Question

If \(P(A)=0.8\), \(P(B)=0.5\) and \(P(B|A)=0.4\), then the value of \(P(A|B)\) is:

• A. \(0.32\)
• B. \(0.64\)
• C. \(0.16\)
• D. \(0.25\)

Hint:

A useful relation is \[ P(A|B) = \frac{P(A)\,P(B|A)}{P(B)}. \] This is the basic form used in Bayes' theorem.

Answer 1

The Correct Option is B

Concept: Conditional probability is defined by \[ P(B|A) = \frac{P(A\cap B)}{P(A)}. \] Hence, \[ P(A\cap B) = P(A)\times P(B|A). \] Also, \[ P(A|B) = \frac{P(A\cap B)}{P(B)}. \]

Step 1: Find \(P(A\cap B)\). Given, \[ P(A)=0.8, \qquad P(B|A)=0.4. \] Therefore, \[ P(A\cap B) = 0.8\times0.4 = 0.32. \]

Step 2: Calculate \(P(A|B)\). Using \[ P(A|B) = \frac{P(A\cap B)}{P(B)}, \] we get \[ P(A|B) = \frac{0.32}{0.5} = 0.64. \] Conclusion: \[ \boxed{0.64} \] Hence, the correct answer is Option (B).