The probability of India winning a test match against England is$\frac{2}{3}$. Assuming independence from match to match, the probability that in a 7 match series India's third win occurs at the fifth match, is

$\frac{16}{81}$

Consider the following events:

A = India wins 2 matches in first 4 matches.

B = India wins fifth match

C = Getting any outcome in sixth and seventh matches.

We have,

$P(A) = {^4C}_2 \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^{4-2}, P(B)=\frac{2}{3}$ and $ P(C) = 1.$

∴ Required probability = P(A ∩ B ∩ C)$

$= P(A) P(B) P(C)$      [∵ A, B, C are independent]

$={^4C}_2  \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^{2}× \frac{2}{3}×1=\frac{16}{81}$