Question

The probability of India winning a test match against the West Indies is 1/2 assuming independence from match-to-match. The probability that in a match series India's second win occurs at the third test, is

• A. $\frac{1}{8}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{2}{3}$

Answer 1

The correct option is B $\frac{1}{4}$

Let $A_1,A_2$ and $A_3$ be the events of match winning in first, second and third matches respectively and whose probabilities are

$P\left(A_1\right)=P\left(A_2\right)=P\left(A_3\right)=\frac{1}{2}$

$P\left({A^{\prime }}_1\right)=P\left({A^{\prime }}_2\right)=P\left({A^{\prime }}_3\right)=\frac{1}{2}$

$\therefore$ Required probability

$=P\left(A_1{A}_2^{\prime }{A}_3\right)+P\left(A_1^{\prime }{A}_2{A}_3\right)$

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

$={\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^3=\frac{1}{8}+\frac{1}{8}=\frac{2}{8}=\frac{1}{4}$