Let A and B two events such that the probability that exactly one of them occurs is `2/5` and the probability that A or B occurs is `1/2`, then probability of both of them occur together is :

Options

A. `0.02`
B. `0.10`
C. `0.01`
D. `0.20`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The probability that exactly one of the events A or B occurs is given as:
\[ P(A \text{ only}) + P(B \text{ only}) = \frac{2}{5} \]
- The probability that either A or B occurs (the union of A and B) is given as:
\[ P(A \cup B) = \frac{1}{2} \]

2. **Expressing the Probabilities**:
- We can express the probabilities of A and B in terms of their intersection:
\[ P(A \text{ only}) = P(A) - P(A \cap B) \]
\[ P(B \text{ only}) = P(B) - P(A \cap B) \]
- Therefore, we can rewrite the first equation as:
\[ (P(A) - P(A \cap B)) + (P(B) - P(A \cap B)) = \frac{2}{5} \]
Simplifying this gives:
\[ P(A) + P(B) - 2P(A \cap B) = \frac{2}{5} \quad \text{(Equation 1)} \]

3. **Using the Union Formula**:
- The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the given value, we have:
\[ P(A) + P(B) - P(A \cap B) = \frac{1}{2} \quad \text{(Equation 2)} \]

4. **Solving the Equations**:
- From Equation 1, we can express \( P(A) + P(B) \):
\[ P(A) + P(B) = \frac{2}{5} + 2P(A \cap B) \quad \text{(Substituting in Equation 2)} \]
- Substitute this into Equation 2:
\[ \left(\frac{2}{5} + 2P(A \cap B)\right) - P(A \cap B) = \frac{1}{2} \]
- Simplifying gives:
\[ \frac{2}{5} + P(A \cap B) = \frac{1}{2} \]
- Rearranging to isolate \( P(A \cap B) \):
\[ P(A \cap B) = \frac{1}{2} - \frac{2}{5} \]

5. **Finding a Common Denominator**:
- The common denominator for 2 and 5 is 10:
\[ P(A \cap B) = \frac{5}{10} - \frac{4}{10} = \frac{1}{10} \]

6. **Final Answer**:
- Therefore, the probability that both A and B occur together is:
\[ P(A \cap B) = \frac{1}{10} \]

### Summary:
The probability that both events A and B occur together is \( \frac{1}{10} \).