The Lottery Paradox

This is an interesting mathematical puzzle involving probability, and how it’s possible for extremely low priors to come true.

Zenia and Yovanni are discussing the lottery. This lottery consists of 1,000,000 people, each of which has one ticket and therefore a 1 in a million chance of winning. Yovanni tells Zenia, “The winner of the lottery was Xavier Williams of California.” Zenia replies, “I don’t believe you. Xavier is one out of a million people, and my prior for him winning is very low. But while you’re generally an honest person, it’s still more likely that you would lie to me than that this one random person would happen to win. If I assume that there’s only about a one in a hundred chance you would lie to me, the chance that Xavier is the winner is still incredibly low, and I shouldn’t believe you.”

This appears at first to be logical reasoning- after all, in Bayesian terms, if you have a 1 in a million starting odds, combined with evidence with a p=0.01 false positive rate, the event still probably didn’t happen. But wait: if this is so unlikely, then that suggests that no matter what Yovanni tells you, you won’t believe him. In fact, it suggests that you should never believe anyone in the world has won the lottery! This is obviously absurd, because somebody always does end up winning.

To think about it another way, if your prior probability for Yovanni lying is 1100, but you are going to update that to near-certainty no matter what evidence you receive, this is highly illogical due to Conservation of Expected Evidence. So clearly one of Zenia’s assumptions, or a step in her reasoning, is false.

So, what’s the resolution to this paradox? I have managed to figure out the answer, but I’m going to wait a few days before I post the solution to see who else can figure it out in the comments.

Edit: Phil’s answer in the comments is exactly correct.