Problem G
Lucky Draw
You 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.
Input
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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 real number $0.1 \leq p \leq 0.9$, with at most two digits after the decimal point, which is the probability the coin lands heads.
Output
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Output a single real number: the probability of the game ending in a draw. Your answer should have an absolute error of at most $10^{-6}$.
| Sample Input 1 | Sample Output 1 |
|---|---|
2 2 0.5 |
0.185185185 |
| Sample Input 2 | Sample Output 2 |
|---|---|
2 2 0.8 |
0.056241426 |
| Sample Input 3 | Sample Output 3 |
|---|---|
5 3 0.85 |
0.045463964 |
