If A and B are two independent events and `P(A)=(1)/(4),P(B)=(1)/(3)`, then :

Options

A. `P(A uu B)=(1)/(5)`
B. `P(A uu B)=1`
C. `P(A uu B)=(1)/(2)`
D. `P(A uu B)=(1)/(3)`

Answer

To solve the problem, we need to find the probability of the union of two independent events A and B, given their individual probabilities.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Probability of event A, \( P(A) = \frac{1}{4} \)
- Probability of event B, \( P(B) = \frac{1}{3} \)

2. **Use the Formula for the Intersection of Independent Events:**
- Since A and B are independent events, the probability of both A and B occurring (the intersection) is given by:
\[ P(A \cap B) = P(A) \times P(B) \]
- Substitute the values:
\[ P(A \cap B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \]

3. **Use the Formula for the Union of Two Events:**
- The probability of the union of two events A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substitute the known values:
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{3} - \frac{1}{12} \]

4. **Finding a Common Denominator:**
- The least common multiple (LCM) of 4, 3, and 12 is 12. Convert each fraction:
\[ P(A) = \frac{1}{4} = \frac{3}{12} \]
\[ P(B) = \frac{1}{3} = \frac{4}{12} \]
\[ P(A \cap B) = \frac{1}{12} \]

5. **Substituting the Converted Values:**
- Now substitute these values back into the union formula:
\[ P(A \cup B) = \frac{3}{12} + \frac{4}{12} - \frac{1}{12} \]

6. **Perform the Addition and Subtraction:**
- Combine the fractions:
\[ P(A \cup B) = \frac{3 + 4 - 1}{12} = \frac{6}{12} \]

7. **Simplify the Result:**
- Simplify \( \frac{6}{12} \):
\[ P(A \cup B) = \frac{1}{2} \]

### Final Answer:
The probability of the union of events A and B is:
\[ P(A \cup B) = \frac{1}{2} \]