Question

A bag contains $w$ white $b$ black balls. Two players $A$ and $B$ alternatively draw one ball each from the bag, replacing the ball after each draw.The person who first draws a white ball is deemed to be winner. The probability that $B$ wins if $A$ begins the game is

• A. $\frac{b}{w+b}$
• B. $\frac{b}{w+2b}$
• C. $\frac{w+b}{w+2b}$
• D. $\frac{2b}{w+2b}$

Answer 1

Answer: (B)

$\frac{b}{w+2b}$

$B$ wins: ($A$ fails and $B$ wins) OR ($A$ fails and $B$ fails and $A$ fails and $B$ wins) OR ... till infinity
= $\frac{b}{w+b}\times \frac{w}{w+b}+{\left(\frac{b}{w+b}\right)}^3\times \frac{w}{w+b}+...\infty =\frac{\frac{b}{w+b}\times \frac{w}{w+b}}{1-{\left(\frac{b}{w+b}\right)}^2}=\frac{b}{w+2b}$