{"problem":{"name":"Dice and Coin","description":{"content":"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: 1.  Throw the die. The current score is the ","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"abc126_c"},"statements":[{"statement_type":"Markdown","content":"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:\n\n1.  Throw the die. The current score is the result of the die.\n2.  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.\n3.  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 $0$.\n\nYou are given $N$ and $K$. Find the probability that Snuke wins the game.\n\n## Constraints\n\n*   $1 ≤ N ≤ 10^5$\n*   $1 ≤ K ≤ 10^5$\n*   All values in input are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$ $K$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"abc126_c","tags":[],"sample_group":[["3 10","0.145833333333\n\n*   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 $\\frac{1}{3} \\times (\\frac{1}{2})^4 = \\frac{1}{48}$.\n*   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 $\\frac{1}{3} \\times (\\frac{1}{2})^3 = \\frac{1}{24}$.\n*   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 $\\frac{1}{3} \\times (\\frac{1}{2})^2 = \\frac{1}{12}$.\n\nThus, the probability that Snuke wins is $\\frac{1}{48} + \\frac{1}{24} + \\frac{1}{12} = \\frac{7}{48} \\simeq 0.1458333333$."],["100000 5","0.999973749998"]],"created_at":"2026-03-03 11:01:14"}}