A bag contains 'a' white and 'b' black balls. Two playe

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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. A begins the game. If the probability of A winning ( that is
drawing a white ball) is twice the probability of B winning, then the ratio a : b is equal to

Options:

A . 1 : 2

B . 2 : 1

C . 1 : 1

D . 1 : 3

Answer: Option C
:
C
Let the event when a white ball is draw be W and for black ball let it be B. So A wins when we get sequence of the form
W or WBW or WBBBW or WBBBBBW......
Probability of getting W is a/a+b . So we get

P (A) = \frac{a}{a + b} + {(\frac{b}{a + b})}^{2} \frac{a}{a + b} + {(\frac{b}{a + b})}^{4} \frac{a}{a + b} + . . . \infty = \frac{a + b}{a + 2 b} S i m i l a r l y w e g e t P (B) = \frac{b}{a + b} . \frac{a}{a + b} + {(\frac{b}{a + b})}^{3} \frac{a}{a + b} + . . . \infty = \frac{b}{a + 2 b} P (A) = 2 P (B) \Rightarrow a + b = 2 b \Rightarrow a = b .

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