace Hearts / Ace Diamonds\r\n",
        "#2.Ace Hearts / Ace Clubs\r\n",
        "#3.Ace Hearts / Ace Spades\r\n",
        "#4.Ace Diamonds / Ace Clubs\r\n",
        "#5.Ace Diamonds / Ace Spades\r\n",
        "#6.Ace Clubs / Ace Spades\r\n",
        "\r\n",
        "#Here n = 4 as only 4 ace cards are there in a pack is 52 and r = 2 as we have to pick 2 cards\r\n",
        "n = 4\r\n",
        "r = 2\r\n",
        "print(\"\\nNumber of Ace cards in a Pack : \",n)\r\n",
        "print(\"\\nNumber of cards to be withdrawn : \",r)\r\n",
        "print(\"\\nPermuation is \",per(n,r))\r\n",
        "\r\n",
        "# Example 2 :\r\n",
        "\r\n",
        "#what is the possible permutation of having a 4-digit pin number from 10 single digit number i.e from 0 to 9 such that it is not repitative.\r\n",
        "\r\n",
        "#Here n = 10 and r = 4\r\n",
        "\r\n",
        "n = 10\r\n",
        "r = 4\r\n",
        "print(\"\\nTotal number of digits : \",n)\r\n",
        "print(\"\\nNumber of Digits for a pin to be created: \",r)\r\n",
        "print(\"\\nPermuation is \",per(n,r))\r\n",
        "\r\n",
        "# Example 3 :\r\n",
        "\r\n",
        "#what is the possible permutation of having a 4-digit pin number from 10 single digit number i.e from 0 to 9 such that it is repitative.\r\n",
        "\r\n",
        "#Here n = 10 and r = 4 \r\n",
        "\r\n",
        "n = 10\r\n",
        "r = 4\r\n",
        "print(\"\\nTotal number of digits : \",n)\r\n",
        "print(\"\\nNumber of Digits for a pin to be created: \",r)\r\n",
        "print(\"\\nPermuation with repitation is \",per_with_rep(n,r))\r\n",
        "\r\n",
        "#Example 4 :\r\n",
        "\r\n",
        "#You have a website on which users can register. They need to provide a password that needs to be exactly 8 characters long and it should be a cbination of characters and numbers such that the characters and numbers cannot repeat. We first need to determine how many characters and digits there are in the English alphabet:\r\n",
        "#the number of letters: 26\r\n",
        "#the number of digits: 10\r\n",
        "#Which is 36 in total. So, n = 36. r would then be 8, because the password needs to be 8 characters long\r\n",
        "\r\n",
        "n = 36\r\n",
        "r = 8\r\n",
        "print(\"\\nTotal number of digits : \",n)\r\n",
        "print(\"\\nLength of password : \",r)\r\n",
        "print(\"\\nPermuation is \",per(n,r))\r\n",
        "\r\n",
        "#Example 5 :\r\n",
        "\r\n",
        "#You have a website on which users can register. They need to provide a password that needs to be exactly 8 characters long and it should be a cbination of characters and numbers such that the characters and numbers can repeat. We first need to determine how many characters and digits there are in the English alphabet:\r\n",
        "#the number of letters: 26\r\n",
        "#the number of digits: 10\r\n",
        "#Which is 36 in total. So, n = 36. r would then be 8, because the password needs to be 8 characters long\r\n",
        "\r\n",
        "n = 36\r\n",
        "r = 8\r\n",
        "print(\"\\nTotal number of digits : \",n)\r\n",
        "print(\"\\nLength of password : \",r)\r\n",
        "print(\"\\nPermuation with repitation is \",per_with_rep(n,r))\r\n",
        "\r\n",
        "#-------------------------------------------------------------------------Combinations--------------------------------------------------------------------\r\n",
        "\r\n",
        "#What is Combination ?\r\n",
        "#A combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter.\r\n",
        "#The formula for Combination (non-repetitive) is :\r\n",
        "#C(n,r)=n!/(r!(n-r)!)\r\n",
        "\r\n",
        "#The formula for Combination (repetitive) is :\r\n",
        "#C(n,r)=(n+r -1)!/(r!(n-1)!)\r\n",
        "\r\n",
        "#--------------------------------------------------NON- REPITATIVE FORMULA FOR COMBINATION---------------------------------------------------------------------\r\n",
        "\r\n",
        "#n=int(input(\"Enter the size of set : \"))\r\n",
        "#r=int(input(\"Enter the size of elements taken from the set : \"))\r\n",
        "\r\n",
        "def com(n,r):\r\n",
        "  return (math.factorial(n)/(math.factorial(r)*(math.factorial(n-r))))\r\n",
        "\r\n",
        "#------------------------------------------------------ REPITATIVE FORMULA FOR COMBINATION---------------------------------------------------------------------\r\n",
        "\r\n",
        "#a=int(input(\"Enter the size of set : \"))\r\n",
        "#b=int(input(\"Enter the size of elements taken from the set : \"))\r\n",
        "\r\n",
        "def com_with_rep(n,r):\r\n",
        "  return ((math.factorial(n+r-1))/(math.factorial(r)*(math.factorial(n-1))))\r\n",
        "\r\n",
        "# Example 1 \r\n",
        "\r\n",
        "#To find the number of permutations of pocket Aces from a pack of cards such that you only pick 2 cards\r\n",
        "\r\n",
        "#In a pack of card, no card is repetated so we will bw using the non repitative formula for combinations.\r\n",
        "\r\n",
        "#in this example,n = 52 as the no of cards in a pack is 52 and r = 2 as only 2 cards have to be withdrawn\r\n",
        "\r\n",
        "n=52\r\n",
        "r=2\r\n",
        "print(\"\\nTotal number of cards in a pack : \",n)\r\n",
        "print(\"\\nNumber of cards to be withdrawn : \",r)\r\n",
        "print(\"\\nCombination is \",com(n,r)) \r\n",
        "\r\n",
        "#example 2 :\r\n",
        "\r\n",
        "#On how many ways can you choose 11 people from a group of 110 for a football team?\r\n",
        "\r\n",
        "#here n= 110 and r = 11\r\n",
        "\r\n",
        "n=110\r\n",
        "r=11\r\n",
        "print(\"\\nTotal number of groups : \",n)\r\n",
        "print(\"\\nNumber of people to be chosen : \",r)\r\n",
        "print(\"\\nCombination is \",com(n,r)) \r\n",
        "\r\n",
        "#Example 3 :\r\n",
        "\r\n",
        "#You’re making a sandwich and you’re only allowed to use 4 ingredients out of 10 possible. However, the ingredients don’t have to be unique i.e the ingredients can be repitated.\r\n",
        "\r\n",
        "#As the ingrediants can be repitated, we wiil be using reitated formula of combination.\r\n",
        "\r\n",
        "#here n= 10 as it is the total no of ingredients available and r = 4 as it is the no of ingredients used.\r\n",
        "\r\n",
        "n=10\r\n",
        "r=4\r\n",
        "print(\"\\nTotal no of ingredients : \",n)\r\n",
        "print(\"\\nTotal no of ingredients to be taken : \",r)\r\n",
        "print(\"\\nCombination with repitation is \",com_with_rep(n,r))\r\n",
        "\r\n",
        "#Example 4 :\r\n",
        "\r\n",
        "#You’re making a sandwich and you’re only allowed to use 4 ingredients out of 10 possible. However, the ingredients have to be unique i.e the ingredients cannot be repitated.\r\n",
        "\r\n",
        "#As the ingrediants cannot be repitated, we wiil be using non repitated formula of combination.\r\n",
        "\r\n",
        "#here n= 10 as it is the total no of ingredients available and r = 4 as it is the no of ingredients used.\r\n",
        "\r\n",
        "n=10\r\n",
        "r=4\r\n",
        "print(\"\\nTotal no of ingredients : \",n)\r\n",
        "print(\"\\nTotal no of ingredients to be taken : \",r)\r\n",
        "print(\"\\nCombination is \",com(n,r))\r\n",
        "\r\n",
        "#Example 5 :\r\n",
        "\r\n",
        "#find all combinations containing exactly 2 letters from the word \"ABCDE\" such that its a unquie cobination i.e the letters should not be repetated\r\n",
        "\r\n",
        "#here n= 5 as the length of word \"ABCDE\" is 5 and r = 2 as we have to create combination with 2 letters of the word\r\n",
        "\r\n",
        "#The possible combinations are :\r\n",
        "\r\n",
        "#A B\r\n",
        "#A C\r\n",
        "#A D\r\n",
        "#A E\r\n",
        "#B C\r\n",
        "#B D\r\n",
        "#B E\r\n",
        "#C D\r\n",
        "#C E\r\n",
        "#D E\r\n",
        "\r\n",
        "n=5\r\n",
        "r=2\r\n",
        "print(\"\\nThe length of word 'ABCDE' : \",n)\r\n",
        "print(\"\\nNumber of letters taken from the word : \",r)\r\n",
        "print(\"\\nPossible Combination are \",com(n,r))\r\n",
        "\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "\n",
            "The size of set :  4\n",
            "\n",
            "The size of elements taken from the set :  2\n",
            "\n",
            "Permuation is  12.0\n",
            "\n",
            "The size of set :  10\n",
            "\n",
            "The size of elements taken from the set :  4\n",
            "\n",
            "Permuation is  5040.0\n",
            "\n",
            "Total number of digits :  10\n",
            "\n",
            "Number of Digits for a pin to be created:  4\n",
            "\n",
            "Permuation with repitation is  10000\n",
            "\n",
            "the size of set :  36\n",
            "\n",
            "the size of elements taken from the set :  8\n",
            "\n",
            "Permuation is  1220096908800.0\n",
            "\n",
            "the size of set :  36\n",
            "\n",
            "the size of elements taken from the set :  8\n",
            "\n",
            "Permuation with repitation is  2821109907456\n",
            "\n",
            "Total number of cards in a pack :  52\n",
            "\n",
            "Number of cards to be drawn :  2\n",
            "\n",
            "Combination is  1326.0\n",
            "\n",
            "the size of set :  110\n",
            "\n",
            "the size of elements taken from the set :  11\n",
            "\n",
            "Combination is  426342151127100.0\n",
            "\n",
            "the size of set :  10\n",
            "\n",
            "the size of elements taken from the set :  4\n",
            "\n",
            "Combination with repitation is  715.0\n",
            "\n",
            "the size of set :  10\n",
            "\n",
            "the size of elements taken from the set :  4\n",
            "\n",
            "Combination is  210.0\n",
            "\n",
            "The length of word 'ABCDE' :  5\n",
            "\n",
            "Number of letters taken from the word :  2\n",
            "\n",
            "Possible Combination are  10.0\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FhZX1icyw-G_"
      },
      "source": [
        "**4. Probability Likelihood**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RKSY2Gicy0Sv"
      },
      "source": [
        "Probability corresponds to finding the chance of something given a sample distribution of the data, while on the other hand, Likelihood refers to finding the best distribution of the data given a particular value of some feature or some situation in the data.\r\n",
        "\r\n",
        "Maximum likelihood estimation(MLE) is a method of estimating the parametersof a statistical model given observations, by finding the parameter values that maximize the likelihood of making the observations given the parameters. MLE can be seen as a special case of the maximum a posteriori estimation (MAP) that assumes a uniform prior distribution of the parameters, or as a variant of the MAP that ignores the prior and which therefore is unregularized.\r\n",
        "\r\n",
        "\r\n",
        "A statistical method for estimating population parameters (such as the mean and variance) from sample data that selects as estimates those parameter values maximizing the probability of obtaining the observed data is called MLE.\r\n",
        "\r\n",
        "The maximum likelihood estimation is a method that determines values for parameters of the model. It is the statistical method of estimating the parameters of the probability distribution by maximizing the likelihood function. The point in which the parameter value that maximizes the likelihood function is called the maximum likelihood estimate. \r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "P2Nhu8MNRMkK",
        "outputId": "f00bac5c-3c88-4884-f455-474165518f13"
      },
      "source": [
        "pip install numdifftools"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Collecting numdifftools\n",
            "\u001b[?25l  Downloading https://files.pythonhosted.org/packages/ab/c0/b0d967160ecc8db52ae34e063937d85e8d386f140ad4826aae2086245a5e/numdifftools-0.9.39-py2.py3-none-any.whl (953kB)\n",
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            "\u001b[?25hInstalling collected packages: numdifftools\n",
            "Successfully installed numdifftools-0.9.39\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "m5kGDFZmQ9yf",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "dc69e028-ec9b-44b0-f14b-6b56495271b1"
      },
      "source": [
        "import pymc3 as pm3\r\n",
        "import numpy as np\r\n",
        "import numdifftools as ndt\r\n",
        "import pandas as pd\r\n",
        "from scipy.stats import norm\r\n",
        "import statsmodels.api as sm\r\n",
        "from statsmodels.base.model import GenericLikelihoodModel\r\n",
        "from scipy.optimize import minimize\r\n",
        "import random\r\n",
        "import math\r\n",
        "import matplotlib.pylab as plt\r\n",
        "%matplotlib inline\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/statsmodels/tools/_testing.py:19: FutureWarning: pandas.util.testing is deprecated. Use the functions in the public API at pandas.testing instead.\n",
            "  import pandas.util.testing as tm\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PvHHD7xHa-oi"
      },
      "source": [
        "eg.1"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "uKv9HoLgHok0"
      },
      "source": [
        "def normal_pdf(x, mu = 1, sigma = 1):\r\n",
        "    return (1./np.sqrt(2*3.14 * sigma**2)) * 2.718**(-(x-mu)**2 / (2. * sigma**2))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "iEP9hKTeHstQ",
        "outputId": "e97ed29e-ae86-46cc-c547-94d37a727dc6"
      },
      "source": [
        "\r\n",
        "x_values = np.linspace(-10,10,100)\r\n",
        "y_values = [normal_pdf(x) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "\r\n",
        "x_values = np.linspace(-10,10,100)\r\n",
        "y_values = [normal_pdf(x, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "y_values = [normal_pdf(x, mu = -5, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f57e92a0990>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 11
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2q-T4zRza7fP"
      },
      "source": [
        "eg.2"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "bFo57BWHSdUz"
      },
      "source": [
        "def normal_pdf(x, mu = 2, sigma = 1):\r\n",
        "    return (1./np.sqrt(2*3.14 * sigma**2)) * 2.718**(-(x-mu)**2 / (2. * sigma**2))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "13NCsz1nR2cK",
        "outputId": "3e828b08-b3df-4eea-fa76-5b03bdb20a7e"
      },
      "source": [
        "x_values = np.linspace(-20,20,150)\r\n",
        "y_values = [normal_pdf(x) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "\r\n",
        "x_values = np.linspace(-10,10,100)\r\n",
        "y_values = [normal_pdf(x, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "y_values = [normal_pdf(x, mu = -5, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f57e8d9c9d0>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 13
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Flmmb_eTbHw2"
      },
      "source": [
        "eg.3"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_35d1uKobGlw"
      },
      "source": [
        "def normal_pdf(x, mu = 1, sigma = 1):\r\n",
        "    return (1./np.sqrt(2*3.14 * sigma**2)) * 2.718**(-(x-mu)**2 / (2. * sigma**2))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "jkDvAZXmbr_X",
        "outputId": "4e5892d2-a568-4a90-997c-9af68812c2da"
      },
      "source": [
        "x_values = np.linspace(-15,15,125)\r\n",
        "y_values = [normal_pdf(x) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "\r\n",
        "x_values = np.linspace(-10,10,100)\r\n",
        "y_values = [normal_pdf(x, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "y_values = [normal_pdf(x, mu = -5, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f57e8d20e50>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 15
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EXYDGtFdcwVw"
      },
      "source": [
        "eg.4"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-nuo5DBtdyTp"
      },
      "source": [
        "import math\r\n",
        "import numpy as np\r\n",
        "from scipy.stats import binom\r\n",
        "\r\n",
        "import statsmodels.api as sm\r\n",
        "import matplotlib.pyplot as plt\r\n",
        "\r\n",
        "%matplotlib inline\r\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 336
        },
        "id": "mHmIFW0nfHTo",
        "outputId": "3fcd2e79-e602-4074-94bc-d18694073e8d"
      },
      "source": [
        "x = np.linspace(0, 10, 11)\r\n",
        "y = [0] * len(x)\r\n",
        "\r\n",
        "fig, ax = plt.subplots(1, 3,sharex=True,sharey=True)\r\n",
        "fig.set_size_inches(20, 5)\r\n",
        "# \r\n",
        "\r\n",
        "# --------------------------------------------------------------------------------\r\n",
        "# First\r\n",
        "# 5 heads out of 10\r\n",
        "# --------------------------------------------------------------------------------\r\n",
        "first = []\r\n",
        "for i in range(11):\r\n",
        "    first.append(binom.pmf(n=10,k=i,p=0.5))\r\n",
        "\r\n",
        "ax[0].grid(\r\n",
        "    color='lightblue',\r\n",
        "    linestyle='--',\r\n",
        "    linewidth=0.5\r\n",
        ")    \r\n",
        "ax[0].plot(\r\n",
        "    [5],\r\n",
        "    [0],\r\n",
        "    linestyle=\"None\",\r\n",
        "    marker='o',\r\n",
        "    markersize=10,\r\n",
        "    color='r'\r\n",
        ")\r\n",
        "ax[0].vlines(x=x, ymin=0, ymax=first, colors='b', linestyles='--', lw=2)\r\n",
        "ax[0].set_title('Initial 5 out of 10 toss')\r\n",
        "\r\n",
        "# --------------------------------------------------------------------------------\r\n",
        "# First + Second likelifood overlay\r\n",
        "# Likelifood for having had 8 heads out of 10\r\n",
        "# --------------------------------------------------------------------------------\r\n",
        "ax[1].grid(\r\n",
        "    color='lightblue',\r\n",
        "    linestyle='--',\r\n",
        "    linewidth=0.5\r\n",
        ")\r\n",
        "ax[1].plot(\r\n",
        "    [5, 8],\r\n",
        "    [0 ,0],\r\n",
        "    linestyle=\"None\",\r\n",
        "    marker='o',\r\n",
        "    markersize=10,\r\n",
        "    color='r'\r\n",
        ")\r\n",
        "likelihood = []\r\n",
        "for i in range(11):\r\n",
        "    likelihood.append(binom.pmf(n=10,k=i,p=0.8))\r\n",
        "\r\n",
        "ax[1].vlines(x=x, ymin=0, ymax=likelihood, colors='r', linestyles='-', lw=2)\r\n",
        "ax[1].vlines(x=x, ymin=0, ymax=first, colors='b', linestyles='--', lw=2)\r\n",
        "ax[1].set_title('8 out of 10 toss observed')\r\n",
        "\r\n",
        "second = np.multiply(first, likelihood)\r\n",
        "second = second / np.sum(second)\r\n",
        "\r\n",
        "# --------------------------------------------------------------------------------\\\r\n",
        "# Posterior\r\n",
        "# --------------------------------------------------------------------------------\r\n",
        "ax[2].grid(\r\n",
        "    color='lightblue',\r\n",
        "    linestyle='--',\r\n",
        "    linewidth=0.5\r\n",
        ")\r\n",
        "ax[2].plot(\r\n",
        "    [5, 8],\r\n",
        "    [0 ,0],\r\n",
        "    linestyle=\"None\",\r\n",
        "    marker='o',\r\n",
        "    markersize=10,\r\n",
        "    color='r'\r\n",
        ")\r\n",
        "ax[2].vlines(x=x, ymin=0, ymax=second, colors='k', linestyles='-', lw=1)\r\n",
        "ax[2].set_title('5 and 8 out of 10 toss')\r\n",
        "\r\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 1440x360 with 3 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WjgJxdEffxFQ"
      },
      "source": [
        "eg.5"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XhEVcsJRfyhB"
      },
      "source": [
        "def normal_pdf(x, mu = 2, sigma = 1):\r\n",
        "    return (1./np.sqrt(2*3.14 * sigma**2)) * 2.718**(-(x-mu)**2 / (2. * sigma**2))"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "id": "blzilz2MgBn3",
        "outputId": "e1710bcd-6f53-40d3-b873-d963c7a6e5e8"
      },
      "source": [
        "x_values = np.linspace(-30,30,200)\r\n",
        "y_values = [normal_pdf(x) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "\r\n",
        "x_values = np.linspace(-10,10,100)\r\n",
        "y_values = [normal_pdf(x, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)\r\n",
        "\r\n",
        "y_values = [normal_pdf(x, mu = -5, sigma = 5) for x in x_values]\r\n",
        "plt.plot(x_values, y_values)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f57e8a09390>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 25
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bxJTZuQksj2v"
      },
      "source": [
        "**5. Probability Expectation**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jW-ez9qOFgkO"
      },
      "source": [
        "**Expectation**\r\n",
        "\r\n",
        "  The expected value of an experiment is the probability-weighted average of all possible values. It is defined mathematically as the following:\r\n",
        "\r\n",
        "  [Math Processing Error]\r\n",
        "\r\n",
        "  The law of large numbers states that the average result from a series of trails will converge to the expected value. Roll the die to see convergence to its expected value. change the theoretical probability of the die to see how that changes the average and expected value."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MS0XYmFGFG7T"
      },
      "source": [
        "**Expectation**\r\n",
        "\r\n",
        "If you have a collection of numbers a1,a2,...,aN, their average is a single \r\n",
        "number that describes the whole collection. Now, consider a random variable X. \r\n",
        "We would like to define its average, or as it is called in probability, its expected value or mean. \r\n",
        "The expected value is defined as the weighted average of the values in the range."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qw2EvdKcAlIr"
      },
      "source": [
        "Expected value (= mean=average):\r\n",
        "\r\n",
        "Definition \r\n",
        "\r\n",
        "Let X be a discrete random variable with range RX={x1,x2,x3,...} (finite or countably infinite). The expected value of X, denoted by EX is defined as\r\n",
        "\r\n",
        "EX= ∑xk P(X=xk)=∑xk PX(xk)    where xk∈RX\r\n",
        "\r\n",
        "consider a discrete random variable with range RX={x1,x2,x3,...}. This random variable is a result of random experiment. Suppose that we repeat this experiment a very large number of times N, and that the trials are independent. Let N1 be the number of times we observe x1, N2 be the number of times we observe x2, ...., Nk be the number of times we observe xk, and so on. Since P(X=xk)=PX(xk), we expect that\r\n",
        "\r\n",
        "PX(x1)≈N1/N,\r\n",
        "\r\n",
        "PX(x2)≈N2/N,\r\n",
        "\r\n",
        "...\r\n",
        "\r\n",
        "PX(xk)≈Nk/N,\r\n",
        "\r\n",
        "...\r\n",
        "\r\n",
        "In other words, we have Nk≈N PX(xk). Now, if we take the average of the observed values of X, we obtain\r\n",
        "\r\n",
        "Average \t= N1x1+N2x2+N3x3+.../N\r\n",
        "\r\n",
        "          ≈ x1NPX(x1)+x2NPX(x2)+x3NPX(x3)+.../N\r\n",
        "          = x1 PX(x1)+x2 PX(x2)+x3 PX(x3)+...\r\n",
        "          = EX.\r\n",
        "\r\n",
        "Thus, the intuition behind EX is that if you repeat the random experiment independently N times and take the average of the observed data, the average gets closer and closer to EX as N gets larger and larger. We sometimes denote EX by μX.\r\n",
        "\r\n",
        "  Different notations for expected value of X: EX=E[X]=E(X)=μX.\r\n",
        "                  "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9OpzvB-9JYdL"
      },
      "source": [
        "**Example 1**\r\n",
        "\r\n",
        "Expectation or expected value of any group of numbers in probability is the long-run average value of repetitions of the experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up in an extremely large number of rolls is close to 3.5. Less roughly, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, or first moment.\r\n",
        "Given an array, the task is to calculate the expected value of the array."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "iMsorL0l_4KU",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "54a9e7c2-1319-453c-bbc3-8ab3d7de8e26"
      },
      "source": [
        "# Example 1 : Find the Expected value of Rolling a dice \r\n",
        "\r\n",
        "# Function to calculate expectation \r\n",
        "def calc_Expectation(a, n): \r\n",
        "\t\r\n",
        "\t# variable prb is for probability \r\n",
        "\t# of each element which is same for \r\n",
        "\t# each element \r\n",
        "\tprb = 1 / n \r\n",
        "\t\r\n",
        "\t# calculating expectation overall \r\n",
        "\tsum = 0\r\n",
        "\tfor i in range(0, n): \r\n",
        "\t\tsum += (a[i] * prb) \r\n",
        "\t\t\r\n",
        "\t# returning expectation as sum \r\n",
        "\treturn float(sum) \r\n",
        "\r\n",
        "\r\n",
        "# Driver program \r\n",
        "n = 6; \r\n",
        "a = [ 1.0, 2.0, 3.0,4.0, 5.0, 6.0 ] \r\n",
        "\r\n",
        "# Function for calculating expectation or expected value\r\n",
        "expected_value = calc_Expectation(a, n) \r\n",
        "\r\n",
        "# Display expectation of given array \r\n",
        "print( \"Expectation of array E(X) is : \", expected_value ) "
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Expectation of array E(X) is :  3.5\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SicrV6XYyOCK"
      },
      "source": [
        "**Method 2**\r\n",
        "\r\n",
        "**Finding Expected Value**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "goKRA2K4yHAl",
        "outputId": "8258d6ad-2de1-42ba-b6dc-c31ea30e1a3f"
      },
      "source": [
        "import random\r\n",
        "def DiceRoll():\r\n",
        "  return random.randint(1,6)  # here we are selecting the integer value from range. \r\n",
        "\r\n",
        "trials=[]\r\n",
        "noOfTrials=100\r\n",
        "for t in range(noOfTrials):\r\n",
        "  trials.append(DiceRoll())\r\n",
        "\r\n",
        "avgs = sum(trials)/float(noOfTrials)\r\n",
        "avgs"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "3.45"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 30
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "K5rV8Fqv1Xkq",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "31ae599a-e379-40d7-a3ec-dd3686cfa934"
      },
      "source": [
        "nTrials=range(100,10000,10) # here we are repeating the experiment for large number of times\r\n",
        "avgs=[]\r\n",
        "for nt in nTrials:\r\n",
        "  trials=[]\r\n",
        "  for t in range(1,nt):\r\n",
        "    trials.append(DiceRoll())\r\n",
        "  avgs.append(sum(trials)/float(nt))\r\n",
        "avgs\r\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
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              " 3.490740740740741,\n",
              " 3.515669205658324,\n",
              " 3.495978260869565,\n",
              " 3.4971769815418026,\n",
              " 3.5053145336225597,\n",
              " 3.492957746478873,\n",
              " 3.5064935064935066,\n",
              " 3.493945945945946,\n",
              " 3.502915766738661,\n",
              " 3.4961165048543688,\n",
              " 3.5060344827586207,\n",
              " 3.495156081808396,\n",
              " 3.494516129032258,\n",
              " 3.4822771213748656,\n",
              " 3.53519313304721,\n",
              " 3.4995712754555197,\n",
              " 3.4946466809421843,\n",
              " 3.489625668449198,\n",
              " 3.5365384615384614,\n",
              " 3.485058697972252,\n",
              " 3.47590618336887,\n",
              " 3.487220447284345,\n",
              " 3.4948936170212765,\n",
              " 3.5149840595111583,\n",
              " 3.50859872611465,\n",
              " 3.4685047720042417,\n",
              " 3.5366525423728814,\n",
              " 3.4843386243386245,\n",
              " 3.5579281183932348,\n",
              " 3.482998944033791,\n",
              " 3.493248945147679,\n",
              " 3.5035827186512116,\n",
              " 3.502842105263158,\n",
              " 3.4772870662460567,\n",
              " 3.4880252100840337,\n",
              " 3.5082896117523608,\n",
              " 3.5224318658280924,\n",
              " 3.506910994764398,\n",
              " 3.4721757322175733,\n",
              " 3.500835945663532,\n",
              " 3.514509394572025,\n",
              " 3.502606882168926,\n",
              " 3.5085416666666664,\n",
              " 3.488449531737773,\n",
              " 3.5136174636174635,\n",
              " 3.4861889927310488,\n",
              " 3.5014522821576763,\n",
              " 3.488082901554404,\n",
              " 3.4976190476190476,\n",
              " 3.481695966907963,\n",
              " 3.490702479338843,\n",
              " 3.49938080495356,\n",
              " 3.4990721649484535,\n",
              " 3.490628218331617,\n",
              " 3.506275720164609,\n",
              " 3.501438848920863,\n",
              " 3.50564681724846,\n",
              " 3.502051282051282,\n",
              " 3.492520491803279,\n",
              " 3.5069600818833164,\n",
              " 3.485071574642127,\n",
              " 3.495914198161389,\n",
              " 3.508469387755102,\n",
              " 3.523343527013252,\n",
              " 3.459775967413442,\n",
              " 3.495320447609359,\n",
              " 3.484959349593496,\n",
              " 3.4936040609137056,\n",
              " 3.5134888438133873,\n",
              " 3.5181357649442755,\n",
              " 3.5121457489878543,\n",
              " 3.4864509605662284,\n",
              " 3.5274747474747477,\n",
              " 3.5078708375378405,\n",
              " 3.4827620967741937,\n",
              " 3.4920443101711984,\n",
              " 3.5261569416498992,\n",
              " 3.4759798994974873,\n",
              " 3.494176706827309,\n",
              " 3.5147442326980944,\n",
              " 3.4698396793587176,\n",
              " 3.5104104104104104]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2aqoYq5d2qS7"
      },
      "source": [
        "Above we are repeating the experiments large number of times Now we analyze the result using plot."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 296
        },
        "id": "8KlRT-Li3NkI",
        "outputId": "1ea59256-1cc5-4adb-c221-7f506292e4be"
      },
      "source": [
        "import matplotlib.pyplot as plt\r\n",
        "plt.plot(nTrials,avgs)\r\n",
        "plt.xlabel('No. of Trials')\r\n",
        "plt.ylabel('Avgs')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0, 0.5, 'Avgs')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 32
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qccGjjT74SLL"
      },
      "source": [
        "Above plot represents the average dice rolls. Here we can see that when we are having 0 number of trials or trials in between o to 100 the values are more flexible there or the values are moving from 3.2 to 3.8 average values and when we are moving towards 2000, 4000 you can see the sample mean or expected value , here sample mean actually we are taking the average or actually we are calculating the sample mean , so we can see in the plot that sampling mean moving towards the expected value. So we can see for 10000 the maximum number of times the value is 3.5. So above the average value is 3.45 for when we are doing it for 100 times. And from graph we are getting the average value as 3.5. \r\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "F5Sus6flE1b-"
      },
      "source": [
        "We can define that if we roll the dice for 10000 times then the Average or expected dice roll is 3.5 or approximately equal to 3.5."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iD7FrRbmE4qW"
      },
      "source": [
        "**Example 2**\r\n",
        "\r\n",
        "The game that will be used in this example is No Limit Texas Hold’em. Played with a standard 52 card deck, Texas Hold’em is the most popular of all the poker variations. Each player tries to make the best five-card hand possible by combining their two cards with the five community cards dealt throughout the hand.\r\n",
        "\r\n",
        "\r\n",
        "The player to the left of the dealer button starts the action and play moves clockwise around the table. The betting is typically started when the player to the left of the dealer button posts a small blind, and the next player to the left posts a big blind. Betting action takes place on multiple streets: preflop, flop, turn, and the river.\r\n",
        "\r\n",
        "\r\n",
        "Here are a few terms we should understand:\r\n",
        "\r\n",
        "\r\n",
        "**Button** - The Button is a circular disk that gets passed around the table clockwise after each hand. In a home game, the person who has the Button deals the hand dealing the first card to the player on their left. At a casino, the Button identifies who is supposed to be the dealer, but the actual dealer is provided by the casino.\r\n",
        "\r\n",
        "**The Blinds** - The Blinds are forced bets by the two players to the left of the Button. They are called “Blinds” because the two players have to bet blind, in that they have to bet before they get to see their cards.\r\n",
        "\r\n",
        "**Streets** - Each round that cards are dealt is referred to as a street.\r\n",
        "Pre-Flop - This is the first betting round. Each player is dealt two cards, and the blinds start the betting action. No community cards have been dealt yet.\r\n",
        "\r\n",
        "**Flop** - This is the second betting round. In this round, three community cards are dealt face up in the middle of the table, followed by a round of betting.\r\n",
        "\r\n",
        "**Turn** - In this round, only one community card is dealt, followed by the third round of betting.\r\n",
        "\r\n",
        "**The River** - The final community card is dealt, followed by the last round of betting.\r\n",
        "\r\n",
        "**Showdown** - After the River is dealt and all betting action has taken place, any remaining players Show their cards to see who has the best hand.\r\n",
        "\r\n",
        "**Check** - If it’s a player's turn to act, and no previous bets have been made during that round of betting, the player can Check which means to choose not to bet, and allow the next player to take their turn.\r\n",
        "\r\n",
        "**Bet or Raise** - Betting or Raising means to increase the size of the wager during a betting round. The first player to put money in the pot each round is making a Bet. If another player increases the size of the bet that round, they are making a Raise.\r\n",
        "\r\n",
        "**Call** - If it’s a player’s turn to act and a bet has already been made during that round, the player can match the previous bet which is known as Calling the bet.\r\n",
        "\r\n",
        "**Fold** -If it’s a player’s turn to act and a bet has already been made during that round, the player can give up and Fold their hand. This player is no longer involved in the hand and losses any money they have put in the pot."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0vXn2ujmS_tC"
      },
      "source": [
        "When playing a game such as poker, we're fairly concerned with questions such as \"how much do I gain - or lose - on average, if I repeatedly play this game?\". We can imagine that this is no different for poker, especially when we're a professional poker player!\r\n",
        "\r\n",
        "Now, if the possible outcomes of the game and their associated probabilities can be described by a random variable, then we can answer the above question by computing its expected value, which is equal to a weighted average of the outcomes where each outcome is weighted by its probability.\r\n",
        "\r\n",
        "Or, in other words, we simply multiply the Total Value times the probability of winning to get our Expected Value:\r\n",
        "\r\n",
        "ExpectedValue=TotalValue×Probability\r\n",
        "\r\n",
        "What is the expected value if there is $$100 (Total Value) in the pot, and your probability of winning the pot is 0.75?\r\n",
        "\r\n",
        "ExpectedValue = $100 * 0.75\r\n",
        "\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "S10Zv9naU0mD",
        "outputId": "3426a7da-bda2-4c3a-c444-e38848d05f69"
      },
      "source": [
        "# Initialize `pot` and `probability` variables\r\n",
        "pot = 100\r\n",
        "probability = 0.75\r\n",
        "\r\n",
        "# Determine expected value\r\n",
        "expected_value = pot * probability\r\n",
        "print(expected_value)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "75.0\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jWkaJcFEVJO0"
      },
      "source": [
        "We're expected value is $$75.Expected value is an important concept in poker. Now let's see how to use expected values to our advantage.\r\n",
        "\r\n",
        "Example :\r\n",
        "\r\n",
        "Your Hand: ('A', '♢'), ('Q', '♢')\r\n",
        "\r\n",
        "Opponents Hand :  (2, '♡'), (2, '♣')\r\n",
        "\r\n",
        "Community Cards : ('J', '♣'), (9, '♢'), (4, '♢'), (2, '♠')\r\n",
        "\r\n",
        "Total Pot = $60\r\n",
        "\r\n",
        "Opponents Bet = $20\r\n",
        "\r\n",
        "\r\n",
        "Your opponent has decided to be helpful and show you his cards, and has a set of 2s. To win the hand on the River, you must hit any Diamond except a Jack or 2. The Jack or 2 of Diamonds would give your opponent a better hand, a full house and four of a kind respectively. You have to call $20 to stay in the hand, and if you win the hand you win $60. If your expected value is greater than $20 you should call the bet, and if not you should fold.\r\n",
        "\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "KPQ6oQtiu9mG",
        "outputId": "c9b86102-f87a-45b2-b70d-76f2039626bc"
      },
      "source": [
        "# Figure out if you should call the bet:\r\n",
        "\r\n",
        "# Sample Space\r\n",
        "cards = 52\r\n",
        "hole_cards = 2\r\n",
        "# Your opponent provided you information... use it!\r\n",
        "opponents_hole_cards = 2 \r\n",
        "\r\n",
        "turn_community_cards = 4\r\n",
        "cards = cards - (hole_cards + opponents_hole_cards + turn_community_cards)\r\n",
        "\r\n",
        "# Outcomes\r\n",
        "diamonds = 13\r\n",
        "diamonds_drawn = 4\r\n",
        "\r\n",
        "# You can't count the two diamonds that won't help you win\r\n",
        "diamond_non_outs = 2 \r\n",
        "\r\n",
        "outs = diamonds - diamonds_drawn - diamond_non_outs\r\n",
        "\r\n",
        "# Determine win probability\r\n",
        "win_probability = outs / cards\r\n",
        "\r\n",
        "# Determine expected value\r\n",
        "pot = 60\r\n",
        "ev = pot * win_probability\r\n",
        "\r\n",
        "# Print ev and appropriate decision\r\n",
        "call_amount = 20\r\n",
        "if ev >= 20:\r\n",
        "    print(round(ev, 2), 'Call')\r\n",
        "else:\r\n",
        "    print(round(ev, 2), 'Fold')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "9.55 Fold\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ehA6gE-evLzv"
      },
      "source": [
        "Your expected value is only $$9.55, which is less than the $20 you would need to risk to get that reward, so you must fold."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Yq_CG_iPT67D"
      },
      "source": [
        "**Example 3:**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WCPD-y0BXyVI"
      },
      "source": [
        "There is a brand of cereal that comes with a single coupon with the purchase of each box. There are a 5 different types of coupons. The probability of getting any one of the 5 coupons with each cereal box purchase is uniform. The objective is to collect all 5 unique coupon types.\r\n",
        "\r\n",
        "Now the questions begs, what is the expected number of boxes of cereal you will need to purchase to collect all 5 unique coupons?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PX7xVwfWY0vY"
      },
      "source": [
        "**solution:**\r\n",
        "\r\n",
        "We will first solve this problem analytically, and then simulate the problem computationally to verify our findings.\r\n",
        "\r\n",
        "Observations:\r\n",
        "\r\n",
        "1. Each cereal box purchased is independent of the last box (or boxes) purchased.\r\n",
        "\r\n",
        "2. The probability of getting any of the 5 coupon types in a given box is the same.\r\n",
        "\r\n",
        "Geometric Random Variables:\r\n",
        "A Geometric random variable is a discrete random variable with a one-parameter distribution as detailed below:\r\n",
        "\r\n",
        "Let's say we have an ever growing sequence of independent Bernoulli trials, each with probability of success p. A geometric random variable X is the number of Bernoulli trails X = x to get to the first success in the sequence.\r\n",
        "\r\n",
        "The probability mass function of X ~ Geometric(p) is:\r\n",
        "\r\n",
        "fx(x:p) = p(1-p)^x-1 for x € {1,2,.......,∞} and 0< p ≤ 1\r\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RzUkge3XddPk"
      },
      "source": [
        "\r\n",
        "Note, that we can view our coupon collecting problem as the sum of the expected value of five geometric random variables, each with different success parameters p:\r\n",
        "\r\n",
        "1st coupons Purchasing:\r\n",
        "\r\n",
        "Prior to purchasing our first box of cereal, we don't yet have any coupons. Therefore, with our first box purchased we are guaranteed to collect a unique coupon we don't yet have with probability 1.\r\n",
        "\r\n",
        "X1 ~ Geometric(p = 5/5)\r\n",
        "\r\n",
        "2nd Unique coupon, after 1st unique coupon has been collected:\r\n",
        "\r\n",
        "Having collected our 1st unique coupon,we know want a second. Note that this will also be a geometric random variable, but with different of success p. Because we have already collected one of the unique coupons, with each box purchased there is a 1/5 probability of getting that same coupon type, and a 4/5 probability of obtaining a new unique coupon we don't yet have.\r\n",
        "\r\n",
        "X2 ~ Geometric(p = 4/5)\r\n",
        "\r\n",
        "Continued Series:\r\n",
        "\r\n",
        "X3 ~ Geometric(p = 3/5)\r\n",
        "\r\n",
        "X4 ~ Geometric(p = 2/5)\r\n",
        "\r\n",
        "X5 ~ Geometric(p = 1/5)\r\n",
        "\r\n",
        "Note, that the expected value of a geometric random variable is 1/p:\r\n",
        "\r\n",
        "\r\n",
        "X ~ Geometric(p) , E[X] = 1/P\r\n",
        "\r\n",
        "Therefore , **Expected number of boxes purchased** \r\n",
        "\r\n",
        "=  E[X1] +  E[X2] + E[X3] +  E[X4] +  E[X5]\r\n",
        "\r\n",
        "=  E[1/(5/5)] +  E[1/(4/5)] + E[1/(3/5)] +  E[1/(2/5)] +  E[1/(1/5)]\r\n",
        "\r\n",
        "≈  11.4167\r\n",
        "\r\n",
        "**Conclusion** : Therefore, rounding up to the nearest integer, the expected number of boxes we will need to purchase to collect all 5 unique coupons is 12 boxes.\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "osfJhttIUAuD",
        "outputId": "49fcbe95-e638-4193-e34c-85f553f0777f"
      },
      "source": [
        "# Now stimulating the above example in python over 10000 iterations.\r\n",
        "\r\n",
        "import numpy as np\r\n",
        "import pandas as pd\r\n",
        "import matplotlib.pyplot as plt\r\n",
        "import warnings\r\n",
        "warnings.filterwarnings('ignore') \r\n",
        "## specify number of coupons, and the probability of getting a coupon\r\n",
        "num_coupons=5\r\n",
        "expected_number_of_trials = 0\r\n",
        "for i in range(num_coupons, 0, -1):\r\n",
        "    expected_number_of_trials = expected_number_of_trials + (1/(i/num_coupons))\r\n",
        "iterations = 10000\r\n",
        "df = pd.DataFrame()\r\n",
        "df['iteration'] = range(1,iterations+1)\r\n",
        "df['average_num_trials_required'] = None\r\n",
        "for i in range(0, iterations):\r\n",
        "    full_set = set(range(0,num_coupons))\r\n",
        "    current_set = set([])\r\n",
        "    total_number_of_coupons_collected = 0\r\n",
        "    while(current_set != full_set):\r\n",
        "        total_number_of_coupons_collected = total_number_of_coupons_collected+1\r\n",
        "        random_coupon = np.asscalar(np.random.randint(low=0, high=num_coupons, size=1))\r\n",
        "        update_set = list(current_set)\r\n",
        "        update_set.append(random_coupon)\r\n",
        "        current_set = set(update_set)\r\n",
        "        del random_coupon, update_set\r\n",
        "    \r\n",
        "    if(i+1 == 1):\r\n",
        "        df.loc[df['iteration']==i+1, 'average_num_trials_required'] = total_number_of_coupons_collected\r\n",
        "    else:\r\n",
        "        df.loc[df['iteration']==i+1, 'average_num_trials_required'] = float(((df.loc[df['iteration']==i, 'average_num_trials_required']*i) + total_number_of_coupons_collected)/(i+1))\r\n",
        "\r\n",
        "average_num_trials_required = df['average_num_trials_required'][i]\r\n",
        "\r\n",
        "print(\"Average or Expected number of trials Requried:\", average_num_trials_required)\r\n",
        "#print(\"Average or Expected number of trials Requried:\", df['average_num_trials_required'][i])"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Average or Expected number of trials Requried: 11.453300000000013\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "EO65msj8rZ77"
      },
      "source": [
        "**Conclusion :**\r\n",
        "\r\n",
        "As shown above, over 10,000 iterations, the sample average or expected value of the number of boxes needed to be purchased to collect all 5 unique coupons was approximately 11.4533. This result coincides with our solution above.\r\n",
        "\r\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_r9dj8dDr12B"
      },
      "source": [
        "**Example 4:**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wEi2Pmgl41aO"
      },
      "source": [
        "**Binomial Distribution:**\r\n",
        "\r\n",
        "A random variable X that has a binomial distribution represents the number of successes in a sequence of n independent yes/no trials, each of which yields success with probability p.\r\n",
        "E(X) = np, Var(X) = np(1−p)\r\n",
        "E(X) is the expected value or mean of the distribution.\r\n",
        "\r\n",
        "Example of binomial distribution: What is the probability of getting 2 heads out of 10 flips of a fair coin? \r\n",
        "In this experiment the probability of getting a head is 0.3,  this means that on an average you can expect 3 coin flips to be heads. I define all the possible values the coin flip can take, k = np.arange(0,11), you can observe zero head, one head all the way upto ten heads. I am using stats.binom.pmf  to calculate the probability mass function for each observation. It returns a list of 11 elements, these elements represent the probability associated with each observation."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UOykh0kIUAzL",
        "outputId": "202b109c-adbe-4230-eebd-082db7207079"
      },
      "source": [
        "import numpy as np\r\n",
        "from scipy.special import erfinv\r\n",
        "from scipy.stats import binom\r\n",
        "import statistics\r\n",
        "import matplotlib.pyplot as plt\r\n",
        "import seaborn as sns\r\n",
        "\r\n",
        "n = 10\r\n",
        "p = 0.3\r\n",
        "k = np.arange(0,21)\r\n",
        "binomial = binom.pmf(k, n, p)\r\n",
        "binomial"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([2.82475249e-02, 1.21060821e-01, 2.33474440e-01, 2.66827932e-01,\n",
              "       2.00120949e-01, 1.02919345e-01, 3.67569090e-02, 9.00169200e-03,\n",
              "       1.44670050e-03, 1.37781000e-04, 5.90490000e-06, 0.00000000e+00,\n",
              "       0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n",
              "       0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,\n",
              "       0.00000000e+00])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 78
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 297
        },
        "id": "23s0YSo1UA1f",
        "outputId": "0c7341f8-2dff-4713-c037-c8a2c6532373"
      },
      "source": [
        "import matplotlib.pyplot as plt\r\n",
        "plt.plot(k, binomial, 'o-')\r\n",
        "plt.title('Binomial: n=%i, p=%.2f' % (n,p),fontsize=15)\r\n",
        "plt.xlabel('Number of Successes')\r\n",
        "plt.ylabel('Probability of Successes', fontsize=15)\r\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GwVyQd2o9PVf"
      },
      "source": [
        "We can simulate a binomial random variable using .rvs. The parameter size specifies how many simulations we want to do. I ask Python to return 10000 binomial random variables with parameters n and p. I am printing the mean and standard deviation of these 10000 random variables. Then I am going to plot the histogram of all the random variables that I simulated."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 313
        },
        "id": "8IMt_kIjUA3-",
        "outputId": "ea09ee75-ad23-46be-fa26-252a60d8ef92"
      },
      "source": [
        "binom_sim = data = binom.rvs(n=10,p=0.3,size=10000)\r\n",
        "print(\"Mean: %g\" % np.mean(binom_sim))\r\n",
        "print(\"SD: %g\" % np.std(binom_sim, ddof=1)) \r\n",
        "plt.hist(binom_sim, bins = 10, density=True, color = \"blue\")  # normed\r\n",
        "plt.xlabel('x')\r\n",
        "plt.ylabel('density')\r\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean: 3.0064\n",
            "SD: 1.46163\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kDquTcYD_7BA"
      },
      "source": [
        "**Example 5:**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "T_7SF4k9_--T"
      },
      "source": [
        "**Poisson Distribution**\r\n",
        "\r\n",
        "A random variable X that has a Poisson distribution represents the number of events occurring in a fixed time interval with a rate parameters λ. λ tells you the rate at which the number of events occur.  The average and variance is λ.\r\n",
        "\r\n",
        "P(X=k) = (λ^k) * (e^-λ) / k!\r\n",
        "\r\n",
        "E(X) = λ, Var(X) = λ"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "UlVlIYRF83Ac",
        "outputId": "fa7e393d-bf1c-44b4-864b-d7383182c4e2"
      },
      "source": [
        "import numpy as np\r\n",
        "from scipy.stats import poisson\r\n",
        "import statistics\r\n",
        "import matplotlib.pyplot as plt\r\n",
        "import seaborn as sns\r\n",
        "\r\n",
        "rate = 2\r\n",
        "n = np.arange(0,10)\r\n",
        "y = poisson.pmf(n, rate)\r\n",
        "y"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([1.35335283e-01, 2.70670566e-01, 2.70670566e-01, 1.80447044e-01,\n",
              "       9.02235222e-02, 3.60894089e-02, 1.20298030e-02, 3.43708656e-03,\n",
              "       8.59271640e-04, 1.90949253e-04])"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 93
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 296
        },
        "id": "URztmFUw83Cl",
        "outputId": "a1323487-6d33-4128-9778-42a9633defec"
      },
      "source": [
        "plt.plot(n, y, 'o-')\r\n",
        "plt.title('Poisson: $\\lambda$ =%i' % rate)\r\n",
        "plt.xlabel('Number of accidents')\r\n",
        "plt.ylabel('Probability of number of accidents')\r\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ocPWv_nNEURR"
      },
      "source": [
        "We can notice that the  number of accidents peaks around the mean. On an average we can expect lambda number of events. Trying different values of lambda and n,  then see how shape of the distribution changes.\r\n",
        "Now I am going to simulate 1000 random variables from a Poisson distribution.\r\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 329
        },
        "id": "-c3PsGTm83Eu",
        "outputId": "cdddaa77-bc71-4545-c9d3-374267d68a14"
      },
      "source": [
        "data = poisson.rvs(mu=2, loc = 0, size = 1000)\r\n",
        "print(\"Mean: %g\" % np.mean(data))\r\n",
        "print(\"SD: %g\" % np.std(data, ddof=1))\r\n",
        "plt.figure()\r\n",
        "plt.hist(data, bins=9, density = True)\r\n",
        "plt.xlim(0,10)\r\n",
        "plt.xlabel('Number of accidents')\r\n",
        "plt.title('Stimulating Poisson Random Variables')\r\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean: 2.053\n",
            "SD: 1.47186\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "B1_kCiLgzV3-"
      },
      "source": [
        "**6. Probability Estimation**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "v2htSJQSjnh0"
      },
      "source": [
        "Estimation refers to the process by which one makes inferences about a population, based on information obtained from a sample. A point estimate, for example, is the single number most likely to express the value of the property. An interval estimate defines a range within which the value of the property can be expected (with a specified degree of confidence) to fall."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pqwpOFUPr-B4"
      },
      "source": [
        "from matplotlib import pyplot\n",
        "from numpy.random import normal\n",
        "from numpy import mean\n",
        "from numpy import std\n",
        "from scipy.stats import norm"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3N-IwXPSr-B6"
      },
      "source": [
        "sample = normal(loc=50, scale=5, size=1000)\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VEAg5-REr-B7",
        "outputId": "64839fa9-0789-4a60-dfbc-06ff689dff71"
      },
      "source": [
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=49.897, Standard Deviation=5.042\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "agKRZ1hAr-B-"
      },
      "source": [
        "dist = norm(sample_mean, sample_std)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VzK_smu6r-CA"
      },
      "source": [
        "values = [value for value in range(30, 70)]\n",
        "probabilities = [dist.pdf(value) for value in values]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "QgVA8vK5r-CA",
        "outputId": "81b00a83-623f-4fde-fa1b-8306ef107f7a"
      },
      "source": [
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x1feab0d7ee0>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3GyTXuzlr-CC",
        "outputId": "8f9a67b8-2488-43f0-fd2a-7f52283887fa"
      },
      "source": [
        "# code of parametric probability estimation\n",
        "# Code 1\n",
        "sample = normal(loc=50, scale=5, size=1000)\n",
        "# calculate parameters\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "# define the distribution\n",
        "dist = norm(sample_mean, sample_std)\n",
        "# sample probabilities for a range of outcomes\n",
        "values = [value for value in range(30, 70)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "# plot the histogram and pdf\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=49.965, Standard Deviation=5.065\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SpbDRZBhr-CE"
      },
      "source": [
        "Running the code first generates the data sample, then estimates the parameters of the normal probability distribution.\n",
        "\n",
        "Note that your results will differ given the random nature of the data sample. Try running the example a few times.\n",
        "\n",
        "In this case, we can see that the mean and standard deviation have some noise and are slightly different from the expected values of 50 and 5 respectively. The noise is minor and the distribution is expected to still be a good fit."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HEBuWj-lr-CF"
      },
      "source": [
        "sample = normal(loc=100, scale=10, size=1000)\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fVZegzhOr-CG",
        "outputId": "b9f21a27-7b87-4662-c128-16dc01034317"
      },
      "source": [
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=100.193, Standard Deviation=9.728\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "_vZdsiayr-CI"
      },
      "source": [
        "dist = norm(sample_mean, sample_std)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "FRvOOY5Tr-CI"
      },
      "source": [
        "values = [value for value in range(30, 200)]\n",
        "probabilities = [dist.pdf(value) for value in values]"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "e4hm6O9Hr-CJ",
        "outputId": "cf8f242f-223d-4898-c737-b4918c8eb20c"
      },
      "source": [
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x1feab33ebe0>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 29
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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AoTl4+ZdH7LGOpNHAvwCfM7NjQfF9wHRgDtAK3F3wxaWlktZLWt/e3h4iXFcSOpozN2aPcgpolp8L4EpUmATQAkzNWZ4C7A9bR9JwMl/+3zWzH2YrmFmbmXWZWRp4gMxQ03nM7H4zm2dm86qrq0OE60pClJeBzldVn7kiqN8g3pWYMAlgHdAgqU5SJXArsCavzhrg9mA20ALgqJm1ShLwILDFzL6R20BSTc7ih4BN/X4XrrR0nYHXdscrAQC85heFc6Wl11lAZnZW0nLgcaACWGVmmyUtC9avBNYCi4Em4ATwyaD59cAngJclbQjKvmRma4GvSZpDZqhoF/DpQXpPLumO7AHrin4GUFbuVNBLr402FucGUa8JACD4wl6bV7Yy57kBdxZo9xsKHx/AzD7Rp0hd+chefjkuPYBxdZlHPw7gSoyfCeziJ3sj9qgvA5FVORIuujRze0jnSognABc/Hc1QORpGxeigv88EciXIE4CLn47m6K8Cms/PBXAlyBOAi5/DO+JzADirqh6OH4TTr0cdiXODxhOAi5eus3AkRlNAs/z+wK4EeQJw8XJ0L6TPxjgB+DCQKx2eAFy8xG0GUFaVTwV1pSfUeQDOFU3E5wDUrnis23XrRlzMk4//ihWPzTxXtuuu9xcjLOeGhPcAXLx0NMPwUTB6YtSRnGeXTaQ21RZ1GM4NGu8BuEjl/+J+cPgz1GgCi7+4tpsW0dltk7g+5ZescqXDewAuVmp1gF0Wv1//ALvSE6lRBxdwOupQnBsUngBcbKRIM1UH2R3XBGCTAJgmHwZypcETgIuNS3WISnWxM/iijZtsz6TWE4ArEZ4AXGxkv1h3p+OZAPYECWCaDkQciXODwxOAi43a4Is1rscAjjGKw3aR9wBcyfAE4GKjXq0ctxG0MS7qULq12yb6MQBXMjwBuNioUys7rYZu7iEUC7tsEtP8XABXIkIlAEkLJW2T1CRpRYH1knRPsL5R0tygfKqkX0jaImmzpM/mtKmS9ISk7cFjfH/2uaKo04HYHgDO2p2eyKUcZgSdUYfi3ID1mgAkVQD3AouA2cBtkmbnVVsENAR/S4H7gvKzwJ+a2SxgAXBnTtsVwJNm1gA8GSy7MjWcs0zVQZqtJupQerTLJpKSMUXtUYfi3ICF6QHMB5rMrNnMOoGHgSV5dZYAD1nGs8BYSTVm1mpmLwKY2evAFmByTpvVwfPVwC0DeysuyS5TGxUydqbjnQB2Bz2UWp8J5EpAmAQwGdibs9zCm1/ioetIqgWuBZ4LiiaaWStA8HhJoReXtFTSeknr29v9V1epqgu+UOM+BOTnArhSEiYBFDoiZ32pI2k08C/A58zsWPjwwMzuN7N5ZjavujpG94h1g6pOrUD8E8ARRnPURnoPwJWEMAmgBZiaszwF2B+2jqThZL78v2tmP8yp0yapJqhTAxzsW+iulNSplUM2hmOMjjqUXohmu5T6IGE5l2RhEsA6oEFSnaRK4FZgTV6dNcDtwWygBcBRM2uVJOBBYIuZfaNAmzuC53cAj/b7XbjEq0/FfwZQVrPVUJ/yBOCSr9cEYGZngeXA42QO4j5iZpslLZO0LKi2FmgGmoAHgM8E5dcDnwBukLQh+FscrLsLuEnSduCmYNmVqTq1xv4AcNaOdA016mAUJ6MOxbkBCXU/ADNbS+ZLPrdsZc5zA+4s0O43dHNWj5kdBm7sS7CuNI3iJBN1JDgJLP6a7VLgzeMWziWVnwnsIpc9oNqcoCEgwI8DuMTzBOAiV39uBlAyegC7bSJdJqb7cQCXcJ4AXOTqdIC0KbY3gsl3mkparJp65U+Gcy5ZPAG4yNWlWtnPeE5TGXUooTVbjQ8BucTzBOAiV6dWmhMyAyjr3LkA6XTUoTjXb54AXMSM+gRcBTTfDruUC9UJx/ZFHYpz/eYJwEVqPMcYoxOJOQCcde6qpYe3RxuIcwPgCcBFqi5hM4CydmSHrA41RRuIcwPgCcBFqi6VrHMAstoZy+t2IRx6NepQnOs3TwAuUvVqpdMq2GdJu9Kr2GE1PgTkEs0TgItUnQ6w2yaRTuBHsdku9SEgl2jJ+1/nSkrmRvDJGv7Jak7XwLEW6DwedSjO9YsnABedrrPU6kDs7wPcnTdnAnkvwCWTJwAXnSO7GaGz7Aiurpk05+I+5McBXDJ5AnDRad8KwPb0lIgD6Z9dNgmQ9wBcYnkCcNFp3waQ2B7AaSph7FTvAbjECpUAJC2UtE1Sk6QVBdZL0j3B+kZJc3PWrZJ0UNKmvDZfkbSvwJ3CXLlo30arVfE6I6OOpP/GN/hUUJdYvSYASRXAvcAiYDZwm6TZedUWAQ3B31Lgvpx13wYWdrP5b5rZnOBvbTd1XKk6tI2mdDJ//Z8zYUZmKqhZ1JE412dhegDzgSYzazazTuBhYElenSXAQ5bxLDBWUg2AmT0FdAxm0K4EpNPQ/ipNNjnqSAZmwuVw5jgc83sDuOQJkwAmA3tzlluCsr7WKWR5MGS0StK4QhUkLZW0XtL69vb2EJt0iXBsH5w5XgIJYEbm8dC2aONwrh/CJIBCN3XP7++GqZPvPmA6MAdoBe4uVMnM7jezeWY2r7o6aZcLcN0KDgBvTyc8AVTPyjwe3BptHM71Q5gE0AJMzVmeAuT3d8PUeQszazOzLjNLAw+QGWpy5SL4xZz4HsDoahg5AQ6+EnUkzvVZmASwDmiQVCepErgVWJNXZw1wezAbaAFw1Mx6vF9e9hhB4EPApu7quhLUvhVGjqeDMVFHMnCXzDp3ToNzSdJrAjCzs8By4HFgC/CImW2WtEzSsqDaWqAZaCLza/4z2faSvgc8A8yU1CLpU8Gqr0l6WVIj8F7gTwbrTbkEaN8GE2ZGHcXguGRWZgjIZwK5hBkWplIwRXNtXtnKnOcG3NlN29u6Kf9E+DBdSTGDg1vgmo9CKVxOv/oK6HwdjrZkTgxzLiH8TGBXfEf3wuljcEn+6SQJdUlwINiHgVzCeAJwxdcWHDCdeGW0cQyW6isyj34g2CWMJwBXfAc3Zx6zv5yTbmQVjJ7kU0Fd4ngCcMV3cAuMmQIXXBx1JIPnklnQviXqKJzrE08ArvjaXoGJJTL+n5WdCZTuijoS50LzBOCKq+sMHHq1dA4AZ028Es6ehI6dUUfiXGieAFxxHdoO6TOlcwA4a+JVmce2l6ONw7k+8ATgiis7U6bUegDVV0BqGBzwBOCSwxOAK662zZkvygkNUUcyuIZfkLky6AG/oolLDk8ArrgONGZ+LQ8bEXUkg2/S1d4DcIniCcAVjxm0boRJ10QdydCYeBW8vh+OH446EudC8QTgiuf1A3C8HWpKNAFMujrz6AeCXUJ4AnDFc6Ax81jztmjjGCrZBODHAVxChLoaqHODojVIANkpkyWgdsVjb1l+dsQ4nv7JT/jTNXUF6++66/3FCMu5ULwH4IqndQNU1cMFJXATmG68kp7GldoddRjOheIJwBXPgcbSHf4JvGx1NKiFCzkVdSjO9SpUApC0UNI2SU2SVhRYL0n3BOsbJc3NWbdK0kFJm/LaVEl6QtL24HHcwN+Oi60THXBkT+nOAAo0puupkHGldkUdinO96jUBSKoA7gUWAbOB2yTln8a5CGgI/pYC9+Ws+zawsMCmVwBPmlkD8GSw7EpVdn58ifcAGtP1AFyT8msCufgL0wOYDzSZWbOZdQIPA0vy6iwBHrKMZ4Gx2Zu+m9lTQEeB7S4BVgfPVwO39CN+lxT7X8w81syJNIyh1s44Wq2Ka1I7og7FuV6FmQU0Gdibs9wCXBeizmSgtYftTjSzVgAza5V0SaFKkpaS6VVw2WWXhQjXRSl/VkzWyuGPcYUm8rt//WyRIyq+xnQ916g56jCc61WYHoAKlFk/6vSLmd1vZvPMbF51dfVgbNJFYE6qiQ02PeowimJjup761AHGcDzqUJzrUZgE0AJMzVmeAuzvR518bdlhouDxYIhYXAJN4jCT9Bob0pdHHUpRvGyZ4wBX+XEAF3NhEsA6oEFSnaRK4FZgTV6dNcDtwWygBcDR7PBOD9YAdwTP7wAe7UPcLkHmBOPh5ZIAzh0I9mEgF3O9JgAzOwssBx4HtgCPmNlmScskLQuqrQWagSbgAeAz2faSvgc8A8yU1CLpU8Gqu4CbJG0HbgqWXQmak2ritA3jFZsWdShFcZTR7ExP5NpUU9ShONejUJeCMLO1ZL7kc8tW5jw34M5u2t7WTflh4MbQkbrEujbVxBabRifDow6laF60GbwntZHMobBCh8ici56fCeyGVAVdXK2dvFQmwz9Z69IzmaBj1OpA1KE41y1PAG5IXaG9jNRpNqTLYwZQ1vr0DADekdoWcSTOdc8TgBtS81NbAHg+PSviSIprh13KERvF2/Vq1KE41y1PAG5IXZfayp50Na2MjzqUojJSvJCewbyUJwAXX54A3BAy5qe28LyV16//rBfSM7g8tZ9xHIs6FOcK8gTghszl2keV3uC59BVRhxKJdemZALw9tT3iSJwrzBOAGzLXpbYC8FyZjf9nNVo9p2w4C1KvRB2KcwV5AnBDZn5qKwdsHHus4HX+St5pKlmfnsH1Kb9HsIsnTwBuiBjXpbbwfPoKyvlEqKfTVzMrtZcJHI06FOfO4wnADYkG7WOSXuPpdOncAL4/nk5fCcDvpDZHHIlz5/ME4IbEu1MbAXiqq7RvAdmbTVbHURvJ7/gwkIshTwBuSLwn1cir6cllN/8/X5oUv01fyTsrNjFIt8hwbtB4AnCD7gJOc11qK79Kl/b9f8N6On0VU3TIrwvkYscTgBt0C1JbGKEzPJUu7+GfrF8F++GG1IZoA3EujycAN+jenWrklA0PZgC5vTaRrempvC/1QtShOPcWngDcIDNuTL3IM+nZnKYy6mBi42fpucxPbYUTHVGH4tw5oRKApIWStklqkrSiwHpJuidY3yhpbm9tJX1F0j5JG4K/xYPzllyUZms301IH+ff0/KhDiZWfdb2dYUpD08+iDsW5c3pNAJIqgHuBRcBs4DZJs/OqLQIagr+lwH0h237TzOYEf2txibew4nm6TDzR9faoQ4mVjVZPu10M2/xj7uIjTA9gPtBkZs1m1gk8DCzJq7MEeMgyngXGSqoJ2daVkEWpdTyXnkUHY6IOJVaMFD/rmgvbfwZnTkUdjnNAuAQwGdibs9wSlIWp01vb5cGQ0SpJ40JH7eLp4FYaUvv4iQ//FLQ2fR10vg7bH486FOeAcAmg0IVc8s9o6a5OT23vA6YDc4BW4O6CLy4tlbRe0vr29vYQ4brIvPIoAD/tmhdxIPH02/SVMHoiND4SdSjOAeESQAswNWd5CrA/ZJ1u25pZm5l1mVkaeIDMcNF5zOx+M5tnZvOqq6tDhOsikU7Dxn/ima7ZtFEVdTSx1EUFXPX7sP2ncPK1qMNxLlQCWAc0SKqTVAncCqzJq7MGuD2YDbQAOGpmrT21DY4RZH0I8IulJNme38Jru3ik6z1RRxJv13wEujrP9Zaci9Kw3iqY2VlJy4HHgQpglZltlrQsWL8SWAssBpqAE8Ane2obbPprkuaQGRLaBXx6EN+XK7aXvgOVF/GTUz7+36OaOTBhBmx8GN7+R1FH48pcrwkAIJiiuTavbGXOcwPuDNs2KP9EnyJ18XXqGGz+V3jbH3Dq6RFRRxNvElz7CXjiL+DAJphU3pfLdtHyM4HdwDV+H86ehDl/GHUkyXDtH8KwC+H5v486ElfmPAG4gUl3wTP3wuR5MMVn/4Qysgqu+Sg0/sAvDeEi5QnADczWH8NrO+F3/jgzvOHCmb8002t68aGoI3FlzBOA6z8zePoeGFcLsz4YdTTJMukqqHsPPPN/ofN41NG4MuUJwPXfjp/DvvXwH5ZDqiLqaJLnvV+C4+3w/ANRR+LKlCcA1z/pLnjiL2HsNJh7e9TRJNNlC2D6jfD0t+D061FH48qQJwDXPxsfhrZNcONfwjCf+tlv7/0ynOyAX38j6khcGQp1HoBzb3HyCPz8r+HSuXDlh6OOJlFqVzx2Xtndw9/FB3/9LRY/OYkmm3KufNdd7y9maK4MeQ/A9d3jX4Y32mDx1yHlH6GB+pszH+c4F/LV4Q8i0lGH48qI/+91ffPq47DhO/DOP4EpftOXwdDBGL569mPMT21jacX5PQTnhooPAbke5Q5ZTFUbj1b+BQdtKjc/cQ2dT/iX1WD5Qdd7eG9qA/992Pd5KX05z9usqENyZcB7AC6U0ZzgweFfJ4Wx7Mzn6GR41CGVGPHnZ5ayxy7h3sp7uExtUQfkyoAnANer0ZxgVeXfUacD/Lczn2OX1fTeyPXZG4xk6ZnPU0EX/1T5N3Bkb++NnBsATwCuR1Uc47uVX+VaNfHZM3fyTPrKqEMqaU02hds7VzCG4/DtxdC2ufdGzvWTHwNw3dv1NGtHfJFxvMGnz/wJP0/PjTqisrDJ6vl455f5h9e+zqj/dwNfOLOUx9LXUfgOq2/yaaOur7wH4M53ogN+/HlY/QFO2Ahu6fyf/uVfZC9bPTef/l802WTurbyHvx/+TabI74ntBpf3ANybXtsN6x6AF1ZnLlD2jv/KzU+9gzcYGXVkZamNKn6v8yt8qmItnx/2z9xQ+RL/3PVuvtN1E5utNurwXAkIlQAkLQS+Rea2jv9gZnflrVewfjGZW0L+kZm92FNbSVXA94FaMreE/KiZ+Z2yi+nMycxdqXY9lZnfv/c5UAVceQu8689g4mzeeMqnekapiwru7/ogj3Zdz7Jh/8bHKn7ObcN+wZb0VJ5Mz+W36St5JT2NI1wUdagugZS5m2MPFaQK4FXgJqCFzI3ebzOzV3LqLAb+mEwCuA74lpld11NbSV8DOszsLkkrgHFm9oWeYpk3b56tX7++n281Qtl9fG5f2/nr+lr2ln+3nLKuzswX+5mTcOYEnD0Fp4/BG+1wvJ37f/IsdTpAg1q4TAdJKdN2c3oaj3Ut4F+7rmc/Ewb2ft2QGcMb3FzxDB+seIa361WGKXPm8D4bz+SZ74Cxl8HoiXDRJBhVDZWjYPhIqBwNlSMzz1PDMldvVQUoFTxP+f0cSpikF8zsvDs2hekBzAeazKw52NDDwBLglZw6S4CHgnsDPytprKQaMr/uu2u7BPjdoP1q4JdAjwmg3/79S/DCtxnSL978spi6vWI4u20im6yWH3W9k202lXXpmRzm4qhDcyEcYzTf6bqJ73TdxBiOc3WqmdnazezUbj50ZC/seQZOHe3n1pWTGPqaDPpYv0/bH8ptJ8gf/CNMv2FQNxkmAUwGcickt5D5ld9bncm9tJ1oZq0AZtYq6ZJCLy5pKbA0WHxD0rYC1SYAh3p/K7ESYcyHeWv+Ds33c3GEjvnlIQ6kD0p6P8fCl2+E/sc8rVBhmARQKJ3m/8ztrk6Ytj0ys/uB+3uqI2l9oe5NnHnMxeExF4fHXByDHXOYaaAtwNSc5SnA/pB1emrbFgwTETweDB+2c865gQqTANYBDZLqJFUCtwJr8uqsAW5XxgLgaDC801PbNcAdwfM7gEcH+F6cc871Qa9DQGZ2VtJy4HEyUzlXmdlmScuC9SuBtWRmADWRmQb6yZ7aBpu+C3hE0qeAPcBHBvA+ehwiiimPuTg85uLwmItjUGPudRqoc8650uSXgnDOuTLlCcA558pU4hKApKmSfiFpi6TNkj4blH9F0j5JG4K/xVHHmkvSLkkvB7GtD8qqJD0haXvwOC7qOLMkzczZlxskHZP0ubjtZ0mrJB2UtCmnrNv9KumLkpokbZP0n2IU899J2iqpUdKPJI0NymslnczZ3ytjFHO3n4UY7+fv58S7S9KGoDwu+7m777eh+UybWaL+gBpgbvD8IjKXmpgNfAX4s6jj6yHuXcCEvLKvASuC5yuAv406zm5irwAOkDmZJFb7GXg3MBfY1Nt+DT4nG4ERQB2wA6iIScz/ERgWPP/bnJhrc+vFbD8X/CzEeT/nrb8b+MuY7efuvt+G5DOduB6AmbVacKE5M3sd2ELmjOMkWkLmMhgEj7dEF0qPbgR2mNnuqAPJZ2ZPAR15xd3t1yXAw2Z22sx2kpm1Nr8YceYqFLOZ/dTMzgaLz5I5ZyY2utnP3Yntfs6SJOCjwPeKGlQvevh+G5LPdOISQC5JtcC1wHNB0fKgC70qTsMpAQN+KumF4PIWkHc5DKDg5TBi4Fbe+h8lzvsZut+v3V2yJG7+M/CTnOU6SS9J+pWkd0UVVDcKfRaSsJ/fBbSZ2facsljt57zvtyH5TCc2AUgaDfwL8DkzOwbcB0wH5gCtZLp3cXK9mc0FFgF3Snp31AGFEZzAdzPwg6Ao7vu5JwO+NMlQk/Rl4Czw3aCoFbjMzK4FPg/8k6QxUcWXp7vPQuz3M3Abb/1RE6v9XOD7rduqBcpC7+tEJgBJw8nsnO+a2Q8BzKzNzLrMLA08QARdzp6Y2f7g8SDwIzLxJeFyGIuAF82sDeK/nwPd7dcwlzWJjKQ7gA8AH7dggDfo2h8Onr9AZox3RnRRvqmHz0Lc9/Mw4MNk7kcCxGs/F/p+Y4g+04lLAMHY3YPAFjP7Rk55TU61DwGb8ttGRdIoSRdln5M54LeJZFwO4y2/lOK8n3N0t1/XALdKGiGpDmgAno8gvvMoc+OkLwA3m9mJnPJqZe6rgaR6MjE3RxPlW/XwWYjtfg68D9hqZi3Zgrjs5+6+3xiqz3TUR737cZT8nWS6OI3AhuBvMfCPZK6O2xjslJqoY82JuZ7MkfqNwGbgy0H5eOBJYHvwWBV1rHlxjyRz7eiLc8pitZ/JJKdW4AyZX0Of6mm/Al8m8+tuG7AoRjE3kRnLzX6mVwZ1fy/4zGwEXgQ+GKOYu/0sxHU/B+XfBpbl1Y3Lfu7u+21IPtN+KQjnnCtTiRsCcs45Nzg8ATjnXJnyBOCcc2XKE4BzzpUpTwDOOVemPAE451yZ8gTgnHNl6v8DvsUW6fMvnRQAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "scrolled": true,
        "id": "NKW7H0nTr-CK",
        "outputId": "b20781e6-9673-43ba-8b0c-2d775dd3b2c5"
      },
      "source": [
        "# code of parametric probability estimation\n",
        "# Code 2\n",
        "sample = normal(loc=100, scale=10, size=1000)\n",
        "# calculate parameters\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "# define the distribution\n",
        "dist = norm(sample_mean, sample_std)\n",
        "# sample probabilities for a range of outcomes\n",
        "values = [value for value in range(30, 200)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "# plot the histogram and pdf\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=99.696, Standard Deviation=10.004\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qBMCPZh_r-CM"
      },
      "source": [
        "Running the code first generates the data sample, then estimates the parameters of the normal probability distribution.\n",
        "\n",
        "Note that your results will differ given the random nature of the data sample. Try running the example a few times.\n",
        "\n",
        "In this case, we can see that the mean and standard deviation have some noise and are slightly different from the expected values of 100 and 10 respectively. The noise is minor and the distribution is expected to still be a good fit."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "P-gT5Jekr-CN",
        "outputId": "d5b974bf-2dbf-4177-ce57-0f2fef2df5b1"
      },
      "source": [
        "sample = normal(loc=200, scale=50, size=1000)\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "dist = norm(sample_mean, sample_std)\n",
        "values = [value for value in range(30, 500)]\n",
        "probabilities = [dist.pdf(value) for value in values]"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=199.280, Standard Deviation=49.744\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XhE8WQe4r-CP",
        "outputId": "05309f30-e1b3-49ce-a1ec-f9c0ca5a1c9a"
      },
      "source": [
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x1feab1d1370>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 35
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xmfNd8I3r-CQ",
        "outputId": "f9e4d525-019d-4873-b222-f19dfaec33f3"
      },
      "source": [
        "# code of parametric probability estimation\n",
        "# Code 3\n",
        "sample = normal(loc=200, scale=50, size=1000)\n",
        "# calculate parameters\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "# define the distribution\n",
        "dist = norm(sample_mean, sample_std)\n",
        "# sample probabilities for a range of outcomes\n",
        "values = [value for value in range(30, 500)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "# plot the histogram and pdf\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=198.202, Standard Deviation=49.452\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-fnKj8Edr-CR"
      },
      "source": [
        "Running the code first generates the data sample, then estimates the parameters of the normal probability distribution.\n",
        "\n",
        "Note that your results will differ given the random nature of the data sample. Try running the example a few times.\n",
        "\n",
        "In this case, we can see that the mean and standard deviation have some noise and are slightly different from the expected values of 200 and 50 respectively. The noise is minor and the distribution is expected to still be a good fit."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZD-He0ucr-CT",
        "outputId": "f0c0e4ec-b825-4235-d540-5b2cdf640f6a"
      },
      "source": [
        "sample = normal(loc=250, scale=70, size=1000)\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "dist = norm(sample_mean, sample_std)\n",
        "values = [value for value in range(30, 600)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=249.647, Standard Deviation=71.204\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x1feac527d90>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 37
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "P0wsxfzZr-CU",
        "outputId": "514c086d-f25c-4c9d-b1aa-8981b3bb828d"
      },
      "source": [
        "# code of parametric probability estimation\n",
        "# Code 4\n",
        "sample = normal(loc=250, scale=70, size=1000)\n",
        "# calculate parameters\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "# define the distribution\n",
        "dist = norm(sample_mean, sample_std)\n",
        "# sample probabilities for a range of outcomes\n",
        "values = [value for value in range(30, 600)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "# plot the histogram and pdf\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=245.684, Standard Deviation=68.823\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9OfCHBzpr-CW"
      },
      "source": [
        "Running the code first generates the data sample, then estimates the parameters of the normal probability distribution.\n",
        "\n",
        "Note that your results will differ given the random nature of the data sample. Try running the example a few times.\n",
        "\n",
        "In this case, we can see that the mean and standard deviation have some noise and are slightly different from the expected values of 250 and 70 respectively. The noise is minor and the distribution is expected to still be a good fit."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HLvBmAiAr-CX",
        "outputId": "aa55f8d2-9a8d-4f34-d056-da8996ced9fc"
      },
      "source": [
        "sample = normal(loc=300, scale=90, size=1000)\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "dist = norm(sample_mean, sample_std)\n",
        "values = [value for value in range(30, 800)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=302.693, Standard Deviation=90.791\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x1feac688fd0>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 40
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "3hmP16rVr-Cb",
        "outputId": "95b11ed3-2a34-4667-aa7d-b8cd990d4ba1"
      },
      "source": [
        "# code of parametric probability estimation\n",
        "# Code 5\n",
        "sample = normal(loc=300, scale=90, size=1000)\n",
        "# calculate parameters\n",
        "sample_mean = mean(sample)\n",
        "sample_std = std(sample)\n",
        "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n",
        "# define the distribution\n",
        "dist = norm(sample_mean, sample_std)\n",
        "# sample probabilities for a range of outcomes\n",
        "values = [value for value in range(30, 800)]\n",
        "probabilities = [dist.pdf(value) for value in values]\n",
        "# plot the histogram and pdf\n",
        "pyplot.hist(sample, bins=10, density=True)\n",
        "pyplot.plot(values, probabilities)\n",
        "pyplot.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Mean=297.163, Standard Deviation=92.014\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FgPNiuAGr-Cc"
      },
      "source": [
        "Running the code first generates the data sample, then estimates the parameters of the normal probability distribution.\n",
        "\n",
        "Note that your results will differ given the random nature of the data sample. Try running the example a few times.\n",
        "\n",
        "In this case, we can see that the mean and standard deviation have some noise and are slightly different from the expected values of 250 and 70 respectively. The noise is minor and the distribution is expected to still be a good fit."
      ]
    }
  ]
}