If A and B are events such that `P(A)=1/2, P(B)=1/3 and P(A cap B)=1/4`, then find
(a) P(A/B)
(b) P(B/A)

Answer

To solve the problem, we need to find the conditional probabilities \( P(A|B) \) and \( P(B|A) \) given the probabilities of events A and B, as well as their intersection.

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

### Step 1: Calculate \( P(A|B) \)

The formula for conditional probability is:

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

Substituting the known values:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{4}}{\frac{1}{3}} \]

### Step 2: Simplify the expression

To divide fractions, we multiply by the reciprocal:

\[ P(A|B) = \frac{1}{4} \times \frac{3}{1} = \frac{3}{4} \]

### Step 3: Calculate \( P(B|A) \)

Using the same formula for conditional probability:

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

Since \( P(A \cap B) = P(B \cap A) \), we can substitute:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{1}{4}}{\frac{1}{2}} \]

### Step 4: Simplify the expression

Again, we multiply by the reciprocal:

\[ P(B|A) = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2} \]

### Final Answers:

- \( P(A|B) = \frac{3}{4} \)
- \( P(B|A) = \frac{1}{2} \)

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