Definition of probability

1. Probability is the chance that something will happen or be confirmed, based on the logical evaluation of a set of information, such as positive and negative similar situations, interference points, space and time. Examples: A) ‘Tomorrow there is a high probability of rain’. B) ‘There is a probability that he is lying’.

2. Mathematics. Calculation on how many possibilities exist for a random event before the total of possible results. The obtained value varies linearly between 0 (impossible) and 1 (totally possible). Example: ‘The probability of winning the lottery is 1 in 50 million’.

Etymology: By the modes of Latin odds, probabilĭtātisbuilt on the adjective probablyof ‘probable’, with respect to the verb I will tryfrom ‘to try’, accompanied by the suffix -ble, with reference in Latin -bileaccording to the deverbal adjective with property of action, the latter combined with the suffix -dad, in Latin -tas, -ātisas -bilidad, according to quality agent depending on the substantivization with respect to the adjective form.

Grammatical category: noun fem.
in syllables: pro-ba-bi-li-dad.

Probability

Basically, probability is defined as the degree of possibility of an event occurring or not. Daniel Kahneman, Nobel laureate in economics, dedicates several passages of his work “Think fast, think slowly” to describe how human beings need to be certain about what will happen. Under this premise, cognitive tools such as heuristics are developed to try to reach clear conclusions about events. However, Kahneman himself declares that many of these conclusions end up being wrong, which turns out to be completely normal, because in his words “it would be a mistake to blame anyone for failing to predict in an improbable world” (Kahneman, p.315, 2021).

Due to this intrinsic need for certainty about events, human beings have seen the need to develop heuristics of thought or ideologies/beliefs that provide certainty, however, when exempted in greater detail, they lack solid support. . So, how could we be more certain about events? The answer to this question can be found in probability, which is a discipline that is typically closely related to statistics.

Although the easiest way to define probability is by describing it as the degree to which it is possible for a certain event to occur, this can be considered somewhat reductionist, so it is necessary to resort to other definitions that allow us to understand it in greater detail. this discipline.

Classic definition of probability

The classical definition states that in a sample space containing a quantity No. of simple events, an event TO can be produced from a number of no different ways; In other words, when carrying out a simulation/experiment there is No. possible outcomes of which no are favorable to the event TO. This definition is accompanied by the following formula.

One of the main criticisms of this theory is the following. When an event A occurs, it can be simple or compound, if the case presented to us corresponds to the second, then determining all the ways in which an event can occur becomes too complex. In addition to the above, the number of elements that make up the sample space can also influence the previously established elements.

Frequentist definition of probability

Previously, Kahneman’s work was mentioned and reference was made to a phrase that allows us to address the frequentist theory (“It would be a mistake to blame anyone for failing in their predictions in an improbable world”). The frequentist definition of probability arises as a consequence of the presence of random factors and elements that make it impossible to accurately determine the probability of an event. That is, when it is complex to determine how many favorable outcomes of an event exist and, consequently, how many possible outcomes can occur.

The probability from the frequentist perspective is obtained by means of the relative frequency, which is obtained in the following way.

Where k is the number of times a given phenomenon is observed, and what is the number of times that a favorable outcome occurs for the event TO. Thus, the probability of the event occurring TO is the result of the observed relative frequency when the observations grow indefinitely.

Axioms of probability

An axiom is defined as a statement that does not require proof, thus, each discipline may have different axioms associated with their fields of study. In the particular case of probability, the Russian mathematician Andrei Kolmogorov (also famous for the univariate normality test) established a series of mathematical axioms referring to this field.

Axiom 1 (absence of negativity): The probability that an event occurs will always be positive or zero; In the event that this last phenomenon occurs, it will be named impossible event.

Axiom 2 (certainty): When an event belongs to E its probability of occurrence is 1, that is, P(E) = 1. It is also called sure event.

Axiom 3 (edit): When two or more incompatible events occur (A1, A2, A3) the probability that each of these events occurs is the sum of the probability that each one occurs separately.

bayesian statistics

Proposed by Thomas Bayes, Bayesian statistics is based on Bayes’ theorem, which very briefly tells us that an event A can occur as a function of an event B. Therefore, in this paradigm statistical inferences are made from the subjective interpretation of probability; that is, based on the evidence, how likely is it that a hypothesis is fulfilled? Or, adding new evidence, how likely is it that a hypothesis is maintained or is it necessary to formulate a new one?

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