If A and B are two events such that `P(A) = 3/8 , P(B) = 5/8 and P(A cup B) = 3/4` then P(A/B) is equal to

Options

A. `6/25`
B. `3/20`
C. `3/8`
D. `2/5`

Answer

To find \( P(A | B) \), we will use the formula:

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

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

We know from the problem statement:

- \( P(A) = \frac{3}{8} \)
- \( P(B) = \frac{5}{8} \)
- \( P(A \cup B) = \frac{3}{4} \)

Using the formula for the union of two events:

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

We can rearrange this to find \( P(A \cap B) \):

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

Substituting the known values:

\[ P(A \cap B) = \frac{3}{8} + \frac{5}{8} - \frac{3}{4} \]

### Step 2: Calculate \( P(A \cap B) \)

First, convert \( \frac{3}{4} \) to have a common denominator with \( \frac{3}{8} \) and \( \frac{5}{8} \):

\[ \frac{3}{4} = \frac{6}{8} \]

Now substituting back:

\[ P(A \cap B) = \frac{3}{8} + \frac{5}{8} - \frac{6}{8} \]

Combine the fractions:

\[ P(A \cap B) = \frac{3 + 5 - 6}{8} = \frac{2}{8} = \frac{1}{4} \]

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

Now that we have \( P(A \cap B) \), we can substitute it back into the formula for conditional probability:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{4}}{\frac{5}{8}} \]

### Step 4: Simplify \( P(A | B) \)

To simplify this, multiply by the reciprocal of \( P(B) \):

\[ P(A | B) = \frac{1}{4} \times \frac{8}{5} = \frac{8}{20} = \frac{2}{5} \]

### Final Answer

Thus, the probability \( P(A | B) \) is:

\[ \boxed{\frac{2}{5}} \]