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Question

Two players A and B toss a coin alternatively, with A beginning the game. The players who first throw a head is deemed to be the winner. B's coin is fair and A's is biased and has a probability p showing a head. Find the value of p so that the game is equiprobable to both the players.

A
12
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B
13
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C
1
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D
16
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Solution

The correct option is C 13
Player "A' wins if he gets head in the first trial or in the third {if B does not get head in his first trial] and so on.
P(A)=p+(1p)×12×p+(1p)×12×(1p)×12×p+...
This is an infinite G.P of the first term p and common ratio (1p)2=p1((1p)2)=2p(p+1)
According to the question,
P(A)=P(B)2p(1+p)=12p(p+1)4p(1+p)=1p=13

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