Discussion
Suppose that probabilities are assigned to the subsets of \(\Omega = \{a, b, c, d, e, f\}\) according to the classical definition of probability. Which of the following statements is/are true for the sets A = {a, b}, B = {b, d} and C = {a, c, d} ?
(I) C and A are independent
(II) C and B are independent
(III) C and \(A \cup B\) are independent
Answer: A
Only (I) and (II) are true.
With six equally likely outcomes, \(P(A) = \frac{2}{6}\), \(P(B) = \frac{2}{6}\) and \(P(C) = \frac{3}{6}\).
(I) \(A \cap C = \{a\}\), so \(P(A \cap C) = \frac{1}{6} = \frac{1}{3} \cdot \frac{1}{2}\): independent.
(II) \(B \cap C = \{d\}\), so \(P(B \cap C) = \frac{1}{6} = \frac{1}{3} \cdot \frac{1}{2}\): independent.
(III) \(A \cup B = \{a, b, d\}\) with probability \(\frac{1}{2}\), and \(C \cap (A \cup B) = \{a, d\}\) with probability \(\frac{1}{3}\). Since \(\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \neq \frac{1}{3}\), these are not independent.
So (I) and (II) hold and (III) fails.
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