LeetCode 837 New21 Game - Medium

837. New21 Game — Medium

Open on LeetCode

Problem

837. 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, W], where W is an integer. Each draw is independent and the outcomes have equal probabilities.

Alice stops drawing numbers when she gets K or more points. What is the probability that she has N or less points?

Example 1: Input: N = 10, K = 1, W = 10 Output: 1.00000 Explanation: Alice gets a single card, then stops.

Example 2: Input: N = 6, K = 1, W = 10 Output: 0.60000 Explanation: Alice gets a single card, then stops. In 6 out of W = 10 possibilities, she is at or below N = 6 points.

Example 3: Input: N = 21, K = 17, W = 10 Output: 0.73278

Note: 0 <= K <= N <= 10000 1 <= W <= 10000 Answers will be accepted as correct if they are within 10^-5 of the correct answer. The judging time limit has been reduced for this question.

Solution

# https://www.youtube.com/watch?v=bd0t6cj7_4E&ab_channel=LeetCode%E5%8A%9B%E6%89%A3
 
class Solution:
    def new21Game(self, N: int, K: int, W: int) -> float:
        dp = [0.0] * (N + W + 1)
        # dp[x] = the answer when Alice has x points
        for k in range(K, N + 1):
            dp[k] = 1.0
 
        S = min(N - K + 1, W)
        # S = dp[k+1] + dp[k+2] + ... + dp[k+W]
        for k in range(K - 1, -1, -1):
            dp[k] = S / float(W)
            S += dp[k] - dp[k + W]
 
        return dp[0]