If `P(A)=1//4, P(B)=1//2, P(A cup B)=5//8, " then " P(A cap B)` is

Options

A. `3//8`
B. `1//8`
C. `2//8`
D. `5//8`

Explanation

To find \( P(A \cap B) \) given \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{2} \), and \( P(A \cup B) = \frac{5}{8} \), we can use the formula for the probability of the intersection of two events:

\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{5}{8} \)

2. **Substitute the Values into the Formula:**
\[ P(A \cap B) = \frac{1}{4} + \frac{1}{2} - \frac{5}{8} \]

3. **Convert All Fractions to a Common Denominator:**
- The least common multiple (LCM) of 4, 2, and 8 is 8.
- Convert \( P(A) \) and \( P(B) \) to have a denominator of 8:
- \( P(A) = \frac{1}{4} = \frac{2}{8} \)
- \( P(B) = \frac{1}{2} = \frac{4}{8} \)

4. **Rewrite the Equation:**
\[ P(A \cap B) = \frac{2}{8} + \frac{4}{8} - \frac{5}{8} \]

5. **Perform the Addition and Subtraction:**
\[ P(A \cap B) = \frac{2 + 4 - 5}{8} = \frac{1}{8} \]

6. **Final Result:**
\[ P(A \cap B) = \frac{1}{8} \]

### Conclusion:
Thus, the probability \( P(A \cap B) \) is \( \frac{1}{8} \).