First slide

Theorems of probability

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

Moderate

A

1:1
B

1:2
C

2:1
D

None of these

Solution

Let W denote the event of drawing a white ball at any draw and B that for a black ball. Then
$P (W) = \frac{a}{a + b} and P (B) = \frac{b}{a + b}$
P(A wins the game)
$\begin{array}{l} = P (W or BBW or BBBBW or \dots) \\ = P (W) + P (BBW) + P (BBBBW) + \dots \\ = P (W) + P (B) P (B) P (W) + P (B) P (B) P (B) P (B) P (W) + \dots \\ = P (W) + P (W) P (B)^{2} + P (W) P (B)^{4} + \dots \\ = \frac{P (W)}{1 - P (B)^{2}} = \frac{a (a + b)}{a^{2} + 2 ab} = \frac{a + b}{a + 2 b} \end{array}$
$Also P (B wins the game) = 1 - \frac{a + b}{a + 2 b} = \frac{a}{a + 2 b}$
According to the given condition,
$\frac{a + b}{a + 2 b} = 3 \frac{b}{a + 2 b} \Rightarrow a = 2 b \Rightarrow a : b = 2 : 1$

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