## 13.5 Outline

- Probabilities with a partition of a sample space
- partition of a set
- probability of a partitioned event

*A posteriori*probabilities*a priori*probabilities*a posteriori*probabilities- Bayes’ theorem

## 13.5 Essential Ideas

**Partition of a set:** A set is said to be partitioned if is divided into subsets satisfying the following two conditions.

- If and be any two subsets, then That is, the members of the subsets are pairwise disjoint.
- The union of all the subsets is ; that is, there are no elements of that are not included in

one of the subsets.

**Probability of a partitioned event:** If *A*_{1}, *A*_{2}, …, *A*_{n} form a partition of a sample space *E* is any event, then

*P(E) =*

*P*(*E*|*A*_{1})

*P*(A

_{1})

+

*P*(

*E*|

*A*

_{2})

*P*(A

_{2})

+ … +

*P*(

*E*|

*A*

_{n})

*P*(A

_{n})

**Bayes’ theorem: **