Exercise 2.7 For any events A, B, and C with P(C) > 0, suppose that

Then, the events A and B are said to be conditionally independent given the event C. Suppose, in addition, that P(C) = 0.5, P(A|C) = P(B|C) = 0.9, P(A|Cc) = 0.2 and P(B|Cc) = 0.1. Prove that A and B are not independent, that is, P(A ∩ B) ̸= P(A)P(B). In general, the conditional independence does not imply independence and vice versa.

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P(ABC) P(A|C)P(B|C).