Question

If \( P(A') = 0.6 \), \( P(B) = 0.8 \) and \( P(B|A) = 0.3 \), then find \( P(A|B) \).

• A. \( \dfrac{7}{20} \)
• B. \( \dfrac{3}{20} \)
• C. \( \dfrac{3}{4} \)
• D. \( \dfrac{9}{20} \)

Hint:

Always compute \( P(A \cap B) \) first using conditional probability before applying Bayes’ theorem.

Answer 1

The Correct Option is B

Step 1: Find \( P(A) \).
\[ P(A') = 0.6 \Rightarrow P(A) = 1 - 0.6 = 0.4 \]

Step 2: Use the conditional probability formula.
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] \[ 0.3 = \frac{P(A \cap B)}{0.4} \Rightarrow P(A \cap B) = 0.12 \]

Step 3: Apply Bayes’ theorem.
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.8} = 0.15 \]

Step 4: Convert to fraction.
\[ 0.15 = \frac{3}{20} \]

Step 5: Conclusion.
\[ \boxed{\frac{3}{20}} \]