Let A and B be the two events such that ` P(A ) = 1/2, P(B ) = 1/3 and P(A nn B ) = 1/4` , find
`P(A uu B )`

Answer

To find the probability of the union of two events A and B, we can use the formula:

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

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{3} \)
- \( P(A \cap B) = \frac{1}{4} \)

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

3. **Find a Common Denominator:**
The least common multiple of the denominators (2, 3, and 4) is 12.

4. **Convert Each Probability to Have the Common Denominator:**
- Convert \( P(A) = \frac{1}{2} \) to twelfths:
\[ P(A) = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} \]
- Convert \( P(B) = \frac{1}{3} \) to twelfths:
\[ P(B) = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]
- Convert \( P(A \cap B) = \frac{1}{4} \) to twelfths:
\[ P(A \cap B) = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \]

5. **Substitute Back into the Formula:**
\[ P(A \cup B) = \frac{6}{12} + \frac{4}{12} - \frac{3}{12} \]

6. **Combine the Fractions:**
\[ P(A \cup B) = \frac{6 + 4 - 3}{12} = \frac{7}{12} \]

### Final Answer:
\[ P(A \cup B) = \frac{7}{12} \]