There are two events X and Y whose probability you’d like to estimate. So you ask a hundred trusted, reasonable people what they think. Half of them say that the probability of X and the probability of Y are both 90%, and the probability of both X and Y occurring is 81%. The other half say that P(X) = 10%, P(Y) = 10%, and P(X and Y) = 1%.
What is your best estimate of P(X), P(Y), and P(X and Y)?
If you said “50%, 50%, 41%,” does it bother you that you deem these events not to be independent, even though every single person you polled said the opposite? If not, what did you say?
(The subtext of this post is: is the “Independence of Irrelevant Alternatives” axiom in Arrow’s theorem a good idea? Feel free to discuss that too.)