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?

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:

  1. 0 <= K <= N <= 10000
  2. 1 <= W <= 10000
  3. Answers will be accepted as correct if they are within 10^-5 of the correct answer.
  4. The judging time limit has been reduced for this question.

Rust Solution

struct Solution;

impl Solution {
    fn new21_game(n: i32, k: i32, w: i32) -> f64 {
        if k == 0 || n > k + w {
            return 1.0;
        }
        let n = n as usize;
        let w = w as usize;
        let k = k as usize;
        let mut dp: Vec<f64> = vec![0.0; n + 1];
        dp[0] = 1.0;
        let mut sum = 1.0;
        let mut res = 0.0;
        for i in 1..=n {
            dp[i] = sum / w as f64;
            if i < k {
                sum += dp[i];
            } else {
                res += dp[i];
            }
            if i >= w {
                sum -= dp[i - w];
            }
        }
        res
    }
}

#[test]
fn test() {
    use assert_approx_eq::assert_approx_eq;
    let n = 10;
    let k = 1;
    let w = 10;
    let res = 1.0;
    assert_approx_eq!(Solution::new21_game(n, k, w), res);
    let n = 6;
    let k = 1;
    let w = 10;
    let res = 0.6;
    assert_approx_eq!(Solution::new21_game(n, k, w), res);
    let n = 21;
    let k = 17;
    let w = 10;
    let res = 0.732777;
    assert_approx_eq!(Solution::new21_game(n, k, w), res);
}

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