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867. New 21 Game

Math Dynamic Programming Sliding Window Probability and Statistics

Problem - New 21 Game

Medium

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 <= 104
  • 1 <= maxPts <= 104

Solutions

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class Solution:
    def new21Game(self, n: int, k: int, maxPts: int) -> float:
        @cache
        def dfs(i : int) -> float:
            if i >= k:
                return int(n >= i)
            if i == k - 1:
                return min(n - k + 1, maxPts) / maxPts

            return dfs(i + 1) + (dfs(i + 1) - dfs(i + maxPts + 1)) /maxPts

        return dfs(0)

Submission Stats:

  • Runtime: 95 ms (5.13%)
  • Memory: 33.4 MB (6.36%)