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

Given that A and B toss a coin until one of them gets a head to win the game.

Let us find the probability of getting the head.

P(AH) = P(A getting a head on tossing a coin)

P(AN) = P(A not getting head on tossing a coin)

P(BH) = P(B getting a head on tossing a coin)

P(BN) = P(B not getting head on tossing a coin)

It is told that A starts the game.

We need to find the probability that B wins the games.

B wins the game only when A losses in1st,3rd,5th,…… tosses.

This can be shown as follows:

P(WB) = P(B wins the game)

P(WB) = P(ANBH) + P(ANBNANBH) + P(ANBNANBNANBH) + …………………

Since tossing a coin by each person is an independent event, the probabilities multiply each other.

P(WB) = (P(AN)P(BH)) + (P(AN)P(BN)P(AN)P(BH)) + ………………

The terms in the bracket resembles the infinite geometric series sequence:

We know that the sum of a Infinite geometric series with first term ‘a’ and common ratio ‘r’ is

The required probability is .

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