Total Probability and Bayes’ Theorem

Events \(E_1,E_2,\dots,E_n\) form a partition of \(S\) if they are pairwise disjoint, have positive probabilities and their union is \(S\). For any event \(A\):

\[P(A)=\sum_{i=1}^nP(E_i)P(A\mid E_i)\qquad\text{(total probability)}\]\[P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_jP(E_j)P(A\mid E_j)}\qquad\text{(Bayes’ theorem)}\]

The \(P(E_i)\) are prior probabilities; \(P(E_i\mid A)\) are posterior probabilities, updated after observing \(A\).

Tip: draw a tree. Multiply along each branch, add the branches that end in \(A\) to get \(P(A)\), then divide the branch you want by that total.