If A and B are two events such that P(A) = 0.5, \(P(A\cup B) = 0.8\) and P(A|B) = 0.4, then P(B) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.5
Let P(B) = x. By the definition of conditional probability:
\(P(A\cap B) = P(B)\,P(A|B) = 0.4x\)
By the addition theorem:
\(P(A\cup B) = P(A)+P(B)-P(A\cap B)\)
\(0.8 = 0.5+x-0.4x\)
\(0.6x = 0.3\)
\(x = 0.5\)
Check: \(P(A\cap B) = 0.2\) and \(0.5+0.5-0.2 = 0.8\).
Why the other options are incorrect
Option (1) 0.3 — Incorrect. This is 0.8 − 0.5, which treats A and B as mutually exclusive.
Option (3) 0.75 — Incorrect. This is \(\frac{0.3}{0.4}\), which divides by P(A|B) instead of by 1 − P(A|B) = 0.6.
Option (4) 0.2 — Incorrect. This is \(P(A\cap B)\), not P(B).