If A and B are two events and `P(A) = (3)/(8)` . `P(B) =(1)/(2), P(A nn B) = (1)/(4)` then `P(A uu B)` =

Options

A. `(3)/(9)`
B. `(3)/(4)`
C. `(1)/(4)`
D. `(5)/(8)`

Answer

To solve the problem, we will 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. **Identify the Given Values**:
- \( P(A) = \frac{3}{8} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cap B) = \frac{1}{4} \)

2. **Substitute the Values into the Formula**:
- Using the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substitute the values:
\[ P(A \cup B) = \frac{3}{8} + \frac{1}{2} - \frac{1}{4} \]

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

4. **Rewrite the Equation**:
\[ P(A \cup B) = \frac{3}{8} + \frac{4}{8} - \frac{2}{8} \]

5. **Combine the Fractions**:
- Now, combine the fractions:
\[ P(A \cup B) = \frac{3 + 4 - 2}{8} = \frac{5}{8} \]

6. **Final Result**:
- Therefore, the probability of the union of events A and B is:
\[ P(A \cup B) = \frac{5}{8} \]

### Answer:
The probability \( P(A \cup B) = \frac{5}{8} \).

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