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Brahma
A and B pick a card at random from a well shuffled pack of cards, one after the other replacing it every time till one of them gets a queen. If A begins the game, then the probability that A wins the game is
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Jay Patel
P(A wins in his first draw)
$=\dfrac{1}{13}$

P(A does not win in his first draw
AND B does not win in his first draw
AND A wins in his second draw)
$=\dfrac{12}{13}.\dfrac{12}{13}.\dfrac{1}{13}$

P(A does not win in his first draw
AND B does not win in his first draw
AND A does not win in his second draw
AND B does not win in his second draw
AND A wins in his third draw
$=\dfrac{12}{13}.\dfrac{12}{13}.\dfrac{12}{13}.\dfrac{12}{13}.\dfrac{1}{13}$

so on

Required probability
$=\dfrac{1}{13}+\dfrac{12}{13}.\dfrac{12}{13}.\dfrac{1}{13}+\dfrac{12}{13}.\dfrac{12}{13}.\dfrac{12} {13}.\dfrac{12}{13}.\dfrac{1}{13}+\cdots\\ =\dfrac{\dfrac{1}{13}}{1-\dfrac{12}{13}×\dfrac{12}{13}}=\dfrac{13}{169-144}=\dfrac{13}{25}$
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