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.

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 [LW(p) · GW(p)] to this example. $$\frac{P\left(x|D\right)}{P\left(y|D\right)}=\frac{P\left(x\right)}{P\left(y\right)}\times \frac{P\left(D|x\right)}{P\left(D|y\right)}$$ Let $x$ = "Xavier won the lottery", $y=\neg x$, $D$ = "Yovanni says Xavier won the lottery". We have $P\left(x\right)/P\left(y\right)\approx 1/1000000$ (for simplicity, let's assume that Xavier couldn't be someone who didn't even enter the lottery), $P\left(D|x\right)\approx 1$. What is $P\left(D|y\right)$? 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 within the 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 $P\left(D|y\right)\approx 1/(1000000\cdot 100)$. Putting this in the formula, we get $P\left(x|D\right)/P\left(y|D\right)\approx 100$, so Xavier is probably the winner after all.

(This is the same as Phil's answer [LW(p) · GW(p)], 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.

1. 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.
2. 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).

-

Yovanni tells Zenia, "The winner of the lottery was Xavier Williams of California."

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.

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