The probabilities of four cricketers A, B, C and D scoring more than 50 runs in a match are `1/2, 1/3, 1/4` and `1/10`. It is known that exactly two of the players more than 50 runs in a particular match. The probability that these players were A and B is

Options

A. `27/65`
B. `5/6`
C. `1/6`
D. None of these

Answer

To solve the problem, we need to find the probability that cricketers A and B are the ones who scored more than 50 runs, given that exactly two players scored more than 50 runs.

### Step-by-Step Solution:

1. **Identify the Probabilities**:
- Let \( P(A) = \frac{1}{2} \) (Probability that A scores more than 50 runs)
- Let \( P(B) = \frac{1}{3} \) (Probability that B scores more than 50 runs)
- Let \( P(C) = \frac{1}{4} \) (Probability that C scores more than 50 runs)
- Let \( P(D) = \frac{1}{10} \) (Probability that D scores more than 50 runs)

2. **Calculate the Probabilities of Not Scoring More than 50 Runs**:
- \( P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \)
- \( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \)
- \( P(D') = 1 - P(D) = 1 - \frac{1}{10} = \frac{9}{10} \)

3. **Identify the Possible Pairs of Players**:
The pairs of players who can score more than 50 runs are:
- (A, B)
- (A, C)
- (A, D)
- (B, C)
- (B, D)
- (C, D)

4. **Calculate the Probability for Each Pair**:
- **For A and B**:
\[ P(A \text{ and } B) = P(A) \cdot P(B) \cdot P(C') \cdot P(D') = \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{3}{4} \cdot \frac{9}{10} \]
\[ = \frac{1 \cdot 1 \cdot 3 \cdot 9}{2 \cdot 3 \cdot 4 \cdot 10} = \frac{27}{240} \]

- **For A and C**:
\[ P(A \text{ and } C) = P(A) \cdot P(C) \cdot P(B') \cdot P(D') = \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{2}{3} \cdot \frac{9}{10} \]
\[ = \frac{1 \cdot 1 \cdot 2 \cdot 9}{2 \cdot 4 \cdot 3 \cdot 10} = \frac{18}{240} \]

- **For A and D**:
\[ P(A \text{ and } D) = P(A) \cdot P(D) \cdot P(B') \cdot P(C') = \frac{1}{2} \cdot \frac{1}{10} \cdot \frac{2}{3} \cdot \frac{3}{4} \]
\[ = \frac{1 \cdot 1 \cdot 2 \cdot 3}{2 \cdot 10 \cdot 3 \cdot 4} = \frac{6}{240} \]

- **For B and C**:
\[ P(B \text{ and } C) = P(B) \cdot P(C) \cdot P(A') \cdot P(D') = \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{1}{2} \cdot \frac{9}{10} \]
\[ = \frac{1 \cdot 1 \cdot 1 \cdot 9}{3 \cdot 4 \cdot 2 \cdot 10} = \frac{9}{240} \]

- **For B and D**:
\[ P(B \text{ and } D) = P(B) \cdot P(D) \cdot P(A') \cdot P(C') = \frac{1}{3} \cdot \frac{1}{10} \cdot \frac{1}{2} \cdot \frac{3}{4} \]
\[ = \frac{1 \cdot 1 \cdot 1 \cdot 3}{3 \cdot 10 \cdot 2 \cdot 4} = \frac{3}{240} \]

- **For C and D**:
\[ P(C \text{ and } D) = P(C) \cdot P(D) \cdot P(A') \cdot P(B') = \frac{1}{4} \cdot \frac{1}{10} \cdot \frac{1}{2} \cdot \frac{2}{3} \]
\[ = \frac{1 \cdot 1 \cdot 1 \cdot 2}{4 \cdot 10 \cdot 2 \cdot 3} = \frac{2}{240} \]

5. **Total Probability of Exactly Two Players Scoring More than 50 Runs**:
\[ P(\text{exactly 2 players}) = P(A \text{ and } B) + P(A \text{ and } C) + P(A \text{ and } D) + P(B \text{ and } C) + P(B \text{ and } D) + P(C \text{ and } D) \]
\[ = \frac{27}{240} + \frac{18}{240} + \frac{6}{240} + \frac{9}{240} + \frac{3}{240} + \frac{2}{240} = \frac{65}{240} \]

6. **Calculate the Required Probability**:
The probability that the two players who scored more than 50 runs are A and B is given by:
\[ P(A \text{ and } B | \text{exactly 2 players}) = \frac{P(A \text{ and } B)}{P(\text{exactly 2 players})} \]
\[ = \frac{\frac{27}{240}}{\frac{65}{240}} = \frac{27}{65} \]

### Final Answer:
The probability that the players who scored more than 50 runs were A and B is \( \frac{27}{65} \).