## What are side events?

The **side events** are the opposite outcomes of a given event in a random experiment. In other words, two events are complementary if one is the opposite result of the other. For example, it rains or it does not rain, it is cold or it is hot.

Their intersection results in the empty set (∅). The sum of the probabilities of two complementary events is equal to **1. **That is to say, that two events with this characteristic completely cover the possibility of events of an experiment.

**What are the complementary events?**

A very useful generic case to understand this type of event is to roll a dice:

When defining the sample space, all the possible cases that the experiment offers are named. This set is known as the universe.

sample space **(S):**

**S : { 1, 2, 3, 4, 5, 6 }**

** **The options not stipulated in the sample space are not part of the possibilities of the experiment. For example, {*let the number seven come out} t*has a probability of zero.

Depending on the objective of the experimentation, sets and subsets are defined if necessary. The set notation to be used is also determined according to the objective or parameter to be studied:

**TO : { Get an even number} = { 2, 4, 6 }**

**B : { get an odd number} = { 1, 3, 5 }**

In this case, **TO **and **B. **They are complementary events, because both sets are mutually exclusive (an even number cannot come out that is odd at the same time) and the union of said sets covers the entire sample space.

Other possible subsets in the above example are:

**C.** : **{ get a prime number} = {2,3,5}**

**D : { x / x Ԑ N ᴧ x ˃ 3 } ** = **{4,5,6}**

the sets **A, B and C **They are written in descriptive and analytical notation, respectively. for the whole **D. **algebraic notation was used, describing the possible results corresponding to the experiment in analytical notation.

It is observed in the first example that, being **TO **and** B. **side events

**TO : { Get an even number} = { 2, 4, 6 }**

**B : { get an odd number} = { 1, 3, 5 }**

The following axioms hold:

**AUB = Yes:** The union of two complementary events is equal to the sample space.

**A∩B = **∅:** **The intersection of two complementary events is equal to the empty set.

**A’ = B ᴧ B’ = A: **Each subset is equal to the complement of its counterpart.

**A’ ∩ A = B’ ∩ B = **∅: Intersecting a set with its complement is equal to empty.

**A’ AU = B’ UB = S: **Uniting a set with its complement is equal to the sample space.

In statistics and probabilistic studies, complementary events are part of the set theory, being very common among the operations that are carried out in this area.

To learn more about complementary events, it is necessary to understand certain terms that help define them conceptually.

**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, the trends in these data are the subject of study for probability.

**What is a plugin?**

With respect to set theory, a complement refers to the portion of the sample space that needs to be added to a set for it to encompass its universe. It is everything that is not part of the set.

A well-known way to denote the complement in set theory is:

**A’ complement of A**

**Venn Diagram**

It is a graphic-analytical scheme of content, widely used in mathematical operations involving sets, subsets and elements. Each set is represented by a capital letter and an oval figure (this characteristic is not mandatory within its use) that contains each and every one of its elements.

The complementary events are directly appreciated in the Venn diagrams, since its graphic method allows to identify the corresponding complements to each set.

Simply, completely visualizing the environment of a set, omitting its border and internal structure, allows us to give a definition to the complement of the studied set.

**Examples of Side Events**

Heads and Tails in a Coin Toss: When tossing a coin, the events “get heads” and “get tails” are complementary. You cannot get both results at the same time. Draw a red card or a black card from a standard deck: If a card is drawn at random from a standard deck of cards (52 cards), the events «draw a red card» and «draw a black card» are complementary, since that a card cannot be red and black at the same time. Rolling an Odd or Even Number on a Die: When rolling a six-sided die, the events “roll an even number” and “roll an odd number” are complementary. An event happening or not happening: for example, “it will rain tomorrow”. The events “rain tomorrow” and “no rain tomorrow” are complementary, since both outcomes cannot occur at the same time in the same place. Draw a white ball or a black ball from an urn: if you have an urn with white and black balls, the events “draw a white ball” and “draw a black ball” are complementary. When a ball is drawn, it can only be one of the two colors, not both.

**Complementary Events Exercises**

**Exercise 1**

Be **S **the universe set defined by all natural numbers less than or equal to ten.

**S : { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }**

The following subsets of **S **

**H : { Natural numbers less than four } = { 0, 1, 2, 3 }**

**J : { Multiples of three } = { 3, 6, 9 }**

**K : { Multiples of five } = { 5 }**

**L : { 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 }**

** M : { 0, 1, 2, 4, 5, 7, 8, 10 }**

** N : {Natural numbers greater than or equal to four} = { 4, 5, 6, 7, 8, 9, 10 }**

Determine:

How many complementary events can be formed by relating pairs of subsets of **S**?

According to the definition of complementary events, the pairs that meet the requirements (mutually exclusive and cover the sample space when joining) are identified. The following pairs of subsets are complementary events**:**

**Exercise 2**

Show that: **( M ∩ K )’ = L**

**{ 0, 1, 2, 4, 5, 7, 8, 10 } ∩ { 5 } = { 5 }. **The intersection between sets yields as a result the common elements between both operating sets. In this way, the **5** is the only common element between **m **and **K.**

**{ 5 }’ = { 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 } = L,** because **L **and **k **are complementary, the third axiom described above holds (*Each subset is equal to the complement of its counterpart.)*

**Exercise 3**

Define : **[ ( J ∩ H ) U N ]’**

**J ∩ H = { 3 }.** In a homologous way to the first step of the previous exercise.

**( J ∩ H ) A** = **{ 3, 4, 5, 6, 7, 8, 9, 10 }. **These operations are known as combined and are usually treated with a Venn diagram.

**[ ( J ∩ H ) U N ]’** = **{ 0, 1, 2 }. **The complement of the combined operation is defined.

**Exercise 4**

Show that: { **[ H U N ] ∩ [ J U M ] ∩ [ L U K ] }’ = **∅

The compound operation described within the braces refers to the intersections between the unions of the complementary events. In this way we proceed to verify the first axiom (The union of two complementary events is equal to the sample space).

**[ H U N ] ∩ [ J U M ] ∩ [ L U K ] = S ∩ S ∩ S = S: **The union and intersection of a set with itself generates the same set.

Then; ** S’ = **∅ ** **By definition of sets.

**Exercise 5**

Define 4 intersections between the subsets, whose results are different from the empty set (∅).

**{ 0, 1, 2, 4, 5, 7, 8, 10 } ∩ { 4, 5, 6, 7, 8, 9, 10 } = { 4, 5, 7, 8, 10 }**

**{ 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 } ∩ { 0, 1, 2, 3 } = { 0, 1, 2, 3 }**

**{ 3, 6, 9 } ∩ { 4, 5, 6, 7, 8, 9, 10 } = { 6, 9 }**

**References**

THE ROLE OF STATISTICAL METHODS IN COMPUTER SCIENCE AND BIOINFORMATICS. Recovered from irina.arhipova.com.

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.