Bayesian Statistics The Fun Way - Python Solutions # CHAPTER 05.ipynb

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   "source": [
    "import math\n",
    "import numpy as np\n",
    "from scipy.stats import beta\n",
    "from scipy.stats import binom"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "# THE BETA DISTRIBUTION"
   ]
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   "source": [
    "Equation: Beta(p; α,β) = (p^(α-1)) * ((1-p)^(β-1))/beta(α,β)\n",
    "\n",
    "where:  \n",
    "\n",
    "p = the probability of an event  \n",
    "α = how many times we observe an event we care about  \n",
    "β = represent how many times event we didn't care about happen"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "This function is called Probability Density Function (PDF)"
   ]
  },
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    "We would like to calculate the probability that rate in which box returns two quarters is less than 0.5 based on the data we have collected: 14 times box returned two quarters and 27 times it didn't"
   ]
  },
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   "source": [
    "In python we can do that using scipy.beta function, just like in the example below:"
   ]
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   "cell_type": "code",
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   "source": [
    "p = 0.5\n",
    "a = 14\n",
    "b = 27"
   ]
  },
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   "source": [
    "beta.cdf(x=p, a=a, b=b)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Another example from the book: based on the information that we have pulled a certain card 5 times out 1200 trials we would like to know how certain we can be that the probability of pulling this card is greater than 0.005"
   ]
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 0.005\n",
    "a = 5\n",
    "b = 1195"
   ]
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   "source": [
    "beta.sf(x=p, a=a, b=b)"
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   "cell_type": "markdown",
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   "source": [
    "Please note that in Python you will use cdf attribute of beta function to calculate probability less than input value. To calculate probability greater than certain value it is handy to use sf attribute of the same function"
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   "source": [
    "### 1. You want to use the beta distribution to determine whether or not a coin you have is a fair coin –meaning that the coin gives you head and tails equally. You flip the coin 10 times and get 4 heads and 6 tails. Using the beta distribution, what is the probability that the coin will land on heads more than 60% of the time?"
   ]
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   "source": [
    "p = 0.6\n",
    "a = 4\n",
    "b = 6"
   ]
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   "source": [
    "beta.sf(x=p, a=a, b=b)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "### 2. You flip the coin 10 more times and now you have 9 heads and 11 tails total. What is the probability that the coin is fair, using our definition of fair, give or take 5 percent?"
   ]
  },
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   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = [0.45, 0.55]\n",
    "a = 9\n",
    "b = 11"
   ]
  },
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   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta.sf(x=p, a=a, b=b)"
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   "source": [
    "As you can see in result we get array which contains probability of being greater than first value in interval we have passed as an argument and probability of being greater than the second value. As a final step we have to subtract these two values to get a probability within the interval."
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    "beta.sf(x=p, a=a, b=b)[0] - beta.sf(x=p, a=a, b=b)[1] "
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   "cell_type": "markdown",
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   "source": [
    "### 3. Data is the best way to become more confident on your assertions. You flip the coin 200 more times and end up with 109 heads and 111 tails. Now what is the probability that the coin is fair, give or take 5 percent?"
   ]
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   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = [0.45, 0.55]\n",
    "a = 109\n",
    "b = 111"
   ]
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     "execution_count": 13,
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   "source": [
    "beta.sf(x=p, a=a, b=b)[0] - beta.sf(x=p, a=a, b=b)[1] "
   ]
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