If A and B are two events such that \(P(A) = \displaystyle\frac{1}{2}, P(B) = \frac{1}{3}\) and \(P(A \cap B) = \displaystyle\frac{1}{4}\), then which of the following statements are true?
(A) A and B are independent events
(B) \(P(A \mid B) = \displaystyle\frac{3}{4}\)
(C) \(P(A' \mid B') = \displaystyle\frac{5}{8}\)
(D) \(P(A' \mid B) = \displaystyle\frac{1}{4}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B), (C) and (D) only
Each conditional probability comes from the conditional probability formula \(P(E \mid F) = \frac{P(E \cap F)}{P(F)}\).
\(\displaystyle P(A)P(B) = \frac{1}{2}\cdot\frac{1}{3} = \frac{1}{6}\)
\(\displaystyle P(A \mid B) = \frac{1/4}{1/3} = \frac{3}{4}\)
The complements need the union first:
\(\displaystyle P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{4} = \frac{7}{12}\)
\(\displaystyle P(A' \cap B') = 1 - P(A \cup B) = \frac{5}{12}\)
\(\displaystyle P(B') = 1 - \frac{1}{3} = \frac{2}{3}\)
\(\displaystyle P(A' \mid B') = \frac{5/12}{2/3} = \frac{5}{8}\)
\(\displaystyle P(A' \cap B) = P(B) - P(A \cap B) = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}\)
\(\displaystyle P(A' \mid B) = \frac{1/12}{1/3} = \frac{1}{4}\)
So \(P(A)P(B) = \frac{1}{6}\), \(P(A \mid B) = \frac{3}{4}\), \(P(A' \mid B') = \frac{5}{8}\) and \(P(A' \mid B) = \frac{1}{4}\).
Checking the statements
(A) A and B are independent events — Incorrect. \(P(A \cap B) = \frac{1}{4}\), but \(P(A)P(B) = \frac{1}{6}\).
(B) \(P(A \mid B) = \frac{3}{4}\) — Correct. \(\frac{1/4}{1/3} = \frac{3}{4}\).
(C) \(P(A' \mid B') = \frac{5}{8}\) — Correct. \(\frac{5/12}{2/3} = \frac{5}{8}\).
(D) \(P(A' \mid B) = \frac{1}{4}\) — Correct. \(\frac{1/12}{1/3} = \frac{1}{4}\).