837. New 21 Game

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?

class Solution {
public:
    double new21Game(int N, int K, int W) {
        if(K == 0 || N >= K + W) return 1.0;
        vector<double> dp(N + 1);
        dp[0] = 1.0;
        double res = 0.0, wSum = 1.0;
        for(int i = 1; i <= N; i++) {
            dp[i] = wSum / W;
            if(i < K) {
                wSum += dp[i];
            } else {
                res += dp[i];
            }
            if(i - W >= 0) wSum -= dp[i - W];
        }
        return res;
    }
};