Question

If \( P(A) = \frac{1}{2}, P(B) = \frac{3}{8} \) and \( P(A \cap B) = \frac{1}{5} \), then \( P(A|B) \) is equal to:

• A. \( \frac{2}{5} \)
• B. \( \frac{8}{15} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{5}{8} \)

Hint:

The probability of the "given" event (the one after the vertical bar) always goes in the denominator.

Answer 1

The Correct Option is B

Step 1: Key Formula or Approach:
The formula for conditional probability \( P(A|B) \) (probability of \( A \) occurring given that \( B \) has already occurred) is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Step 2: Detailed Explanation:
Given:
\( P(A \cap B) = \frac{1}{5} \)
\( P(B) = \frac{3}{8} \)
Applying the values to the formula:
\[ P(A|B) = \frac{1/5}{3/8} \] To divide by a fraction, multiply by its reciprocal:
\[ P(A|B) = \frac{1}{5} \times \frac{8}{3} = \frac{8}{15} \] Step 3: Final Answer:
The value of \( P(A|B) \) is \( \frac{8}{15} \).