A thousand people as a frequency tree: 100 sick, 90 of them test positive — and so do 90 of the 900 well. The two positive boxes are the same size, which is the theorem's whole surprise drawn as area. Evidence: the posterior climbing with each further test; the ladder is base rates on a log scale.
Your dog barks at cats — and at the mail
Your dog barks whenever a cat walks past. But she also barks at the mail carrier, the wind, and her own reflection. So when you hear a bark, is there a cat? Only sometimes! To guess well you have to remember two things: how often cats actually come by, and how often the dog barks at nothing at all.
A bark means more when the dog rarely barks for no reason.
P(H|E) = P(E|H) · P(H) / P(E)
Common cold
in winter
1 IN 5
Colour blindness
in men
1 IN 12
Rare disease
screening
1 IN 10,000
Jackpot
one ticket
1 IN 14M
Bars are log-scale odds against — each step is 10×; the rarer the prior, the more a positive result must overcome