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 following steps:

### Step 1: Find \( P(A \cup B) \)
We know the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

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

First, we need to convert these fractions to a common denominator to perform the addition and subtraction. The least common multiple of 8, 3, and 4 is 24.

Converting each probability:
- \( P(A) = \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \)
- \( P(B) = \frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24} \)
- \( P(A \cap B) = \frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24} \)

Now substituting these values into the formula:
\[ P(A \cup B) = \frac{9}{24} + \frac{8}{24} - \frac{6}{24} = \frac{9 + 8 - 6}{24} = \frac{11}{24} \]

### Step 2: Find \( P(\bar{A} \cap \bar{B}) \)
We can use the relationship between the union and the complements:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]

Substituting the value we found for \( P(A \cup B) \):
\[ P(\bar{A} \cap \bar{B}) = 1 - \frac{11}{24} = \frac{24}{24} - \frac{11}{24} = \frac{13}{24} \]

### Final Answer
Thus, the probability \( P(\bar{A} \cap \bar{B}) \) is:
\[ \boxed{\frac{13}{24}} \]

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