The probability of India winning a test match against west Indies is 1/2. assuming independence from match to match, the probability that in a 5 match series India's second win occurs at third test is

Options

A. `1//8`
B. `1//4`
C. `1//2`
D. `2//3`

Answer

To solve the problem of finding the probability that India's second win occurs in the third test of a 5-match series against West Indies, we can use the concept of binomial probability.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the probability that India wins exactly 2 matches out of the first 3 matches, and wins the 3rd match.

2. **Define the Probabilities**:
Let \( p = \frac{1}{2} \) (the probability of India winning a match) and \( q = 1 - p = \frac{1}{2} \) (the probability of India losing a match).

3. **Set Up the Conditions**:
For India's second win to occur in the third match, the following must happen:
- India must win the 3rd match.
- India must win exactly 1 of the first 2 matches.

4. **Calculate the Probability of Winning the 3rd Match**:
The probability that India wins the 3rd match is:
\[ P(\text{Win 3rd match}) = p = \frac{1}{2} \]

5. **Calculate the Probability of Winning Exactly 1 of the First 2 Matches**:
The number of ways to choose 1 win from 2 matches is given by \( \binom{2}{1} = 2 \).
The probability of winning 1 match and losing 1 match is:
\[ P(\text{1 win and 1 loss in first 2 matches}) = \binom{2}{1} \cdot p^1 \cdot q^1 = 2 \cdot \left(\frac{1}{2}\right)^1 \cdot \left(\frac{1}{2}\right)^1 = 2 \cdot \frac{1}{2} \cdot \frac{1}{2} = 2 \cdot \frac{1}{4} = \frac{1}{2} \]

6. **Combine the Probabilities**:
Now, we combine the probabilities of winning exactly 1 of the first 2 matches and winning the 3rd match:
\[ P(\text{Second win in 3rd match}) = P(\text{1 win in first 2}) \cdot P(\text{Win 3rd match}) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

7. **Final Answer**:
The probability that India's second win occurs at the third test is:
\[ \boxed{\frac{1}{4}} \]