The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match, find the probability that in a match series Indias second win occurs at the third test.

Explanation

To solve the problem, we need to find the probability that India's second win occurs at the third test match. We denote the events as follows:

- Let \( A_1 \) be the event that India wins the first test.
- Let \( A_2 \) be the event that India wins the second test.
- Let \( A_3 \) be the event that India wins the third test.

We know that the probability of India winning a test match is \( P(A) = \frac{1}{2} \) and the probability of India losing a test match is \( P(A') = 1 - P(A) = \frac{1}{2} \).

### Step-by-Step Solution:

1. **Understanding the requirement**: We need to find the probability that India's second win occurs at the third test. This means that:
- India must win the third test \( (A_3) \).
- India must have exactly one win in the first two tests.

2. **Possible scenarios**: The scenarios that satisfy the condition of having exactly one win in the first two tests are:
- Win the first test and lose the second test: \( (A_1 \cap A_2') \)
- Lose the first test and win the second test: \( (A_1' \cap A_2) \)

3. **Calculating probabilities**:
- For the first scenario \( (A_1 \cap A_2' \cap A_3) \):
\[ P(A_1) \times P(A_2') \times P(A_3) = P(A) \times P(A') \times P(A) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

- For the second scenario \( (A_1' \cap A_2 \cap A_3) \):
\[ P(A_1') \times P(A_2) \times P(A_3) = P(A') \times P(A) \times P(A) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

4. **Total probability**: Since these two scenarios are mutually exclusive, we can add their probabilities:
\[ P(\text{Second win at third test}) = P(A_1 \cap A_2' \cap A_3) + P(A_1' \cap A_2 \cap A_3) = \frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \]

### Final Answer:
The probability that India's second win occurs at the third test is \( \frac{1}{4} \).

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