The complementary events They are defined as any group of mutually exclusive events, where the union of them is capable of completely covering the sample space or possible cases of an experiment (they are exhaustive).
Their intersection results in the empty set (∅). The sum of the probabilities of two complementary events is equal to 1. That is, 2 events with this characteristic completely cover the possibility of events of an experiment.
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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 Has a probability of zero.
According to 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 : Leave an even number = 2, 4, 6
B: Get an odd number = 1, 3, 5
In this case TO Y B They 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 : Leave a prime number = 2, 3, 5
D: x / x Ԑ N ᴧ x ˃ 3 = 4, 5, 6
The sets A, B and C are written in notation Descriptive Y Analytics respectively. For the whole D algebraic notation was used, then the possible results corresponding to the experiment were described in notation Analytics.
It is observed in the first example that being TO Y B complementary events
TO : Leave an even number = 2, 4, 6
B: Get an odd number = 1, 3, 5
The following axioms hold:
In statistics and probabilistic studies, complementary events are part of the theory of the whole, being very common among the operations carried out in this area.
To learn more about the complementary events, it is necessary to understand certain terms that help to define them conceptually.
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 the probability.
Examples of events are:
Regarding set theory. A Complement refers to the portion of sample space that needs to be added to a set so that it encompasses its universe. It is everything that is not part of the whole.
A well-known way to denote the complement in set theory is:
A 'Complement of A
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 complementary events can be seen directly in Venn diagrams, since its graphical method allows to identify the complements corresponding to each set.
Simply completely visualizing the environment of a set, omitting its boundary and internal structure, allows to give a definition to the complement of the studied set..
Are examples of complementary events success and defeat in an event where equality cannot exist (A baseball game).
Boolean variables are complementary events: True or false, alike right or wrong, closed or open, on or off.
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
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). They are complementary events the following pairs of subsets:
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 the 5 is the only common element between M Y K.
5 '= 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 = L; Because L Y K are complementary, the third axiom described above is fulfilled (Each subset is equal to the complement of its counterpart)
Define: [(J ∩ H) U N] '
J ∩ H = 3 ; In a homologous way to the first step of the previous exercise.
(J ∩ H) U N = 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.
Show that: [H U N] ∩ [J U M] ∩ [L U K] '= ∅
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 equals 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.
Later; S '= ∅ By definition of sets.
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
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