Exercise 2.14 (2.14)

Question

Consider the following scenario, from Tversky and Kahneman [27]:
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|B
c
)? Explain your intuition.
(b) Intuitively, which do you think is bigger, P(B|A) or P(B|A
c
)? Explain your intuition.
(c) Show that for any events A and B (with probabilities not equal to 0 or 1), the
inequality P(A|B) > P(A|B
c
) is equivalent to P(B|A) > P(B|A
c
).
(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|B
c
) and P(B|A) < 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?

fifthist x · Accepted Answer

\begin{enumerate}[label=(\alph*)]

\item Intuitively, $P(A|B) > P(A|B^{c})$, since Peter will be in a rush to 
install his alarm if he knows that his house will be burglarized before the 
end of next year.

\item Intuitively $P(B|A^{c}) > P(B|A)$, since Peter is more likely to be 
robbed if he doesn't have an alarm by the end of the year.

\item  See \url{https://math.stackexchange.com/a/3761508/649082}.

\item An explanation might be that in part $a$, we assume Peter to be driven 
to not let burglars rob him, but in part $b$ we assume the burglars to not 
necessarily be as driven, since we assume that if the burglers know that 
Peter will install an alaram before the end of the next year 
they might not rob him. If the burglers are driven, they might actually be 
more inclined to rob Peter sooner, before he actually installs the alarm.
\end{enumerate}

Exercise 2.14 (2.14)

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Exercise 2.14 (2.14)

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