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 A1, A2, …, An form a partition of a sample space E is any event, then
P(E) = P(E|A1)P(A1)
+ P(E|A2)P(A2)
+ … + P(E|An)P(An)
+ P(E|A2)P(A2)
+ … + P(E|An)P(An)
Bayes’ theorem: