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?

Xwins when the outcome is one of the following set of outcomes:H,TTH,TTTTH,Since subsequent tosses are independent, the probability thatXwins isp+p4+p16+...=4p3SimilarlyYwins if the outcome is one of the following:TH,TTTH,TTTTTH,...So, the probability thatYwins is1−p2+1−p8+1−p32=2(1−p)3SinceXandYwin with equal probability, we have4p3=2(1−p)3⇒p=13So, option A is the correct answer.