Question

Clearly explain what it means for a set of events to form a partition of a sample space.

Give an example of a random experiment which has a sample space for which you can define five events which together form a partition. Clearly state your choice of experiment, the resulting sample space and the five events which form the partition. State any assumptions you have made.

Answer 1

A partition of a sample space is a set events that are pairwise disjoint and together form the sample space and the probability of each event is greater than 0.

Example of a random experiment:

Consider a family with 5 members. Write all the 5 names on different papers, fold them similarly and put into a hat or box. Then draw one paper randomly.

Drawing each name is an event. So, there are 5 events, say, A, B, C, D, E. Sample space, S is the set of all 5 names.

Now, the events are pairwise disjoint because if we consider A and B, the occurrence of event A prevents the occurrence of event B and vice versa as we are drawing only one name.

The probability of each event is greater than 0 because P(A) =P(B) =.... =P(E) =1/5 =0.2

These 5 events together form the sample space, S.

Thus, this set of 5 events, A, B, C, D, E is the partition of the sample space, S.

Note: No assumptions are made here.