Tree diagrams, total probability and Bayes' theorem
Using tree diagrams for multi-stage experiments, the law of total probability, and Bayes' theorem with spam filters and medical tests.
Tree diagrams
For an experiment with stages, a tree diagram shows every path. Each branch carries the probability of that step given what happened before.
The law of total probability
\(20\%\) of a company's emails are spam. The word "free" appears in \(60\%\) of spam emails and in \(5\%\) of legitimate ones. What fraction of all emails contain "free"?
Seeing the word raised the probability of spam from \(20\%\) to \(75\%\). That's exactly how simple spam filters work: Bayes' theorem updates a probability in light of new evidence.
The medical test surprise
Practice
Quick check
In a tree diagram, how do you find the probability of one complete path?
Each branch is a conditional probability, so the multiplication rule applies.
Quick check
What does Bayes' theorem let you find?
It reverses a conditional probability using the base rates.
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