If A and B be two events such that `P(A)= (1)/(2)P(B) = (1)/(3) and P((A)/(B))=(1)/(4)` then`P(AnnB)` equals

Options

A. `(1)/(12)`
B. `(3)/(4)`
C. `(1)/(4)`
D. `(3)/(16)`

Answer

To solve the problem, we need to find \( P(A \cap B) \) given the probabilities \( P(A) = \frac{1}{2} \), \( P(B) = \frac{1}{3} \), and \( P(A|B) = \frac{1}{4} \).

### Step-by-Step Solution:

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

2. **Use the Formula for Conditional Probability:**
The formula for conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Rearranging this formula gives us:
\[ P(A \cap B) = P(A|B) \times P(B) \]

3. **Substitute the Known Values:**
Now, substitute the values of \( P(A|B) \) and \( P(B) \) into the rearranged formula:
\[ P(A \cap B) = \left(\frac{1}{4}\right) \times \left(\frac{1}{3}\right) \]

4. **Calculate \( P(A \cap B) \):**
Perform the multiplication:
\[ P(A \cap B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} \]

5. **Final Result:**
Thus, the value of \( P(A \cap B) \) is:
\[ P(A \cap B) = \frac{1}{12} \]

### Conclusion:
The required value of \( P(A \cap B) \) is \( \frac{1}{12} \).