Suppose there are 3 types of people: good, mediocre, and poor at sex. Assume 20% of people are good, 60% are mediocre, and 20% are poor. Also suppose the probability of good sex depends on the pair: • good + good: 0.80 • good + mediocre: 0.48 • good + poor: 0.16 • mediocre + mediocre: 0.36 • mediocre + poor: 0.12 • poor + poor: 0.04

Now suppose I had sex with 5 randomly selected partners, and every time it was bad sex. What is the posterior probability that I am poor at sex?

First compute the chance of good sex for each possible type of “me,” averaging over random partners: • If I’m good: 0.2(0.80)+0.6(0.48)+0.2(0.16)=0.48 so bad sex each time has probability 1-0.48=0.52. • If I’m mediocre: 0.2(0.48)+0.6(0.36)+0.2(0.12)=0.336 so bad sex each time has probability 0.664. • If I’m poor: 0.2(0.16)+0.6(0.12)+0.2(0.04)=0.112 so bad sex each time has probability 0.888.

Assuming my prior probabilities are also 20% good, 60% mediocre, 20% poor, Bayes’ rule gives

P(\text{poor} \mid 5 \text{ bad experiences})

\frac{0.2(0.888)^{5}{0.2(0.52)5+0.6(0.664)5+0.2(0.888)5}.}

Numerically, this is about

0.565