A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of ' p ' is

Step-by-step solution

If the outcome is one of the following: H, T T H, T T T T H, , then X wins As subsequent tosses are independent, so the probability that X wins is p+ p 4 + p 16 + = 4 p 3 . Similarly Y wins if the outcome is one of the following: T H, T T T H, T T T T T H, Therefore, the probability that Y wins is 1-p 2 + 1-p 8 + 1-p 32 = 2(1-p) 3 Since, the probability of winning the game by both the players is equal then, we have 4 p 3 = 2(1-p) 3 p= 1 3