Exhaustive Events – Definition & Examples

Consider an experiment whose outcome is not predictable with certainty in advance. Although the outcome of the experiment will not be known in advance, let us suppose that the set of all possible outcomes is known. This set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by S.

Definition:

A collection of events is said to be exhaustive/collectively exhaustive if the union of those events is equal to the sample space. This means that the given set of events includes every possible outcome of the random experiment. If in addition to being exhaustive, the events are all disjoint from each other then we say that the events are mutually exclusive and exhaustive. The sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1.

Examples of Exhaustive Events:

1. When tossing a coin if we define two events A and B as getting a heads and getting a tails respectively then the events A and B are collectively exhaustive. This is because the sample space is S = {H, T} and we have that A U B = S. The events A and B cover all possible outcomes of the random experiment.
2. Suppose a die is thrown. The sample space is equal to S = {1, 2, 3, 4, 5, 6}. We define the events A and B as getting an even number and getting an odd number respectively. Then A = {2, 4, 6} and B = {1, 3, 5}. The events A and B are exhaustive becasuse A U B = S.
3. Once again consider the event where a single dice is thrown. Let A be the event that we get a number less than or equal to 5. Let B be the event that we get an odd number. Here A = {1, 2, 3, 4, 5} and B = {2, 4, 6} and the events are exhaustive becasuse A U B = S. Notice that in this example the two events are not disjoint.
4. When two dice are thrown together there are 6²=36 possible exhaustive outcomes.
5. If the outcome of an experiment consists in the determination of the sex of a newborn child, then the events G and B are mutually exclusive and exhaustive. The event G occurs if a girl is born and the event B occurs if a boy is born.
6. When picking a card from a pack of 52 playing cards the events A= Getting a face card and B = Getting a number card are mutually exclusive and exhaustive.