Suppose you are increasingly annoyed by the amount of spam among your e-mails, which makes up 20% of all the e-mails you receive.

Suppose you are increasingly annoyed by the amount of spam among your e-mails, which makes up 20% of all the e-mails you receive. To block spam you install a filter, which filters all e-mails you receive that contain the phrase "lottery win." The filter is 99% reliable, which means two things: 99% of the spam you receive contains the phrase "lottery win" and so gets filtered, and 99% of the e-mails you receive that are not spam do not contain the phrase "lottery win" and so do not get filtered. The formal language we consider is built up from a vocabulary containing two sentence letters: s says that the next e-mail you receive is spam, f says that the next e-mail you receive gets filtered. Based on the information above you think the probabilities are as follows: Pr(s) = .2, Pr f | s = .99, and Pr ¬f | ¬s = .99. What is the conditional probability that the next e-mail you receive is spam given that it gets filtered, Pr s | f ? You may assume that for all sentences α and γ, LE Pr(α) = Pr γ , if α and γ are logically equivalent, N Pr(¬α) = 1 − Pr(α) and Pr ¬α | γ = 1 − Pr α | γ , and BT Pr α | γ = Pr(α)·Pr(γ|α) Pr(γ|α)·Pr(α)+Pr(γ|¬α)·Pr(¬α) , if Pr(α) > 0, Pr(¬α) > 0, and Pr γ > 0 (this is, of course, just Bayes' theorem for sentences).