837. New 21 Game

Problem

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 <= 10^4

• 1 <= maxPts <= 10^4

Solution (Java)

class Solution {
    public double new21Game(int n, int k, int w) {
        if (n >= k + w || k == 0) {
            return 1.0;
        }
        double[] dp = new double[n + 1];
        dp[0] = 1.0;
        double res = 0.0;
        double runningSum = dp[0];
        for (int i = 1; i <= n; i++) {
            dp[i] = runningSum / w;
            if (i < k) {
                runningSum += dp[i];
            } else {
                res += dp[i];
            }
            if (i - w >= 0) {
                runningSum -= dp[i - w];
            }
        }
        return res;
    }
}

Explain:

nope.

Complexity:

• Time complexity : O(n).
• Space complexity : O(n).