This highlights an interesting case where pure Bayesian reasoning fails. While the chance of it occurring randomly is very low (but may rise when you consider how many chances it has to occur), it is trivial to construct. Furthermore, it potentially applies in any case where we have two possibilities, one of which continually becomes more probable while the other shrinks, but persistently doesn't become disappear.

Suppose you are a police detective investigating a murder. There are two suspects: A and B. A doesn't have an alibi, while B has a strong one (time stamped receipts from a shop on the other side of town). A belonging of A's was found at the crime scene (which he claims was stolen). A has a motive: he had a grudge against the victim, while B was only an acquaintance.

A naive Bayesian (in both senses) would, with each observation, assign higher and higher probabilities to A being the culprit. In the end, though, it turns out that B commited the crime to frame A. He chose someone B had a grudge against, planted the belonging of A's, and forged the receipts.

It's worth noting that, assuming your priors are accurate, given enough evidence you *will* converge on the correct probabilities. Actually acquiring that much evidence in practice isn't anywhere near guaranteed, however.

They seem to forget to first condition on the fact that the threshold must be an integer. This narrows the possibility space to have size countable infinit rather than uncountable infinit. Meaning they need to do a completely different mahtematics, which gives the correct ressult