Bob is taking a test. There are two questions he is stumped on and he decides to guess. Let A be the event that he gets the first question right; let B be the event he gets the second question right (adapted from Blom et al., 1991).
(a) Obtain an expression for p₁, the probability that he gets both questions right conditional on getting the first question right.
(b) Obtain an expression for p₂, the probability that he gets both questions right conditional on getting either of the two questions right (A or B).
(c) Show that p₂ ≤ p₁. This may seem paradoxical. Knowledge that A or B has taken place makes the conditional probability that A and B happens smaller than when we know that A has happened. Can you untangle the paradox?