If `P(A) =(1)/(4),P(B) = (1)/(2),P(A uu B) = (5)/(8)` , then `P(A nn B)` is equal to .

Options

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

Answer

To find \( P(A \cap B) \) given the probabilities \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{2} \), 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 a common denominator:**
- The common denominator for \( \frac{1}{4} \) and \( \frac{1}{2} \) is 4.
- Convert \( \frac{1}{2} \) to \( \frac{2}{4} \).
- Now the equation looks like this:
\[ \frac{5}{8} = \frac{1}{4} + \frac{2}{4} - P(A \cap B) \]

4. **Combine the fractions on the right side:**
\[ \frac{5}{8} = \frac{3}{4} - P(A \cap B) \]

5. **Convert \( \frac{3}{4} \) to have a denominator of 8:**
- \( \frac{3}{4} = \frac{6}{8} \).
- Now the equation is:
\[ \frac{5}{8} = \frac{6}{8} - P(A \cap B) \]

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

7. **Perform the subtraction:**
\[ P(A \cap B) = \frac{1}{8} \]

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