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. 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

• A. 1 : 2
• B. 2 : 1
• C. 1 : 1
• D. 1 : 3

Answer 1

The correct option is C 1 : 1

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
$P\left(A\right)=\frac{a}{a+b}+{\left(\frac{b}{a+b}\right)}^2\frac{a}{a+b}+{\left(\frac{b}{a+b}\right)}^4\frac{a}{a+b}+...\infty =\frac{a+b}{a+2b}SimilarlywegetP\left(B\right)=\frac{b}{a+b}.\frac{a}{a+b}+{\left(\frac{b}{a+b}\right)}^3\frac{a}{a+b}+...\infty =\frac{b}{a+2b}P\left(A\right)=2P\left(B\right)\Rightarrow a+b=2b\Rightarrow a=b.$