If `P(A)=0.8, P(B)=0.5` , then `P(AcapB)` lies in the interval

Options

A. `[0.2, 0.5]`
B. `[0.2, 0.3]`
C. `[0.3, 0.5]`
D. `[0.1, 0.5]`

Answer

To find the interval in which \( P(A \cap B) \) lies, we can use the properties of probabilities.

### Step-by-Step Solution:

1. **Identify Given Probabilities**:
- \( P(A) = 0.8 \)
- \( P(B) = 0.5 \)

2. **Use the Formula for Intersection**:
- The probability of the intersection of two events \( A \) and \( B \) can be expressed using the formula:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

3. **Determine the Maximum Value of \( P(A \cap B) \)**:
- The maximum value of \( P(A \cap B) \) cannot exceed the probability of the least likely event. Therefore:
\[ P(A \cap B) \leq \min(P(A), P(B)) = \min(0.8, 0.5) = 0.5 \]

4. **Determine the Minimum Value of \( P(A \cap B) \)**:
- The minimum value of \( P(A \cap B) \) can be derived from the fact that:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]
- Substituting the values:
\[ P(A \cap B) \geq 0.8 + 0.5 - 1 = 0.3 \]

5. **Conclusion**:
- Therefore, combining both results, we find:
\[ 0.3 \leq P(A \cap B) \leq 0.5 \]
- Hence, the interval in which \( P(A \cap B) \) lies is \( [0.3, 0.5] \).

### Final Answer:
The interval for \( P(A \cap B) \) is \( [0.3, 0.5] \).

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