Partition of the sample space

Sometimes it is useful to divide the sample space into “boxes” that cover it entirely without overlapping. This subdivision is called a partition.

Definition

The events $A_1, A_2, \ldots, A_n$ form a partition of the sample space if:

• their union is the whole of $\Omega$: $\bigcup_{i=1}^{n} A_i = \Omega$;
• they are pairwise disjoint: $A_i \cap A_j = \emptyset$ for every $i \neq j$.

In a partition, one and only one of the events $A_i$ occurs.

The two conditions guarantee, respectively, that no outcome is left uncovered and that no outcome belongs to two different events.

Links

Topics: Probability
Concepts: Disjoint events · Partition · Sample space