Let A and B be two events with P(A) = 0.2 and P(B) = 0.5. Consider the following statements:
(A) If A and B are mutually exclusive, then \(P(A\cup B) = 0.7\).
(B) If A and B are independent, then \(P(A\cup B) = 0.6\).
(C) If \(A\subset B\), then P(A|B) = 0.4.
(D) If A and B are independent, then P(A|B) = 0.5.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
In each case \(P(A\cap B)\) is found first; then the addition theorem or the definition of conditional probability is used.
Mutually exclusive: \(P(A\cap B) = 0\)
Independent: \(P(A\cap B) = 0.2\times 0.5 = 0.1\)
\(A\subset B\): \(P(A\cap B) = P(A) = 0.2\)
So \(P(A\cup B)\) is 0.7 for mutually exclusive events and \(0.7-0.1 = 0.6\) for independent events, P(A|B) is \(\frac{0.2}{0.5} = 0.4\) when \(A\subset B\), and P(A|B) = P(A) = 0.2 for independent events.
Checking the statements
(A) If A and B are mutually exclusive, then \(P(A\cup B) = 0.7\). — Correct. \(P(A\cup B) = 0.2+0.5-0 = 0.7\).
(B) If A and B are independent, then \(P(A\cup B) = 0.6\). — Correct. \(P(A\cup B) = 0.2+0.5-0.1 = 0.6\).
(C) If \(A\subset B\), then P(A|B) = 0.4. — Correct. \(P(A|B) = \frac{P(A)}{P(B)} = \frac{0.2}{0.5} = 0.4\).
(D) If A and B are independent, then P(A|B) = 0.5. — Incorrect. \(P(A|B) = \frac{0.1}{0.5} = 0.2\), which is P(A), not P(B).