If P(A) = `3/8`, P(B) = `1/3` and `P(AcapB)=1/4` then `P(barAcapbarB)` equals

Options

A. `5/12`
B. `7/12`
C. `13/24`
D. `17/24`

Answer

To find \( P(\bar{A} \cap \bar{B}) \), we can use the relationship between the union and intersection of events. Here are the steps:

### Step-by-Step Solution:

1. **Given Values**:
- \( P(A) = \frac{3}{8} \)
- \( P(B) = \frac{1}{3} \)
- \( P(A \cap B) = \frac{1}{4} \)

2. **Find \( P(A \cup B) \)**:
We use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the given values:
\[ P(A \cup B) = \frac{3}{8} + \frac{1}{3} - \frac{1}{4} \]

3. **Finding a Common Denominator**:
The least common multiple of the denominators (8, 3, and 4) is 24. We convert each fraction:
\[ P(A) = \frac{3}{8} = \frac{9}{24}, \quad P(B) = \frac{1}{3} = \frac{8}{24}, \quad P(A \cap B) = \frac{1}{4} = \frac{6}{24} \]

4. **Substituting into the Formula**:
Now substitute these values back into the equation:
\[ P(A \cup B) = \frac{9}{24} + \frac{8}{24} - \frac{6}{24} \]
Simplifying this gives:
\[ P(A \cup B) = \frac{9 + 8 - 6}{24} = \frac{11}{24} \]

5. **Finding \( P(\bar{A} \cap \bar{B}) \)**:
We know that:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]
Thus,
\[ P(\bar{A} \cap \bar{B}) = 1 - \frac{11}{24} = \frac{24 - 11}{24} = \frac{13}{24} \]

### Final Answer:
\[ P(\bar{A} \cap \bar{B}) = \frac{13}{24} \]