Question

A and B toss a coin alternatively till one of them gets a head and wins the game. If A starts the game, find the probability of his winning at his third toss.

Solution

Let P(A), $straight P left parenthesis straight A with bar on top right parenthesis$ be the probabilities of A's getting the head and not getting the head respectively.
Then $straight P left parenthesis straight A right parenthesis space equals 1 half comma space space straight P left parenthesis straight A with bar on top right parenthesis space equals space 1 minus straight P left parenthesis straight A right parenthesis space equals space 1 minus 1 half space equals space 1 half.$
Similarly, P(B) = $1 half comma space space space space straight P left parenthesis straight B with bar on top right parenthesis space equals space 1 half$
Now A starts the game and wins in his third toss i.e., in 5th row
$therefore$ required probability = $straight P left parenthesis straight A with bar on top right parenthesis thin space straight P left parenthesis straight B with bar on top right parenthesis space straight P left parenthesis straight A with bar on top right parenthesis thin space straight P left parenthesis straight B with bar on top right parenthesis thin space straight P left parenthesis straight A right parenthesis$
$equals space 1 half cross times 1 half cross times 1 half cross times 1 half cross times 1 half space equals space 1 over 32$

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Probability

A and B toss a coin alternatively till one of them gets a head and wins the game. If A starts the game, find the probability of his winning at his third toss.

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