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837. New 21 Game

{leetcode}/problems/new-21-game/[LeetCode - 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.

link:{sourcedir}/_0837_New21Game.java[role=include]