AtCoder Beginner Contest 126
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Problem Statement
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Snuke has a fair $N$-sided die that shows the integers from $1$ to $N$ with equal probability and a fair coin. He will play the following game with them:
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- Throw the die. The current score is the result of the die.
- As long as the score is between
$1$and$K-1$(inclusive), keep flipping the coin. The score is doubled each time the coin lands heads up, and the score becomes$0$if the coin lands tails up. - The game ends when the score becomes
$0$or becomes$K$or above. Snuke wins if the score is$K$or above, and loses if the score is$0$.
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You are given $N$ and $K$. Find the probability that Snuke wins the game.
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Constraints
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$1 ≤ N ≤ 10^5$$1 ≤ K ≤ 10^5$- All values in input are integers.
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Input
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Input is given from Standard Input in the following format:
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N K翻译中…
Output
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Print the probability that Snuke wins the game. The output is considered correct when the absolute or relative error is at most $10^{-9}$.
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样例
Input #1
3 10
Output #1
0.145833333333
- If the die shows $1$, Snuke needs to get four consecutive heads from four coin flips to obtain a score of $10$ or above. The probability of this happening is $\frac{1}{3} \times (\frac{1}{2})^4 = \frac{1}{48}$.
- If the die shows $2$, Snuke needs to get three consecutive heads from three coin flips to obtain a score of $10$ or above. The probability of this happening is $\frac{1}{3} \times (\frac{1}{2})^3 = \frac{1}{24}$.
- If the die shows $3$, Snuke needs to get two consecutive heads from two coin flips to obtain a score of $10$ or above. The probability of this happening is $\frac{1}{3} \times (\frac{1}{2})^2 = \frac{1}{12}$.
Thus, the probability that Snuke wins is $\frac{1}{48} + \frac{1}{24} + \frac{1}{12} = \frac{7}{48} \simeq 0.1458333333$.
Input #2
100000 5
Output #2
0.999973749998
题目信息
题面与样例版权归 AtCoder 及出题人所有；中文翻译由 AI 生成（CI 管线），如有出入请以原文为准。