In deterministic logic, the statement “A implies B” is equivalent to its contrapositive, “not B implies not A”. In this problem we will consider analogous statements in probability, the logic of...

In deterministic logic, the statement “A implies B” is equivalent to its contrapositive, “not B implies not A”. In this problem we will consider analogous statements in probability, the logic of uncertainty. Let A and B be events with probabilities not equal to 0 or 1. (a) Show that if P(B|A) = 1, then P(Ac|Bc) = 1. Hint: Apply Bayes’ rule and LOTP. (b) Show however that the result in (a) does not hold in general if = is replaced by ˜. In particular, ?nd an example where P(B|A) is very close to 1 but P(Ac|Bc) is very close to 0.
Hint: What happens if A and B are independent?