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

⇒

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

⇒

⇒ P(B_H) = P(B getting a head on tossing a coin)

⇒

⇒ P(B_N) = 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 in1^st,3^rd,5^th,…… tosses.

This can be shown as follows:

⇒ P(W_B) = P(B wins the game)

⇒ P(W_B) = P(A_NB_H) + P(A_NB_NA_NB_H) + P(A_NB_NA_NB_NA_NB_H) + …………………

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

⇒ P(W_B) = (P(A_N)P(B_H)) + (P(A_N)P(B_N)P(A_N)P(B_H)) + ………………

⇒

⇒

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

⇒

⇒

⇒

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∴ The required probability is .