Let `P(A) = (7)/(13) , P(B) = (9)/(13) and P (A cap B) = (4)/(13)` . Then `P (A | B) ` is equal to

Options

A. `(6)/(13)`
B. `(4)/(13)`
C. `(4)/(9)`
D. `(5)/(9)`

Answer

To find \( P(A | B) \), we can use the formula for conditional probability:

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

Given:
- \( P(A) = \frac{7}{13} \)
- \( P(B) = \frac{9}{13} \)
- \( P(A \cap B) = \frac{4}{13} \)

Now, we can substitute the values into the formula.

### Step 1: Substitute the values into the formula
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{4}{13}}{\frac{9}{13}} \]

### Step 2: Simplify the fraction
When dividing fractions, we multiply by the reciprocal of the denominator:
\[ P(A | B) = \frac{4}{13} \times \frac{13}{9} \]

### Step 3: Cancel out the common terms
The \( 13 \) in the numerator and denominator cancels out:
\[ P(A | B) = \frac{4}{9} \]

Thus, the final answer is:
\[ P(A | B) = \frac{4}{9} \]