If A and B are two events such that P(A|B) = 0.6 and \(P(A'\cap B) = 0.16\), then P(B) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 0.40
Given B, either A happens or A does not happen, so the two conditional probabilities add up to 1:
\(P(A'|B) = 1-P(A|B)\)
\(P(A'|B) = 1-0.6 = 0.4\)
By the multiplication theorem, \(P(A'\cap B) = P(B)\,P(A'|B)\). So
\(0.16 = P(B)\times 0.4\)
\( \displaystyle P(B) = \frac{0.16}{0.4} = 0.40 \)
Why the other options are incorrect
Option (1) 0.24 — Incorrect. This is \(0.4\times 0.6 = P(A\cap B)\), the part of B where A also happens.
Option (2) 0.84 — Incorrect. This is \(1-0.16\), the probability that \(A'\cap B\) does not happen.
Option (4) 0.76 — Incorrect. This is \(0.6+0.16\), which adds a conditional probability to an ordinary one.