Alice plays the following game, loosely based on the card game "21".
Alice starts with 0 points and draws numbers while she has less than k points. During each draw, she gains an integer number of points randomly from the range [1, maxPts], where maxPts is an integer. Each draw is independent and the outcomes have equal probabilities.
Alice stops drawing numbers when she gets k or more points.
Return the probability that Alice has n or fewer points.
Answers within 10-5 of the actual answer are considered accepted.
Example 1:
Input: n = 10, k = 1, maxPts = 10 Output: 1.00000 Explanation: Alice gets a single card, then stops.
Example 2:
Input: n = 6, k = 1, maxPts = 10 Output: 0.60000 Explanation: Alice gets a single card, then stops. In 6 out of 10 possibilities, she is at or below 6 points.
Example 3:
Input: n = 21, k = 17, maxPts = 10 Output: 0.73278
Constraints:
0 <= k <= n <= 1041 <= maxPts <= 104
function new21Game(n: number, k: number, maxPts: number): number {
if (!k) return 1.0;
let dp = new Array(k + maxPts).fill(0.0);
for (let i = k; i <= n && i < k + maxPts; i++) {
dp[i] = 1.0;
}
dp[k - 1] = (1.0 * Math.min(n - k + 1, maxPts)) / maxPts;
for (let i = k - 2; i >= 0; i--) {
dp[i] = dp[i + 1] - (dp[i + maxPts + 1] - dp[i + 1]) / maxPts;
}
return dp[0];
}