Let A and B be two events such that the probability that exactly one of them occurs is `1/5` and the probability that A or B occurs is `1/3` , then the probability of both of them occur together is :

Options

A. `2/15`
B. `4/15`
C. `1/15`
D. `7/15`

Explanation

To solve the problem, we need to find the probability of both events A and B occurring together, denoted as \( P(A \cap B) \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The probability that exactly one of the events A or B occurs is given as:
\[ P(A \text{ only}) + P(B \text{ only}) = \frac{1}{5} \]
- The probability that either A or B occurs (the union of A and B) is given as:
\[ P(A \cup B) = \frac{1}{3} \]

2. **Express the Probabilities**:
- We can express the probabilities in terms of \( P(A) \), \( P(B) \), and \( P(A \cap B) \):
- The probability of exactly one of them occurring can be expressed as:
\[ P(A \text{ only}) + P(B \text{ only}) = P(A) + P(B) - 2P(A \cap B) \]
- Thus, we have:
\[ P(A) + P(B) - 2P(A \cap B) = \frac{1}{5} \quad (1) \]

3. **Using the Union Formula**:
- The probability of the union of A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Therefore, we can write:
\[ P(A) + P(B) - P(A \cap B) = \frac{1}{3} \quad (2) \]

4. **Set Up the Equations**:
- Now we have two equations:
- From (1): \( P(A) + P(B) - 2P(A \cap B) = \frac{1}{5} \)
- From (2): \( P(A) + P(B) - P(A \cap B) = \frac{1}{3} \)

5. **Subtract the Two Equations**:
- Subtract equation (1) from equation (2):
\[ (P(A) + P(B) - P(A \cap B)) - (P(A) + P(B) - 2P(A \cap B)) = \frac{1}{3} - \frac{1}{5} \]
- This simplifies to:
\[ P(A \cap B) = \frac{1}{3} - \frac{1}{5} \]

6. **Calculate the Right Side**:
- To compute \( \frac{1}{3} - \frac{1}{5} \), we need a common denominator:
\[ \frac{1}{3} = \frac{5}{15}, \quad \frac{1}{5} = \frac{3}{15} \]
- Therefore:
\[ P(A \cap B) = \frac{5}{15} - \frac{3}{15} = \frac{2}{15} \]

7. **Final Result**:
- Thus, the probability of both events A and B occurring together is:
\[ P(A \cap B) = \frac{2}{15} \]