A and B toss a coin alternately till one of them gets a head and wins the game. if A starts first , find the probability that B will win the game.

U.k. Akshay posted a question in Class XI-XII Tuition (PUC) Question: A and B toss a coin alternately till one of them gets a head and wins the game. if A starts first , find the probability that B will win the game. Answer: If ‘H’ denotes the event of head and ‘T’ denotes the event of tail, then the sample space for B getting a head is an infinite series: T H [T (when A tossed the coin) followed by H (when B tossed the coin)] T T T H [T (when A tossed the coin) followed by T (when B tossed the coin) T (when A tossed the coin for second time) followed by H (when B tossed the coin for a second time)] T T T T T H T T T T T T T H … As the probability of obtaining T and H are identically (1/2) at a tossing of the coin, the overall probability that B will obtain head first is P = [(1/2)*(1/2)] + [(1/2)*(1/2)* (1/2)*(1/2)] + [(1/2)*(1/2)* (1/2)*(1/2)* (1/2)*(1/2)*] + … = (1/2)^2 + (1/2)^4+(1/2)^6 + … = (1/4) + (1/4)^2+(1/4)^3 + … (sum of successive, infinite terms of a geometric progression with first member =1/4 and common factor also =1/4) Therefore P = (1/4)*[1/(1-1/4)]= (1/3) read less