Which of the following statements are correct?
(A) If E and F are independent events then \(P(E\cap F) = P(E) P(F)\)
(B) If E and F are mutually exclusive events, then \(P(E\cup F) = P(E) + P(F) - P(E).P(F)\)
(C) The conditional probability of an event E, given the occurrence of the event F is given by \(\displaystyle\frac{P(E\cap F)}{P(F)},P(F)\ne 0\)
(D) If E and F be the events associated with the sample space S of an experiment, then \(P (\bar{E}|F) = 2 - P(E|F)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
(A) If E and F are independent events then \(P(E\cap F) = P(E) P(F)\) — Correct. This is the definition of independent events.
(B) If E and F are mutually exclusive events, then \(P(E\cup F) = P(E) + P(F) - P(E).P(F)\) — Incorrect. For mutually exclusive events \(E \cap F = \phi\), so \(P(E \cap F) = 0\) and the addition rule gives
\(P(E\cup F) = P(E) + P(F) - P(E \cap F) = P(E) + P(F)\)
The printed formula, with \(P(E)P(F)\) subtracted, holds for independent events, not for mutually exclusive ones.
(C) The conditional probability of an event E, given the occurrence of the event F is given by \(\displaystyle\frac{P(E\cap F)}{P(F)},P(F)\ne 0\) — Correct. This is the definition of conditional probability.
(D) If E and F be the events associated with the sample space S of an experiment, then \(P (\bar{E}|F) = 2 - P(E|F)\) — Incorrect. Given F, the events E and \(\bar{E}\) are complementary, so
\(P(\bar{E}|F) = 1 - P(E|F)\)