Set Theory: Definitions and Types of Sets

• A set is denoted by a capital letter
• The element by a lowercase letter.
• The elements are enclosed in braces {} and separated by commas.
• The set D whose elements are the numbers that appear when rolling a die: D= {1,2,3,4,5,6}.
• The set of the days of the week: S = { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday }.
• When defining a set, it can be done in 2 ways: by extension or numbering.
• In this method a list of its elements is made, examples previous.

Method of Comprehension

• A property preserved by all its members is described, but not by non members, example:
• The set of vowels: V= { x|x is a vowel }.

Types of Sets

• Finite sets: are those that can count its elements.
• Ex Conjunto de datos que aparecen al lanzar un dado,
• Infinite sets: Cannot count or have a limited number of elements.
• Ex Set of real numbers.
• Universal set: It is the set of all the elements considered in a given problem or situation. -Ex Set of positive real numbers U = R += {0,∞}.
• Empty Set: It is a set that does not have elements and is denoted by Ø or {}.
• Ex set of months that only have 27 days.

Venn Diagram

• Any closed geometric figure (circle, rectangle, oval, etc.) serve to graphically represent the operations between sets
• These diagrams are called Venn Diagrams.
• Usually the universal set is represented with a rectangle and the sets with a circle or ellipse.
• A Venn diagram at no time constitutes a mathematical proof
• It does, however, allow an intuitive vision of the relationship that may exist between sets.

Union

• A union is the set of all the elements that belong to A or B
• The union of sets is when the elements are combined, giving rise to a new set.
• AuB

Intersection Intersecaón

• An intersection is the set of all the elements that belong to A and B
• AnB

Difference

• The set of all the elements of A that do not belong to B.
• A-B

Complement

• All the sets of the universe that are not A.
• A' or Ac

Basic and Conditional Probability

• Probability: It is the measure of the uncertainty of an event.
• Probability is used to express how likely an event that has been determined to occur is.
• Event: Any set of results of procedures or experiment.
• Experiment: Set of tests or the performance of a process that leads to a result and observation of which is not safe.

Procedure, Sample Space and Event relationship

• Rolling a coin, possible result is face or tails, the success is face
• Torsing 2 coins, possible results are face, face, -face, cross,- cross, face -cross, cross, successful outcome is face, cross
• Sample space: For an experiment , is a set of all experimental outcomes, ie when all possible results have been specified
• The probability of an event can take values that range from 0 to 1.

Terminology for understanding Probability

• Improbable: 0
• Probable: 0.5
• Safe: 1
• If A is an event, its probability is determined by P(A).
• Probability of the event A (ie already is likelihood or whether a coin falls face).

Formulas for Probability

• 0≤P(A)≤1
• The probability of the event will be less than or equal to 1.

Classic Probability Method

• If a procedure has n simple different events with the same probability of occurring
• Then: P(A) = number of times that can occur / number of simple diverse events. -Example, tossing a coin.
• The simple success simple probability face or tails.

Empirical Probability Method

• Repeat a procedure a large number of times.
• Count the number of times the event A occurred.
• then P(A)
• Number of times that event A occurred, number of times the procedure was repeated.

Subjective Probability Method

• The probability of the event A is estimated based on knowledge of relevant circumstances for the question success.
• Example:
• P (rain): Estimate based on knowledge of the time.
• time of year is a part
• has rained on previous days.
• the geografical

Counting techniques

• Mathematical strategies used in probability and statistics to determine the total number of results that may occur from making combinations within a set of objects.
• A tree diagram a type of graphic or diagram that enables relationships to be established in a hierarchical manner between concepts.
• There are 4 chairs yellow, red, blue and green, as combinations of 3 of them can be made to arrange one next to another:

Calculate Probabilities Using a Tree Diagram

• For the probability calculation we will use a trick
• If to calculate certain probability, we have to advance to the right, then it will be multiplied.
• On the other hand if to calculate we have to advance upwards, then it will be added.
• There is an explicit example available in the training data

Permutations and Combinations Definition

• Some probability situations involve multiple events.
• When one of the events affects others, they are called dependent events.
• For example, when objects are taken from a list or group and are not returned, that selection reduces the options for future elections.
• Two ways to order or combine dependent events, this is a permutation
• Permutations are groupings in which the order of objects is important.

Definition of Combinations

• Combinations are groupings in which the content is important, but not the order.

Definition of Permutations

• A permutation of a set of elements is an arrangement of these elements.
• Considers the order.
• Consider permutations of "n" elements taken from "k" to "k".

Formula for Permutations

Formula to calculate a premutation n!/(n-k)!

Definition of Combinations with Repetitions

• If used in a total of "n" elements, with one kind or element repeating "a" times, the the second "b" times etc
• The order is considered
• There are repeated elements
• All elements participate

Formula for Permutations with Repetitions

• n!/a!⋅b!⋅c!.

Definition of Circular Premutations

• As scenario when elements are being used to form a cirle which would the order in which they are organised matters

Formula for Circular Permutations

P(C,n) = (n-1)!

Formula for Binomial Distribution, where f(x) = P(x = x)

(nx)(P^x)((1-p)^(n-x))

• n = number of ensayo
• P = probabilidad de exito
• X = variable aleatorra binomial - media,