# 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 ^{1}⁄_{100}, 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.

The logic would be correct if, when Yovanni lied, he would always say it was Xavier Williams. In that case, there would be (roughly)

^{1}⁄_{100}“Yovanni lies and says it was Xavier” for every^{1}⁄_{1,000,000}“Yovanni tells the truth and says it was Xavier.”But if Yovanni lies randomly, and you have no prior that he would lie and say Xavier any more than he would lie and say anyone else, you have

^{1}⁄_{100}*^{1}⁄_{1,000,000}“Yovanni lies and also Yovanni says it was Xavier” for every^{99}⁄_{100}*^{1}⁄_{1,000,000}“Yovanni tells the truth and says it was Xavier,” which is 99% truth.Right, which is why the claim is immediately more suspect if Xavier is a close friend/relative/etc.

Crazy story: one time I went to random.org and generated a 20 digit string and it was 89921983981118509034.

Let’s apply Bayes formula in odds form to this example.P(x|D)P(y|D)=P(x)P(y)×P(D|x)P(D|y) Let x = “Xavier won the lottery”, y=¬x, D = “Yovanni says Xavier won the lottery”. We have P(x)/P(y)≈1/1000000 (for simplicity, let’s assume that Xavier couldn’t be someone who didn’t even enter the lottery), P(D|x)≈1. What is P(D|y)? Given that someone other than Xavier won the lottery, what is the probability that Yovanni would claim that it was Xavier in particular who did? While Yovanni might have a reason to single out Xavier, Zenia doesn’t, so from the hypothesis y they would predict that anyone could be wrongly named the winner, which gives picking Xavier one in a million chance (also, the Yovanni

withinthe hypothesis y won’t pay attention to y in particular, so can’t anchor to Xavier based on that). Then, Yovanni must make a mistake or decide to lie, with probability of, say, 1%. In total, we have P(D|y)≈1/(1000000⋅100). Putting this in the formula, we get P(x|D)/P(y|D)≈100, so Xavier is probably the winner after all.(This is the same as Phil’s answer, formulated a little bit differently.)

Phil is right, and also “don’t believe him” is a mapping of “the thing he said is x% likely” onto a single believe/don’t with a cutoff at t%. So it can be true that both A) you update to believe it is more than

^{1}⁄_{1,000,000}that Xavier won, but also B) your level still falls below t%, so you don’t believe it.As I’ve noted before, this is actually how real people behave. They know the odds of winning are super low, so when they get a bit of evidence that says “I won” they seek out independent bits of evidence (eg, asking a second person to look at the ticket and confirm the numbers; checking their bank account after putting in the claim to confirm the balance was updated) before acting on the reality of winning.

The description doesn’t fully specify what’s happening.

Yovanni is answering questions in form of “Did X win the lottery”. And gives correct answer 99% of the time. In that case you shouldn’t believe that Xavier won the lottery. If you asked the question for all the participants you’d end up with list of (in expectation) about 10′000 people for which Yovanni claimed they won the lottery.

Yovanni is making 1 claim about who won the lottery. And for questions like that Yovanni gives correct answers 99% of the time. In that case Phil got it and probability that Xavier won is 99%.

Also I think it’s better to avoid using humans in examples like that and try to use something else / not agenty. Because humans can strategically lie (for example somebody can reach very high accuracy in statements they make by talking a lot about simple arithmetic operations. If they later say you should give them money and will receive 10x as much in return then you shouldn’t conclude that there is 99+% chance this will work out and you should give them a lot of money).

People don’t always win the lottery, a jackpot consists of prize money rolled over. Furthermore, reported news and actual reality don’t have to relate in any way.

When evaluating the veracity of this statement, Zenia is not evaluating the odds of someone with that name winning a lottery. Someone won the lottery and their name has been made publicly available. What she is evaluating is Yovanni’s general trustworthiness in this kind of situations. Does Yovanni like to make stuff up? To play pranks? Has reading comprehension issues? The odds of the lottery have zero to do with this. They would only matter if Yovanni could have some useful additional information, such as if it was a raffle at a party, and you knew that no one named Xavier Williams was invited.

Scenario 1: A friend who you know to generally be fairly trustworthy tells you that there is a snowstorm in New York City.

Scenario 2: A friend who you know to generally be fairly trustworthy tells you that there is a snowstorm in Ecuador.

Scenario 3: A friend who you know to generally be fairly trustworthy tells you that there is a snowstorm on the surface of Venus, which they spotted by looking up into the sky with their keen vision after Batman showed up and used his power ring to grant them super senses.

As I understand your comment above, it sounds like you’re saying you will evaluate the veracity of the three scenarios above the same way, caring only about the friend’s trustworthiness and not at all about how implausible the story sounds. This seems very strange—am I misunderstanding?

The ‘paradox’ being mentioned in the post is that Xavier Williams winning the lottery seems like it should be a plausible-sounding story (a la Scenario A), but a naive mathematical analysis (missing Phil’s point above) makes it seem like a very implausible-sounding one.

No, these are completely different. NYC and Equador are not random samples of places where a snowstorm is equally likely. A more charitable comparison would be “It’s winter in the Northern Hemisphere, and there was a snowstorm in one of the major cities. Yovanni checks the weather app and says that it’s in NYC.”

If you’re one in a million, there are 9 of you in Manhattan, and 1400 of you in China.

Also 200 in Nigeria. One of them won the lottery but is too poor to pay the related tax, so he is willing to share the winnings with you 50:50 if you help him pay the tax first.