Question

The probability of India winning a test match against South Africa is $\frac{1}{2}$ assuming independence from match to match played. The probability that in a $5$ match series India’s second win occurs at the $3^{rd}$ 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

$\frac{1}{4}$

Calculate the probability

Let $A_{1,}{A}_2\&A_3$be the events of match that India winning in first, second and third matches respectively.

Now the winning probabilities are,

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

And losing probabilities are

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

Here, out of $5$ matches, India's second win occur in $3^{rd}$ match

Hence, the required probability

$$\begin{gathered} P\left(A_1{A^{\text{'}}}_2{A}_3\right)+P\left({A^{\text{'}}}_1{A}_2{A}_3\right) \\ =P\left(A_1\right)P\left({A^{\text{'}}}_2\right)P\left(A_3\right)+P\left({A^{\text{'}}}_1\right)P\left(A_2\right)P\left(A_3\right) \\ ={\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^3 \\ =\frac{1}{8}+\frac{1}{8} \\ =\frac{1+1}{8} \\ =\frac{2}{8} \\ =\frac{1}{4} \end{gathered}$$

Therefore, the probability that in a $5$ match series India’s second win occurs at the $3^{rd}$ match is $\frac{1}{4}$, so, option (B) is the correct answer.