Mutually Exclusive and Exhaustive events

It needs to be taken into account that when two events do not occur simultaneously or at the same time, then the two events can be called mutually exclusive events. This theory of probability needs to be taken into consideration to discuss mutually exclusive events. In the case of mutually exhaustive events, the events do not occur at the same time or simultaneously, instead one of the events takes place. There needs to be an extensive discussion about how mutually exclusive events occur and how mutually exhaustive events are different from mutually exclusive events. These events are associated with the outcome of an experiment so that more specific results can be achieved with respect to a mathematical or statistical conclusion. 

Mutually Exclusive Events in Statistics

The simple question that comes to the mind of beginners is that what are mutually exhaustive events? The answer to this question is very simple, for mutually exclusive events is a statistical term t6hat plays a key role in defining two or more events that cannot take place at the same time. In a more lucid language, it can be said that the term mutually exclusive events when a situation where one event supersedes or surpasses another event, is described. 

A simple example can be cited to further explain this term mutually exclusive events. Suppose the event of a person being an adult is considered. In this case, the case is either the person is an adult or does not fit into the category of being an adult. Therefore, these events can be considered mutually exclusive. It can be further said that the events can be said to be mutually exclusive when the occurrence of another event is not feasible due to the occurrence of an event. This is an either-or circumstance, where there is no scope for union or concurrence. In the theory of probability, it needs to be taken into account that the example of dice is best suited to define what are mutually exclusive events? When a dice is rolled into the plane, the dice can offer an outcome that is set even or odd numbers, either the dice will represent a set of odd numbers or a set of even numbers. Therefore, (1, 3, 5) and (2, 4, 6) can be considered as mutually exclusive events. 

In this respect, it is essential to know if we consider two events such as a and b, whether there is any possibility to know that a and b are mutually exclusive. Considering the discussion above, it can be stated that if a and b are mutually exclusive events only when both the events do not take place at the same time. If a and b are mutually exclusive events then, they are not supposed to share the consequences. In statistical expression, it can be stated as P (A AND B) = 0. Therefore, if a and b are mutually exclusive events then, the probability of both the events taking place simultaneously or at the very same time is equal to zero.

General discussion on Mutually Exclusive and Exhaustive Events

What are mutually exclusive events? The answer can be supplemented with more interesting examples that will enable the readers to grasp the idea of the events in the field of statistics. Even though it is a statistical expression, mutually exclusive events take place in day-to-day lives despite the fact that no layman can have an idea regarding the terminologies associated with the common events that fall under the mutually exclusive events. On tossing a coin, either heads or tails are observed as two separate events that do not take place simultaneously. On the other hand, one of the interesting examples is that in the game of cards, the events of Kings and Aces are mutually exclusive events.

If a and b is mutually exclusive events then, then a statistical expression can be stated to define the phenomenon more clearly as: 

P(A∩B)=ϕ⇒P(A∪B)≤1⇒P(A)+P(B)−P(A∩B)≤1P(A∩B)=0,∵(A∩B)=ϕ⇒P(A)≤1−P(B)⇒P(A)≤P(B)

What are mutually exclusive events? The answer to this question has been given in the previous section in an extensive manner. However, it is also important to touch upon mutually exhaustive events. Considering the aspects of probability, events can be collectively termed as exhaustive when they occupy the entire probability space. In simpler words, it can be said that in the case of mutually exhaustive events, the probability of the occurrence of any one event is 100%. In a set that is considered to be mutually exhaustive, it can be inferred that at least one of the sets is definitely true and valid. In the case of mutually exclusive events, one of the two events has to take place, wherein in the case of mutually exhaustive events the probability of one of the events is a must.

Conclusion

The above discussion shows that the concepts of mutually exclusive and exhaustive events are quite interlinked. These statistical terms and connotations can be expressed with the help of statistical formulas to achieve mathematical validation. However, it needs to be taken into account that the examples cited in the above discussion clearly indicate that these events occur every month, and the chapter on probability has a wide range of use in the daily lives of the people.