The probability of India winning a test match againest England is `(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

Options

A. `(8)/(27)`
B. `(16)/(81)`
C. `(8)/(81)`
D. `(32)/(81)`

Explanation

To solve the problem, we need to find the probability that India's third win occurs at the fifth match in a series of seven matches. We can use the concept of the negative binomial distribution for this scenario.

### Step-by-Step Solution:

1. **Identify the probabilities**:
- The probability of India winning a match, \( p = \frac{2}{3} \).
- The probability of India losing a match, \( q = 1 - p = \frac{1}{3} \).

2. **Understand the scenario**:
- We want the third win to occur on the fifth match. This means that in the first four matches, India must have exactly 2 wins and 2 losses.

3. **Calculate the number of ways to arrange the wins and losses**:
- The number of ways to choose 2 wins from the first 4 matches can be calculated using the binomial coefficient:
\[ \text{Number of ways} = \binom{4}{2} = \frac{4!}{2! \cdot 2!} = 6 \]

4. **Calculate the probability of a specific arrangement**:
- The probability of winning 2 matches and losing 2 matches in the first four matches is given by:
\[ P(\text{2 wins, 2 losses}) = p^2 \cdot q^2 = \left(\frac{2}{3}\right)^2 \cdot \left(\frac{1}{3}\right)^2 = \frac{4}{9} \cdot \frac{1}{9} = \frac{4}{81} \]

5. **Include the probability of winning the fifth match**:
- Since the third win occurs at the fifth match, we multiply the probability of the first four matches by the probability of winning the fifth match:
\[ P(\text{3rd win at 5th match}) = \text{Number of ways} \cdot P(\text{2 wins, 2 losses}) \cdot P(\text{win at 5th match}) \]
\[ = 6 \cdot \frac{4}{81} \cdot \frac{2}{3} \]

6. **Calculate the final probability**:
- Now calculate the final probability:
\[ = 6 \cdot \frac{4}{81} \cdot \frac{2}{3} = 6 \cdot \frac{8}{243} = \frac{48}{243} = \frac{16}{81} \]

### Final Answer:
The probability that India's third win occurs at the fifth match is \( \frac{16}{81} \).