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Question

A bag contains a white and b black balls. Two players, A and B alternately draw a ball from the bag, replacing the ball each time after the draw till one of them draws a white ball and wins the game. A begins the game. If the probability of A winning the game is three times that of B, the ratio a:b is

A
1:1
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B
1:2
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C
2:1
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D
None of these
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Solution

The correct option is B 2:1
Let W denote the event of drawing a white ball at any draw and B that for a black ball. Then
P(W)=aa+b and P(B)=ba+b

P(A wins the game)=P(W or BBW or BBBBW or ...)
=P(W)+P(BBW)+P(BBBBW)+...
=P(W)+P(B)P(B)P(W)+P(B)P(B)P(B)P(B)P(W)+...

=P(W)+P(W).P(B)2+P(W).P(B)4+...

=P(W)1P(B)2=a(a+b)a2+2ab=a+ba+2b

Also P(B wins the game )=1a+ba+2b=ba+2b
According to the given conditions,
a+ba+2b=3.ba+2ba=2ba:b=2:1

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