If A and B are independent events such that P(A) = 0.6 and \(P(A\cap B) = 0.18\), then \(P(A'\cap B')\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0.28
For independent events, \(P(A\cap B) = P(A)\,P(B)\):
\(0.18 = 0.6\times P(B)\)
\(P(B) = 0.3\)
The complements of independent events are also independent, so
\(P(A'\cap B') = P(A')\,P(B')\)
\(P(A'\cap B') = 0.4\times 0.7 = 0.28\)
Why the other options are incorrect
Option (2) 0.42 — Incorrect. This is \(0.6\times 0.7 = P(A\cap B')\), the probability that A occurs and B does not.
Option (3) 0.12 — Incorrect. This is \(0.4\times 0.3 = P(A'\cap B)\), the probability that B occurs and A does not.
Option (4) 0.72 — Incorrect. This is \(P(A\cup B) = 1-0.28\), the probability that at least one occurs.