Mutually exhaustive events are a set of events where at least one of the events must occur. In other words, there is no possibility that none of the events will happen.
Definition:
A set of events {E1, E2, ..., En} is said to be mutually exhaustive if their union covers the entire sample space (S). This can be expressed mathematically as:
```
E1 ∪ E2 ∪ ... ∪ En = S
```
Example:
Consider the event of tossing a coin. The possible outcomes are heads (H) or tails (T). The events {H} and {T} are mutually exhaustive because it is guaranteed that the outcome will be either heads or tails. There is no other possibility.
Characteristics of Mutually Exhaustive Events:
* No overlap: The events cannot occur simultaneously.
* Complete coverage: The events together cover all possible outcomes.
* Probability of 1: The sum of the probabilities of all mutually exhaustive events is equal to 1.
Note:
Mutually exhaustive events do not have to be mutually exclusive. Mutually exclusive events cannot happen at the same time, while mutually exhaustive events cover all possibilities.
In summary:
Mutually exhaustive events are a set of events that cover all possible outcomes in a sample space, ensuring that at least one of the events must occur.