2.4.3 Bayes' Theorem

The computation of a posterior probability $P\left(A_i\mid B\right)$ from given prior probabilities $P\left(A_i\right)$ and conditional probabilities $P\left(B\mid A_i\right)$ occupies a central position in elementary probability.

• The general rule for such computations, which is really just a simple application of the multiplication rule, goes back to Reverend Thomas Bayes, who lived in the eighteenth century. To state it we first need another result.
• Recall that events $A_1,\dots ,A_k$ are mutually exclusive if no two have any common outcomes.
• The events are exhaustive if one $A_i$ must occur, so that $A_1\cup \dots \cup A_k=\mathcal{E}$ .

The Law of Total Probability

EX 2.30 spam email

Bayes’ theorem

• As long as there are relatively few events in the partition,
  • a tree diagram (as in Example 2.29) can be used as a basis for calculating posterior probabilities without ever referring explicitly to Bayes’ theorem.

EX 2.31 Incidence of a rare disease