Bayes’ Theorem a Triviality?
As is so often the case, micro-examples make the otherwise abstract obvious. Almost anyone will be able to reason upward from the following adaptation of a famous example to a general statement of the theorem. For some reason, the little example is so easy to understand and to remember.
Imagine you have a school of 200 students with 60% boys and 40% girls; 90% of the boys wear pants, 10% wear kilts; 35% of the girls wear pants and 65% wear skirts. You see a student in the distance and can tell that he or she is wearing a kilt or a skirt, but you can’t tell whether the student is a boy or a girl. What’s the probability the student is a girl?
This is only easy if you just write out the counts, as in 120 boys: 108 in pants, 12 in kilts; 80 girls: 28 in pants, 52 in skirts. If you try to work in fractions you will go insane with arithmetic.
But Bayes’ theorem becomes a triviality: 64 (12+52) students are in skirts or kilts; 136 (108+28) in pants (marginal counts). Just write out the ratios. Probability that student is girl given that we see skirt or kilt is P(girl|skirt) = P(girl & skirt) / P(skirt) = 52 / 64. Prob that student is in skirt given she is a girl is P(skirt|girl) = P(girl & skirt) / P(girl) = 52 / 80. P(girl & skirt) = P(skirt|girl)*P(girl) = P(girl|skirt)*P(skirt). That’s the usual statement of the theorem.
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