If A and B are two events such that P(B) = 0.8 and P(A|B) = 0.35, then \(P(A'\cap B)\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0.52
Given that B has happened, either A happens or it does not, so
\(P(A'|B) = 1-P(A|B)\)
\(P(A'|B) = 1-0.35 = 0.65\)
By the multiplication theorem:
\(P(A'\cap B) = P(B)\,P(A'|B)\)
\(P(A'\cap B) = 0.8\times 0.65 = 0.52\)
Why the other options are incorrect
Option (2) 0.28 — Incorrect. This is \(P(A\cap B) = 0.8\times 0.35\), the part where A does happen.
Option (3) 0.65 — Incorrect. This is \(P(A'|B)\); it still has to be multiplied by P(B).
Option (4) 0.45 — Incorrect. This is 0.8 − 0.35, which subtracts a conditional probability from P(B) directly.