I.

"Transitivity" is a property in mathematics and logic. Put simply, if something is transitive it means that there's a relationship between things where when x relates to y, and y relates to z, there's the same relationship between x and z. For a more concrete example, think of size. If my car is bigger than my couch, and my couch is bigger than my hat, you know that my car is bigger than my hat.

(Epistemic status: I am not a math major, and if there's a consensus in the comments that I'm using the wrong term or otherwise making math mistakes I can update the post.)

This is a neat property. Lots of things do not have it.

II.

Consider the following circumstance: Bob is traveling home one night, late enough there isn't anyone else around. Bob sees a shooting star growing unusually bright, until it resolves into a disc-shaped machine with lights around the edges. He finds himself levitated up into the machine, gets poked and prodded by the creatures inside for a while, and then set back down on the road.

Assuming Bob is a rational, rationalist, well-adjusted kind of guy, he now has a problem. Almost nobody in his life is going to believe a word of this.

From Bob's perspective, what happened? He might not be certain aliens are real (maybe he's just had a schizophrenic break, or someone slipped him some interesting drugs in his coffee) but he has to be putting a substantially higher percentage on the idea. Sure, maybe he hallucinated the whole thing, but most of us don't have psychotic breaks on an average day. Break out Bayes.

[WARNING: There's a discussion in the comments suggesting I'm doing the setup and math wrong. Seems plausible, this is my first time doing Bayes with an audience.]

What are Bob's new odds aliens abduct people, given that his experiences? Let's say his prior probability on alien abductions being real was 1%, about one in a hundred. (That's P(A).) He decides the sensitivity of the test - that he actually got abducted, given he had this experience - is 5% since he knows he doesn't have any history of drug use, mental illness, or prankish friends with a lot of spare time and weird senses of humour. (That's P(B|A).) If you had asked him before his abduction what the false positive rate was - that is, how often people think they've been abducted by aliens even though they haven't - he'd say .1%, maybe one in a thousand people have seemingly cause...