Question

The probability of India wining a test match against West Indies is $1/2$. Assuming independence from match to match the probability that in a $5$ match series Indias second win occurs at the third test is

• A. $1/8$
• B. $1/4$
• C. $1/2$
• D. $1/3$

Answer 1

The correct option is B $1/4$

$A_1\to$ India wins 1st test $\to P(A_1=\frac{1}{2})$

$A_2\to$ India wins 2nd test $\to P(A_2=\frac{1}{2})$

$A_3\to$ India wins 3rd test $\to P(A_3=\frac{1}{2})$

Probability of India's second win at third test:

$=P(A_1^{\prime }\cap A_2\cap A_3)+P(A_1\cap A_2^{\prime }\cap A_3)$

$=P\left(A_1^{\prime }\right)P\left(A_2\right)P\left(A_3\right)+P\left(A_1\right)P\left(A_2{)}^{\prime }P\right(A_3)$

$=(1-\frac{1}{2})\times \frac{1}{2}\times \frac{1}{2}+\frac{1}{2}\times (1-\frac{1}{2})\times \frac{1}{2}$

$=\frac{1}{2^3}+\frac{1}{2^3}$

$=\frac{1}{8}+\frac{1}{8}$

$=\frac{1}{4}$