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, 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?
Example1
Input: N = 10, K = 1, W = 10
Output: 1.00000
Explanation: Alice gets a single card, then stops.
Example2
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.
Example3
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.
Solution
題都看不懂
class Solution {
public:
double new21Game(int N, int K, int W) {
if (K == 0) {
return 1.0;
}
vector<double> dp(K + W);
for (int i = K; i <= N && i < K + W; i++) {
dp[i] = 1.0;
}
dp[K - 1] = 1.0 * min(N - K + 1, W) / W;
for (int i = K - 2; i >= 0; i--) {
dp[i] = dp[i + 1] - (dp[i + W + 1] - dp[i + 1]) / W;
}
return dp[0];
}
};