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 calculate the probability that India's third win occurs in the fifth match of a 7-match series against England, given that the probability of India winning any single match is \( \frac{2}{3} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the probability that India wins exactly 2 out of the first 4 matches and wins the 5th match, which will be their 3rd win.

2. **Define Events**:
- Let event A be that India wins exactly 2 matches in the first 4 matches.
- Let event B be that India wins the 5th match.

3. **Calculate the Probability of Event A**:
To find the probability of India winning exactly 2 matches out of the first 4 matches, we can use the binomial probability formula:
\[ P(A) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:
- \( n = 4 \) (total matches),
- \( k = 2 \) (wins),
- \( p = \frac{2}{3} \) (probability of winning),
- \( 1 - p = \frac{1}{3} \) (probability of losing).

Thus, we have:
\[ P(A) = \binom{4}{2} \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^{4-2} \]

4. **Calculate the Binomial Coefficient**:
The binomial coefficient \( \binom{4}{2} \) is calculated as:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

5. **Substituting Values**:
Now substituting the values into the probability formula:
\[ P(A) = 6 \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^2 = 6 \cdot \frac{4}{9} \cdot \frac{1}{9} = 6 \cdot \frac{4}{81} = \frac{24}{81} \]

6. **Calculate the Probability of Event B**:
The probability that India wins the 5th match is simply:
\[ P(B) = \frac{2}{3} \]

7. **Combine the Probabilities**:
Since events A and B are independent, the total probability that India wins exactly 2 of the first 4 matches and wins the 5th match is:
\[ P(A \cap B) = P(A) \times P(B) = \frac{24}{81} \times \frac{2}{3} \]

8. **Final Calculation**:
\[ P(A \cap B) = \frac{24}{81} \times \frac{2}{3} = \frac{24 \times 2}{81 \times 3} = \frac{48}{243} \]
Simplifying \( \frac{48}{243} \) gives us \( \frac{16}{81} \).

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