Bayes' theorem

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   An email inbox contains $75$ emails, of which $24$ are spam and the remaining $51$ are legitimate. Of the $24$ spam emails, $7$ contain the word "free"; of the $51$ legitimate emails, $11$ contain the word "free". An email containing the word "free" is chosen at random. Using Bayes' theorem, find the probability that it is spam. Give your answer as a fraction.

2. Warm-up

   An email inbox contains $90$ emails, of which $22$ are spam and the remaining $68$ are legitimate. Of the $22$ spam emails, $11$ contain the word "free"; of the $68$ legitimate emails, $2$ contain the word "free". An email containing the word "free" is chosen at random. Using Bayes' theorem, find the probability that it is spam. Give your answer as a fraction.

3. Warm-up

   An email inbox contains $90$ emails, of which $28$ are spam and the remaining $62$ are legitimate. Of the $28$ spam emails, $10$ contain the word "free"; of the $62$ legitimate emails, $5$ contain the word "free". An email containing the word "free" is chosen at random. Using Bayes' theorem, find the probability that it is spam. Give your answer as a fraction.

Problems shown under their original terms: Generated (template, solver-computed).