2.probability of india winning test match against England is 2/3. Assuming independence from match to match, probability that in a 7 match series india’s third win occurs at 5 th match
1.8/27
2.16/81
3.8/81
4.32/81
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Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{If} \text{'}\mathrm{w}\text{'} \mathrm{represents} \mathrm{win} \mathrm{and} \text{'}\mathrm{l}\text{'} \mathrm{represents} \mathrm{loss}, \mathrm{then} \\ \mathrm{P}\left(\mathrm{w}\right)=\frac{2}{3} \\ \mathrm{P}\left(\mathrm{l}\right)=1-\frac{2}{3}=\frac{1}{3} \\ \mathrm{Required} \mathrm{Probability} \\ =\mathrm{India} \mathrm{wins} \mathrm{any} 2 \mathrm{of} \mathrm{first} 4 \mathrm{matches} \left(\mathrm{and} \mathrm{loses} \mathrm{other} 2\right) \mathrm{AND} \left(\mathrm{wins} \mathrm{fifth} \mathrm{match}\right) \\ =\left\{{}^4{\mathrm{C}}_2 {\left(\frac{2}{3}\right)}^2{\left(\frac{1}{3}\right)}^2\right\}\left\{\frac{2}{3}\right\} \\ =\frac{4\times 3}{2\times 1}\times \frac{8}{81\times 3} \\ \mathbf{=}\frac{\mathbf{16}}{\mathbf{81}}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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