We often use the word “probability“ in ordinary language; so, for example: “It is likely that tomorrow can not come”

   When we say these kinds of sentences, we want to express that we are not totally sure of the affirmation, but that we have some confidence that this will happen. As it is logical, what interests us is to see if these situations can be mathematized in some way.

   For a long time it was thought that it was improbable to treat these situations from a mathematical point of view; it even seems difficult to establish a historical chronology of the development of the concept of probability, even though its origin was undoubtedly due to gambling. 

   It is important to emphasize that the concept of probability, like any other mathematical concept, is an elaboration of the thought in which each generation builds its efforts on the pillars constructed by its predecessors.

Events

By repeating the same experiment under equal conditions, it is always possible to obtain the same result, or else it is unpredictable. In the first case, we say that the experiment is deterministic, while in the second case, we can say that it is random.

   The first step that must be done to study a random experiment is to determine the set of possible outcomes. We call each of them elementary event.

   To the set of all the possible results we call it show space and we represent it by the letter E, or by the Greek letter Ω .

For example: in the random experiment that consists of throwing a dice and observing its score, the sample space is:

Ω = {1,2,3,4,5,6}

Now we can consider the situation A: Obtain an even number. We can express it by the set A: {2,4,6}, which is a subset of Ω.

We call event to any subset of Ω, that is, any set of possible outcomes. The events are represented by uppercase letters, and we say that an event is fulfilled or occurs when performing a random experiment if the result obtained is part of that event.

Secure event and impossible event

Among the events that we can consider when performing a random experiment, there are some that have special characteristics.

   If we study these events taking as an example the release of a die we see that:

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Operations with events

The usual operations that we can perform with the subsets of the sample space Ω, are:

• Union: We call the union of events A and B to the event formed by all the results that are in A or B. It is represented by A U B and it is fulfilled if A or B are fulfilled.
• Intersection: We call the intersection of events A and B the event formed by all the results that are in A and B at the same time. It is represented by A Π B and it is fulfilled if both A and B are met simultaneously.
• Difference: We call the difference between the event A and the event B to the event formed by all the results that are in A, but not in B. We represent it as A-B and it is fulfilled if A is fulfilled but it is not fulfilled B.
• Complement:  We call complement or opposite of event A and it is represented by Ā or A, to the event formed by all the results of the experiment that are not in A, that is, to the difference Ω-A. The event A^c is fulfilled if it is not fulfilled A.

As a scheme we can see the different operations together:

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The operations mentioned above have the following properties:

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Supported events and incompatible events

Two or more events are compatible, if they can be fulfilled simultaneously; that is, if they have at least one common result. Conversely, they are incompatible or mutually exclusive and their intersection is the empty set Ø.

 If we again focus on the experiment of throwing a die. Events:

A = {2,3}     B = {1,2}     C = {4,5}

they comply with the following: 

A and B are compatible, and B and C are incompatible, B Π C = Ø

“We can affirm that three or more events are incompatible two to two if any couple that can be formed between them is incompatible”.

Complete system of events

If in the die toss experiment, the events:

G = {1,2,3}     H = {4,5}     I = {6}

they comply with the following:

• Its union is the sample space: G U H U I = Ω
• They are incompatible in pairs:  G Π H = Ø, G Π I = Ø,  H Π I = Ø we say that G, H and I form a complete system of events.

If Ω is the simple space of a fandom experiment, events  A₁, A₂,….A_n form a complete system of events only if the following conditions are met: 

• A1 U ….. U An = Ω
• A1, ….., An they are incompatible two to two.

Probability

Initially, we refer to probability as a measure of the degree of certainty about the occurrence of an event or not. We should recall the experimental and axiomatic definitions of probability.

Experimental definition

We made several series of N realizations of a random experiment:

– The number of times an event A is fulfilled in each of them is called “absolute frequency of A” and it is symbolized by n_A. 

– The ratio between the absolute frecuencias and the number of realizations, N, of the experiment we call “relative frequency of event A” and it is symbolized by f_A. 

   As the number of realizations of the experiment increases, the relative frequencies of an event tend toward a certain value. This property allows us, therefore, to affirm that: 

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Axiomatic definition

Although the experimental definition seems to satisfy intuition, it has a series of drawbacks:

• It would be necessary to repeat the experiment infinitely to know the limit of the relative frequencies, which is not feasible. 

• Nothing assures us that the regularity of the relative frequencies is true for any number of repetitions of the experiment. 

   To overcome these problems, we give an axiomatic definition of the probability of an event, mathematically much more rigorous.

Given the sample space Ω associated with a random experiment, we call probability a function:

P : P(Ω)→ℜ  A → P(A)

which it associates to each event A real number called probability of A, P (A), and that it fulfills the following affirmations or axioms: 

• The probability of any event A is positive or zero: P(A) ≥ 0

• The probability of the sure event is worth 1: P(Ω) = 1

• The probability of the union of a set (finite or infinite) of incompatible events two to two, A1, …, Ai, …, is the sum of the probabilities of the events:P(A1 U A2 U ….U An) = P(A1) + P(A2) + … + P(An)

Properties of probability

The main properties of probability are: 

• The probabilities of the events A and Ā add 1:  P(A) + P(Ā) = 1
• The probability of an event A contained in another event B is less than or equal to the probability of B: A ⊂ B ⇒ P(A) ≤ P(B)
• The probability of the union of two compatible events A and B is the sum of their probabilities minus their intersection: P(A U B) = P(A) + P(B) – P(A Π B)

Conditioned probability

Sometimes, being able to have prior information about an event causes its probability to vary.

For example: In a school with N students, where n_o don’t have a computer and n_m are girls. When choosing a student at random, we are interested in two events:

• to choose a student who has a computer.
• to choose a girl. 

Properties of conditioned probability

The main properties of conditioned probability are:

• • If an event B is contained in an event C, then the probability of B given A is less than or equal to the probability of C given A. B ⊂ C ⇒ P(B | A) ≤ P(C | A)
  • If an event B is contained in another event A, then the probability of the event B given A is the quotient between the probability of the event B and the probability of the event A.B ⊂ A ⇒ P(B | A) = P(A)/P(B)
  • If an event A is contained in another event B, then the probability of the event B given A is equal to one. A ⊂ B ⇒ P(B | A) = 1

Dependent events independent events When throwing a die twice, the fact that it is verified or not to obtain an even number in the first throw does not influence the obtaining of an odd number in the second throw or vice versa. In this case, the probability of an event occurring is not conditioned to the occurrence of the other. Thus:P(A | B) = P(A); P(B | A)  = P(B)

From the beginning of the composite probability, we can affirm that:

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Bayes theorem

This theorem, in the theory of probability, is a proposition proposed by the mathematician Thomas Bayes in which he expresses that since an event B has occurred we can deduce the probabilities of the event A.

For example: We have three urns:

• A with 3 red balls and 5 black ones
• B with 2 red balls and 1 black ball
• C with 2 red balls and 3 black ones

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We choose one of the urns at random and we extract a single ball. If the ball drawn is red, what is the probability of choosing the urn A at random?

   If we call R to draw a red ball and N to draw a black ball, the probability of having chosen the ballot box A at random is P (A | R), which using the “Bayes theorem” would be:

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Random variables

 If we consider now the random experiment of throwing two coins (each with one face (c) and a cross (+).) The sample space of this experiment is:

Ω = [(c, c), (c, +), (+, c), (+, +)]

We can assign a numerical value to each of the possible results, such as, for example, the number of faces obtained. This assignment defines a function of the sample space in the set of real numbers. 

   To the functions defined in this way, we call them “random variables”, and they are represented by X, therefore: 

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Types of random variable

A random variable is a function, so it has a domain and a path. The domain is the sample space Ω associated with a random experiment and the path is a subset of R.The random variables that we have considered so far only take values ​​within a finite set of elements: {0,1,2}, {-1,1}, … so we will say that they are discrete random variables.    If, for example, we now perform the experiment of choosing a class student at random and the random variable that assigns each one of them is its height, said variable can take, in principle, any value within a range of the set of numbers real R, so we say that it is a continuous random variable, since its path is one or several intervals of R. Otherwise, we will say that it is discrete.

You can download the App BioProfe READER to practice this theory with self-corrected exercises.