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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 :
  1. 13
  2. 15
  3. 25
  4. 14


Solution

The correct option is A 13
Case1: If X is winner
Outcomes=H,TTH,TTTTH,...
P(X)=p+(1p)×12×p+(1p)2×(12)2×p+...
         =p112(1p)=2pp+1
Case2: If Y is the winner
Outcomes=TH,TTTH,TTTTTH,...
P(Y)=1p2+(1p)24+(1p)38+...
         =(1p)21(1p)2=(1p)p+1
Since, the probability of winning the game by both the players is equal
2pp+1=(1p)p+1
p=13

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