If A and B are two events such that P(A) = 0.25, P(B) = 0.4 and P(A|B) = 0.5, then P(B|A) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0.8
By the multiplication theorem, the probability of both events is
\(P(A\cap B) = P(B)\,P(A|B)\)
\(P(A\cap B) = 0.4\times 0.5 = 0.2\)
Now the given event is A, so P(A) goes in the denominator:
\( \displaystyle P(B|A) = \frac{P(A\cap B)}{P(A)} \)
\( \displaystyle = \frac{0.2}{0.25} = 0.8 \)
Why the other options are incorrect
Option (2) 0.2 — Incorrect. This is \(P(A\cap B)\), not yet divided by P(A).
Option (3) 0.625 — Incorrect. This is \(\frac{0.25}{0.4} = \frac{P(A)}{P(B)}\); the probability of both events must be used.
Option (4) 0.5 — Incorrect. This is P(A|B); in general P(B|A) is not equal to P(A|B).