Question

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_______
A. 1/3
B. 1/5
C. 1/4
D. 2/5

Answer 1

To find the value of 'p' for the biased coin so that player X and player Y have an equal chance of winning the game, we consider the geometric probability series and set up an equation. After solving, we find that p is 1/3, meaning option A is the correct answer.

Step-by-step explanation:

The student's question concerns a game involving two players, X and Y, each with a different coin. Player X has a biased coin with probability p of showing heads, while player Y has a fair coin. The game is played alternatively until someone gets a head, and they are declared the winner.

To find the probability p such that both players have an equal chance of winning the game, we need to consider the geometric probability series. Let's calculate it step-by-step:

1. The probability that X wins in the first round is simply p.
2. The probability that neither wins in the first round is (1-p)*0.5.
3. X can also win in the second round after both have failed in the first round. So we multiply the probability of neither winning in the first round by p to get their chance of winning subsequently.
4. This process continues infinitely as there is no limit on the number of turns.

The total probability that X wins is given by p + (1-p)*0.5*p + (1-p)*0.5*(1-p)*0.5*p + ... This infinite geometric series has a sum formula of p/(1 - (1-p)*0.5).

Setting this equal to 0.5 (since the probability of winning needs to be equal for both players) and solving for p, we get:

p = 0.5 / (1 + 0.5 - 0.5p)
Solving this equation for p, we find that p = 1/3 (Option A).