Let A and B be two events. Then which of the following statements are TRUE?
(A) \(P(B|A) =\displaystyle\frac{P(A\cap B)}{P(A)}\), provided \(P(A) \ne 0\)
(B) \(P(B')=1+ P(B)\)
(C) \(P(A\cup B) = P(A) + P(B) + P(A\cap B)\)
(D) \(P(A\cap B) = P(A).P(B)\) If A and B are independent events
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (D) only
The basic rules of probability are the definition of conditional probability, the complement rule, the addition rule and the product rule for independent events:
\( \displaystyle P(B|A) = \frac{P(A\cap B)}{P(A)},\quad P(A) \ne 0 \)
\(P(B') = 1 - P(B)\)
\(P(A\cup B) = P(A) + P(B) - P(A\cap B)\)
\(P(A\cap B) = P(A)\,P(B)\) for independent A and B
So \(P(B') = 1 - P(B)\) and \(P(A\cup B) = P(A) + P(B) - P(A\cap B)\).
Checking the statements
(A) \(P(B|A) =\frac{P(A\cap B)}{P(A)}\), provided \(P(A) \ne 0\) — Correct. This is the definition of the conditional probability of B given A.
(B) \(P(B')=1+ P(B)\) — Incorrect. \(P(B') = 1 - P(B)\), not \(1 + P(B)\).
(C) \(P(A\cup B) = P(A) + P(B) + P(A\cap B)\) — Incorrect. \(P(A\cap B)\) is subtracted, not added.
(D) \(P(A\cap B) = P(A).P(B)\) If A and B are independent events — Correct. This is the definition of independent events.