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
Answer & explanation
Correct answer: option 2
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]
$={^4C}_2 \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^{2}× \frac{2}{3}×1=\frac{16}{81}$