레이팅 미적용You and your friends at the Betting against All Probability Club are visiting a casino where the following game is played.
Each of the \(n\) players starts with \(k\) lives and puts in a fixed amount of money. In each round of the game, each player flips a biased coin and loses a life if she gets tails. The game ends when only one player remains, in which case this person wins, or ends in a draw if all remaining players lose their last life in the same round. If there is a winner, she wins \(n\) times her original bet. In case of a draw, no one wins anything.
Being a BAPC member you quickly realize the casino has an edge here: whenever the game ends in a draw all of the contestants lose the money they bet. You are now wondering what exactly is the probability that this game ends in a draw, so you can figure out how much the casino profits on average.
- One line containing two integers, \(2\leq n\leq 50\), the number of players, \(1\leq k\leq 50\), the number of lives each player has, and a floating point number \(0.1 \leq p \leq 0.9\), the probability the coin lands heads.
- Output a single floating point number: the probability of the game ending in a draw. Your answer should have an absolute error of at most \(10^{-6}\).
2 2 0.5
0.185185185
2 2 0.8
0.056241426
5 3 0.85
0.045463964