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 union of two events:

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

### Step-by-Step Solution:

1. **Write down the formula for the union of two events:**
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

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

3. **Convert the fractions to have a common denominator:**
- The least common multiple (LCM) of 4 and 2 is 8.
- Convert \( P(A) \) and \( P(B) \):
\[ P(A) = \frac{1}{4} = \frac{2}{8} \]
\[ P(B) = \frac{1}{2} = \frac{4}{8} \]

4. **Substitute these values back into the equation:**
\[ \frac{5}{8} = \frac{2}{8} + \frac{4}{8} - P(A \cap B) \]

5. **Combine the fractions on the right side:**
\[ \frac{5}{8} = \frac{2 + 4}{8} - P(A \cap B) \]
\[ \frac{5}{8} = \frac{6}{8} - P(A \cap B) \]

6. **Rearrange the equation to solve for \( P(A \cap B) \):**
\[ P(A \cap B) = \frac{6}{8} - \frac{5}{8} \]
\[ P(A \cap B) = \frac{1}{8} \]

### Final Answer:
\[ P(A \cap B) = \frac{1}{8} \]