## What are mutually exclusive events?

The **mutually exclusive events** they happen when both cannot occur simultaneously in the result of an experimentation. They are also known as incompatible or disjoint events. That is, if one occurs, the other cannot occur.

For example, by rolling a die, you can separate the possible outcomes as odd or even numbers, each of these events excluding the other (you cannot roll both an odd and an even number).

Going back to the example of the dice, only one face will be up and we will obtain an integer value between **one** and **six**. This is a simple event, as it only has one outcome chance. All simple events are **mutually exclusive** by not admitting another event as a possibility.

**When do they appear?**

They arise as a result of operations carried out in the Theory of sets, where groups of elements constituted in sets and subsets, are grouped or demarcated according to relational factors: union ( U ), intersection ( ∩ ) and complement ( ‘ ) among others.

They can be treated from different branches (mathematics, statistics, probability and logic, among others) but their conceptual composition will always be the same.

**What are the events?**

They are possibilities and events resulting from experimentation, capable of offering results in each of its iterations. The **events **generate the data to be recorded as elements of sets and subsets, trends in these data are reason for study for probability.

Examples of events are:

The coin pointed heads.

The match resulted in a draw.

The chemical reacted in 1.73 seconds.

The velocity at the maximum point was 30 m/s.

The dice marked the number 4.

Two mutually exclusive events can also be considered as complementary events, if they span the sample space with their union, thus covering all the possibilities of an experiment.

For example, the experiment based on tossing a coin has two possibilities, heads or tails, where these results cover the entire sample space. These events are incompatible with each other and at the same time, collectively exhaustive.

Every dual element or variable of Boolean type is part of the mutually exclusive events, this characteristic being the key to define their nature. The absence of something governs its state, until it presents itself and is no longer absent. The dualities of good or bad, right and wrong operate under the same principle, where each possibility is defined by excluding the other.

**Properties of mutually exclusive events**

Let A and B be two mutually exclusive events.

A ∩ B = B ∩ A = ∅

If A = B’ are complementary events and AUB = S (Sample space)

P (A ∩ B) = 0. The probability of simultaneous occurrence of these events is null.

Resources like the **Venn Diagram** greatly facilitate the classification of **mutually exclusive events**among others**, **since it allows to completely visualize the magnitude of each set or subset.

The sets that do not have common events or are simply separated, will be considered as incompatible and mutually exclusive.

**Examples of mutually exclusive events**

In a soccer game, a team can win, lose, or tie. You cannot win and lose at the same time. In an exam, a student passes or fails. In some surveys, respondents must choose Yes or No. They cannot choose both. When a person dies, his organism stops working after a few minutes. You can’t be alive and dead at the same time. When it’s hot, it can’t be cold in the same place. Send an email without internet. If it is night it cannot be day in the same place.

**References**

Colin, GG Statistics and the Evaluation of Evidence for Forensic Scientists. Second Edition. The University of Edinburgh.

Robert B. Ash. BASIC PROBABILITY THEORY, Department of Mathematics. University of Illinois

Mario F. Triola. Elementary STATISTICS. Tenth Edition. Boston St.