The complete group of all possible outcomes of a random experiment given an exhaustive set of events.

Definition of Exhaustive Set of Events

In probability theory, an exhaustive set of events is a set of events that includes all possible outcomes of a random experiment. This means that every possible outcome of the experiment belongs to one of the events in the set.

For example, if we flip a coin, the exhaustive set of events would be "heads" and "tails". There are no other possible outcomes.

The Complete Group of All Possible Outcomes

When we have an exhaustive set of events, we can determine the complete group of all possible outcomes by listing all the events in the set. Each event represents a possible outcome of the experiment, so the complete group of all possible outcomes is simply the set of events.

For example, if we roll a six-sided die, the exhaustive set of events is {1, 2, 3, 4, 5, 6}. The complete group of all possible outcomes is simply this set.

Mutually Exclusive vs Exhaustive Sets

It's important to note that an exhaustive set of events is not necessarily mutually exclusive. Mutually exclusive events are events that cannot happen at the same time. For example, if we roll a die, the events "rolling a 1" and "rolling a 2" are mutually exclusive because we cannot roll both a 1 and a 2 at the same time.

In contrast, the events "rolling an even number" and "rolling a number less than 4" are not mutually exclusive, but they are exhaustive. The exhaustive set of events in this case would be {1, 2, 3, 4, 5, 6}, but the events are not mutually exclusive because the number 2 belongs to both events.

Conclusion

In summary, the complete group of all possible outcomes of a random experiment given an exhaustive set of events is simply the set of events. An exhaustive set of events includes all possible outcomes of the experiment, but the events may or may not be mutually exclusive.

Union of 2 sets results into sample space then both events is called as exhaustive event
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