A and B are independent events such that P(A) = 0.5 and \(P(A\cap B) = 0.15\). Then \(P(B|A')\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.3
For independent events, \(P(A\cap B) = P(A)P(B)\). So
\(0.15 = 0.5\times P(B)\)
\(P(B) = 0.3\)
When A and B are independent, A′ and B are also independent, so knowing that A has not happened does not change the chance of B:
\(P(B|A') = P(B) = 0.3\)
Check from the definition: \(P(A'\cap B) = P(B)-P(A\cap B) = 0.3-0.15 = 0.15\) and \(P(A') = 0.5\), so
\( \displaystyle P(B|A') = \frac{0.15}{0.5} = 0.3 \)
Why the other options are incorrect
Option (1) 0.15 — Incorrect. This is \(P(A'\cap B)\), not yet divided by P(A′).
Option (3) 0.7 — Incorrect. This is P(B′), the chance that B does not happen.
Option (4) 0.5 — Incorrect. This is P(A′), the probability of the given event, not of B.