837. New 21 Game

Problem Link

Description

---

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

Solution

---

Python3

class Solution:
    def new21Game(self, n: int, k: int, maxPts: int) -> float:
        if k == 0:
            return 1.0
        dp = [1] + [0] * n
        window = 1
        for p in range(1, n + 1):
            dp[p] = window / maxPts
            if p < k:
                window += dp[p]
            if 0 <= p - maxPts < k:
                window -= dp[p - maxPts]
        return sum(dp[k:])