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 ((barA )/( barB ))`

Answer

To solve the problem, we need to find \( P(\bar{A} | \bar{B}) \), which is the probability of the complement of event A given the complement of event B. We can use the formula for conditional probability:

\[ P(\bar{A} | \bar{B}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})} \]

### Step 1: Find \( P(A \cup B) \)

We start by calculating \( P(A \cup B) \) using the formula:

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

Given:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{3} \)
- \( P(A \cap B) = \frac{1}{4} \)

Substituting the values:

\[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \]

To perform the addition, we need a common denominator. The least common multiple of 2, 3, and 4 is 12.

Converting each term:
- \( P(A) = \frac{1}{2} = \frac{6}{12} \)
- \( P(B) = \frac{1}{3} = \frac{4}{12} \)
- \( P(A \cap B) = \frac{1}{4} = \frac{3}{12} \)

Now substituting these values:

\[ P(A \cup B) = \frac{6}{12} + \frac{4}{12} - \frac{3}{12} = \frac{6 + 4 - 3}{12} = \frac{7}{12} \]

### Step 2: Find \( P(\bar{B}) \)

Next, we find \( P(\bar{B}) \):

\[ P(\bar{B}) = 1 - P(B) \]

Substituting \( P(B) \):

\[ P(\bar{B}) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 3: Find \( P(\bar{A} \cap \bar{B}) \)

Using the complement rule, we can find \( P(\bar{A} \cap \bar{B}) \):

\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]

Substituting \( P(A \cup B) \):

\[ P(\bar{A} \cap \bar{B}) = 1 - \frac{7}{12} = \frac{5}{12} \]

### Step 4: Calculate \( P(\bar{A} | \bar{B}) \)

Now we can substitute into the conditional probability formula:

\[ P(\bar{A} | \bar{B}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})} \]

Substituting the values we found:

\[ P(\bar{A} | \bar{B}) = \frac{\frac{5}{12}}{\frac{2}{3}} \]

To divide fractions, we multiply by the reciprocal:

\[ P(\bar{A} | \bar{B}) = \frac{5}{12} \times \frac{3}{2} = \frac{5 \times 3}{12 \times 2} = \frac{15}{24} \]

Simplifying \( \frac{15}{24} \):

\[ P(\bar{A} | \bar{B}) = \frac{5}{8} \]

### Final Answer

Thus, the final answer is:

\[ P(\bar{A} | \bar{B}) = \frac{5}{8} \]