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:
0 <= K <= N <= 10000
1 <= W <= 10000
Answers will be accepted as correct if they are within 10^-5 of the correct answer.
The judging time limit has been reduced for this question.
class Solution:
    def new21Game(self, N: int, K: int, W: int) -> float:
        if K==0 or K-1+W<=N:
            return 1
        dp=[1]+[0]*N
        cursum=1
        for i in range(1,N+1):
            # dp[i]=sum(dp[max(0,i-W):min(i,K)])*(1/W)
            dp[i]=cursum*(1/W)
            if i<K:
                cursum+=dp[i]
            if i>=W:
                cursum-=dp[i-W]
        return sum(dp[K:])
class Solution:
    def new21Game(self, N: int, K: int, W: int) -> float:
        if K==0 or K-1+W<=N:
            return 1
        dp=[1]+[0]*N
        cursum=1
        for i in range(1,N+1):
            # dp[i]=sum(dp[max(0,i-W):min(i,K)])*(1/W)
            dp[i]=cursum*(1/W)
            if i<K:
                cursum+=dp[i]
            if i>=W:
                cursum-=dp[i-W]
        return sum(dp[K:])