COMPLEMENTARY EVENTS: WHAT THEY CONSIST OF AND EXAMPLES - MATHS - 2025

The additional events are defined as any group of mutually exclusive events each other, where the union of them is able to fully cover the sample space or possible cases of experimentation (are exhaustive).

Their intersection results in the empty set (∅). The sum of the probabilities of two complementary events is equal to 1. In other words, 2 events with this characteristic completely cover the possibility of events of an experiment.

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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 {the number seven comes up} It has a probability of zero.

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

A: {Output an even number} = {2, 4, 6}

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

In this case A and B are Complementary Events. Because both sets are mutually exclusive (an even number that is odd in turn cannot come out) and the union of these sets covers the entire sample space.

Other possible subsets in the example above are:

C: {Output a prime number} = {2, 3, 5}

D: {x / x Ԑ N ᴧ x ˃ 3} = {4, 5, 6}

Sets A, B, and C are written in Descriptive and Analytical notation respectively. For the set D algebraic notation was used, and the possible results corresponding to the experiment were described in Analytical notation.

It is observed in the first example that since A and B are complementary events

A: {Output an even number} = {2, 4, 6}

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

The following axioms hold:

1. AUB = S; The union of two complementary events is equal to the sample space
2. A ∩B = ∅ ; The intersection of two complementary events is equal to the empty set
3. A '= B ᴧ B' = A; Each subset is equal to the complement of its homolog
4. A '∩ A = B' ∩ B = ∅; Intersect a set with its complement equals empty
5. A 'UA = B' UB = S; Joining a set with its complement equals the sample space

In statistics and probabilistic studies, complementary events are part of the whole theory, being very common among the operations 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 their iterations. The events generate the data to be recorded as elements of sets and sub-sets, the 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 speed at the maximum point was 30 m / s
• The die marked the number 4

What is a plugin?

Regarding 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 whole.

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

A 'Complement of A

Venn Diagram

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It is a graphical - content analytical scheme, widely used in mathematical operations involving sets, sub-sets 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 additional events are seen directly Venn diagrams, as its graphical method to identify the corresponding adders to each set.

Simply completely visualizing the environment of a set, omitting its boundary and internal structure, allows a definition to be given to the complement of the studied set.

Examples of complementary events

Examples of complementary events are success and defeat in an event where equality cannot exist (A baseball game).

Boolean variables are complementary events: True or false, likewise right or wrong, closed or open, on or off.

Complementary event exercises

Exercise 1

Let S be 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 are defined

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}

Decide:

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 are identified (mutually exclusive and cover the sample space when joining). The following pairs of subsets are complementary events :

• H and N
• J and M
• L and K

Exercise 2

Show that: (M ∩ K) '= L

{0, 1, 2, 4, 5, 7, 8, 10} ∩ {5} = {5}; The intersection between sets yields the common elements between both operant sets. In this way 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 is fulfilled (Each subset is equal to the complement of its homologue)

Exercise 3

Define: '

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

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

' = {0, 1, 2}; The complement of the combined operation is defined.

Exercise 4

Prove that: { ∩ ∩} '= ∅

The compound operation described within the curly 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).

∩ ∩ = 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 subsets, whose results are different from the empty set (∅).

• M ∩ N

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

• L ∩ H

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

• J ∩ N

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

References

1. THE ROLE OF STATISTICAL METHODS IN COMPUTER SCIENCE AND BIOINFORMATICS. Irina Arhipova. Latvia University of Agriculture, Latvia.
2. Statistics and the Evaluation of Evidence for Forensic Scientists. Second Edition. Colin GG Aitken. School of Mathematics. The University of Edinburgh, UK
3. BASIC PROBABILITY THEORY, Robert B. Ash. Department of Mathematics. University of Illinois
4. Elementary STATISTICS. Tenth Edition. Mario F. Triola. Boston St.
5. Mathematics and Engineering in Computer Science. Christopher J. Van Wyk. Institute for Computer Sciences and Technology. National Bureau of Standards. Washington, DC 20234
6. Mathematics for Computer Science. Eric Lehman. Google Inc.

   F Thomson Leighton Department of Mathematics and the Computer Science and AI Laboratory, Massachussetts Institute of Technology; Akamai Technologies