leetcode 837 New 21 Game

solution 1 dynamic programming

This problem is similar to the idea of the classic dynamic programming problem "Climbing Stairs".
Let dp[i] represents the probability that Alice can accumulate to i points. Think about how we can reach i points from the last state, then we can conclude that dp[i] = dp[i-W] * 1/W + ... + dp[i-1] * 1/W. Now we can implement the algorithm based on the above formula.

    double new21Game(int N, int K, int W) {
        if(K == 0 || N >= K + W) return 1.0;
        vector<double> dp(N + 1, 0);
        dp[0] = 1.0;
        double wSum = 1.0, res = 0.0;
        for(int i = 1; i <= N; i++) {
            dp[i] = wSum / W;
            if(i < K) {
                wSum += dp[i];
            } else {
                res += dp[i]; //we will only accumulate the probability for points that are no less than K
            }
            if(i - W >= 0) {
                wSum -= dp[i-W];
            }
        }
        return res;
    }