1. Conditional Probability More often than not, we wish to express probabilities conditionally. i.e. we specify the assumptions or conditions under which it was measured Pr{H|O} ≡ probability that event H occurred in the subset of cases where event O also occurred Conditional statements of probability take the form:

2. Conditional Probability This can be represented as a Venn diagram The intersection H ∩ O represents events where both H and O occurred S H O

3. Conditional Probability More often than not, we wish to express probabilities conditionally. i.e. we specify the assumptions or conditions under which it was measured We can express the idea shown in the Venn diagram as: Note that in the “universe of possibilities”, O has effectively replaced S. Our probability for event H is now conditional on the assumption that event O has already taken place. P(H | O ) = P(H & O) / P(H)

4. Tree Diagram Representations S O ~O H & O ~H & O H & ~O ~H & ~O See www.mathsisfun.com/data/probability-events-conditional.html P(H & O) = P(H|O) P(O)

5. Bayes’ Theorem Suppose we observe H but do not observe whether O occurred or not. In order to infer O we may use the previous tree diagram S O ~O H & O ~H & O H & ~O ~H & ~O P(O|H) = P(H & O) / (P(H & O) + P(H & ~O)) = P(H|O)P(O) / (P(H|O)P(O) + P(H|~O)P(~O)) See www.digitalbiologist.com/2014/01/a-theorem-for-all-seasons.html