Two persons a and B toss an unbiased coin alternatively on the understanding that the first who gets the head wins.If A starts the game, find their respective chances of winning.

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