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${\text{A}}{\text{. }}\dfrac{1}{8} \\ {\text{B}}{\text{. }}\dfrac{1}{4} \\ {\text{C}}{\text{. }}\dfrac{1}{2} \\ {\text{D}}{\text{. }}\dfrac{2}{3} \\$

Hint: - AS given in the question India’s second win should come in the third test that means either India should win the first and third match or it should win the second and third match, so we have to find the probability of these two cases.

Let ${A_1},{A_2}{\text{ and }}{A_3}$be the events of match winning in first , second and third matches respectively and whose probabilities are
$P\left( {{A_1}} \right) = P\left( {{A_2}} \right) = P\left( {{A_3}} \right) = \dfrac{1}{2}$
$P\left( {{A_1}'} \right) = P\left( {{A_2}'} \right) = P\left( {{A_3}'} \right) = \dfrac{1}{2}$
$= P\left( {{A_1}{A_2}'{A_3}} \right) + P\left( {{A_1}'{A_2}{A_3}} \right)$
$= P\left( {{A_1}} \right)P\left( {{A_2}'} \right)\left( {{A_3}} \right) + P\left( {{A_1}'} \right)p\left( {{A_2}} \right)p\left( {{A_3}} \right)$
$= {\left( {\dfrac{1}{2}} \right)^3} + {\left( {\dfrac{1}{2}} \right)^3} = \dfrac{1}{8} + \frac{1}{8} = \dfrac{1}{4}$