Question: Consider the following scenario, from Tversky and Kahneman [30]: Let A be the event that before the end of next year, Peter will have installed a burglar alarm system in his home. Let B denote the event that Peter's home will be burglarized before the end of next year.
(a) Intuitively, which do you think is bigger, P(A|B) or P(A|Bc)? Explain your intuition.
(b) Intuitively, which do you think is bigger, P(B|A) or P(B|Ac)? Explain your intuition.
(c) Show that for any events A and B (with probabilities not equal to 0 or 1), P(A|B) > P(A|Bc) is equivalent to P(B|A) > P(B|Ac).
(d) Tversky and Kahneman report that 131 out of 162 people whom they posed (a) and (b) to said that P(A|B) > P(A|Bc) and P(B|A) c). What is a plausible explanation for why this was such a popular opinion despite (c) showing that it is impossible for these inequalities both to hold?