Probability and Statistics

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Probability and Statistics

When we talk about probability and statistics, we commonly refer to the study of chance from a mathematical perspective. That is, to the study of the formal laws that govern it, from two clearly differentiated points of view:

Probability is understood as the degree of certainty that one has regarding whether an event will occur or not, and it also constitutes a discipline responsible for creating predictive models for random phenomena, in order to anticipate them and study their logical consequences.

Statistics, on the other hand, offers its own methods and techniques to understand what these models mean, since it is an independent discipline, a branch of mathematics, focused on the study of variability.

Probability and statistics are closely linked, given that they are the two great tools available to humanity to confront random phenomena. That is, they study those whose patterns of occurrence are beyond our understanding or involve calculations that are too large and with too much margin of error to be addressed concretely.

Probability

Probability is the expression of the degree of certainty that something will occur. That is, it is the measure of the possibility of an event occurring, often expressed in percentage or fractional terms.

Thus, when studying the probability of something, we are studying the frequency with which it occurs and this is theoretically expressed through a number between 1 (total possibility of occurrence) and 0 (absolute impossibility of occurrence). Thus, depending on whether the event is more or less likely, it is expressed with a number closer to one extreme or the other.

The discipline responsible for this type of study is probability theory, a branch of mathematics. It is widely used as an auxiliary discipline by other natural and social sciences, as it allows them to handle possible scenarios based on generalizations.

The first formal considerations on this subject come from the 17th century, specifically from the correspondence between Pierre de Fermat and Blaise Pascal in 1654, or from the studies of Christiaan Huygens in 1657 and Juan Caramuel's Kybeia in 1649, a text now lost.

Types of Probability

There are different types of probability, according to the type of recurrence they express:

Frequency. Determines the number of times a phenomenon can occur, considering a given number of opportunities, through experimentation.

Mathematics. Belongs to the field of arithmetic and aims at calculating in numbers the probability of certain random events occurring, based on formal logic and not experimentation.

Binomial. Studies the success or failure of an event, or any other type of probable scenario that has only two possible outcomes.

Objective. Knows in advance the frequency of an event, and simply makes known the probable cases of said event occurring.

Subjective. Is based on certain eventualities that allow us to infer the probability of an event, although far from a certain or calculable probability. Hence its subjectivity.

Hypergeometric. It is obtained thanks to sampling techniques, creating groups of events based on their occurrence.

Logical. Its characteristic feature is that it establishes the possibility of an event occurring based on the laws of inductive logic.

Conditional. It is used to understand the causality between two different events, when the occurrence of one can be determined after the occurrence of the other.

Formula to calculate probability

Probability = Favorable cases / possible cases x 100 (to convert it to a percentage)

With this formula, you can, for example, calculate the probability of a coin coming up heads in a single toss, assuming that only one head (1) can come up out of the two available (2), that is, 1 / 2 x 100 = 50% probability (0.5)

However, if you want to calculate how many times the same head will come up in two consecutive tosses, you should consider the favorable case (heads and heads, or tails and tails) to be one in four possible outcomes (heads and heads, heads and tails, tails and heads, tails and tails). Therefore, 1/4 x 100 = 25% probability (0.25).

Applications of Probability

The calculation of probability has numerous applications in everyday life, such as:

In business risk analysis. The probability of a stock price falling is estimated, and an attempt is made to predict whether or not it is advisable to invest in a particular company.

In the statistical analysis of behavior. Important for sociology, it evaluates the possible behavior of certain sectors of the population and thus predicts trends in thought or opinion. It is commonly used in election campaigns.

In determining warranties and insurance. It studies the probability of product failure or the reliability of a service (or an insured party, for example) to determine how long a warranty should be offered, or who should be insured and for how much.

In locating subatomic particles. It uses the Heisenberg Uncertainty Principle, which states that we cannot know where a subatomic particle is at a given time and at the same time how fast it is moving. Calculations in the field are usually done in probabilistic terms: there is an X percent chance that the particle is there.

In biomedical research. The success and failure rates of medical drugs or vaccines are calculated to determine whether they are reliable or not, whether they should be mass-produced, or what percentage of the population they might experience certain side effects.

In weather forecasting. By studying atmospheric conditions over the past few days, meteorologists calculate the probability of a phenomenon occurring.

Examples of Probability

Financial speculation involving the buying and selling of stocks on the stock market, depending on the probability that they will rise or fall.

Weather forecasts on smartphone apps, which indicate the percentage chance of rain.

Statistics

Statistics is a formal and deductive scientific discipline, often considered a branch of mathematics, that studies variability and the laws of probability, using diverse tools, both conceptual and sampling.

The field of statistics encompasses the methods and procedures necessary to collect information from reality and organize, contextualize, and classify it in order to obtain viable conclusions, expressed mathematically. It can be said that it is the science of data management.

Thus, statistics contemplates four levels of data measurement, known as statistical measurement scales, which are:

Nominal: Describes variables whose difference between them lies more in quality than in quantity.

Ordinal: Describes variables on a continuum in which their values can be ordered, that is, assigning a hierarchy or order to the data.

Interval: Describes variables whose values establish recognizable intervals.

Rational: Describes variables with equal intervals and that allow an absolute zero to be located, such that it represents the absence of characteristics.

Although statistics constitutes a field of study in itself, it is characterized by its transversal nature, that is, by serving as a tool for many other disciplines and sciences, regardless of their specific fields of knowledge: biology, economics, demography, etc.

Branches of Statistics

Descriptive statistics is dedicated to the visualization, classification, and numerical or graphical presentation of data arising during the study. Its objective is to facilitate the handling of large volumes of data, such as those found in population pyramids, histograms, or pie charts.

Inferential statistics is dedicated to generating models and predictions from the studied phenomena, taking into account their random dynamics. Through these mathematical models, it aims to arrive at useful conclusions or predictions that transcend the merely descriptive realm.

Importance of Statistics

Statistics has immense relevance in the modern world, which transcends the specific needs of states to organize their populations.

The latter, however, linked to control and decision-making, as well as the implementation of public policies, are fundamental subjects for approaching the thinking and way of life of populations.

However, statistics also serves as an information processing tool for numerous disciplines, both in the natural sciences and the social sciences, since it allows the collection of information regarding Objects of any nature.

Activity. Answer the following questions.

Remember to review the answers to the open-ended questions at the bottom of this page.

1. What does the probability of an event represent?

2. What branch of statistics is responsible for the visualization and presentation of data?

3. Which of the following examples applies probability calculus to everyday life?

Once you click this button, the questions will close and you will not be able to change your answer.

4. There are 3 red balls and 7 blue balls in a bag. What is the probability of randomly picking a red ball?

5. A die has 6 faces, numbered 1 to 6. What is the probability of rolling an even number?

6. There is an urn with 4 red balls, 5 green balls, and 6 blue balls. If a ball is drawn at random, what is the probability that it is green?

7. What is inferential statistics and how is it used to make predictions?

8. In a standard deck of cards (52 cards), what is the probability of drawing a card that is a king?

9. If we roll two dice, what is the probability that the sum of the two dice is 7?

10. How can statistics be used to improve decisions in business?

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Answers to open-ended questions:

4. Probability = Favorable cases / Possible cases = 3 / (3 + 7) = 3/10 = 0.3 or 30%

6. Probability = Favorable cases / Possible cases = 5 / (4 + 5 + 6) = 5 / 15 = 1/3 ≈ 0.33 or 33%

7. Inferential statistics uses mathematical models to make predictions about random phenomena based on data samples. From these models, conclusions can be generalized beyond the observed data, allowing estimates and forecasts to be made about the entire population.

9. Favorable cases: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) → 6 favorable combinations.

10.Statistics can help businesses make informed decisions by analyzing historical data, such as sales, costs, and market trends. Using data analysis techniques and predictive models, businesses can anticipate product demand, identify investment opportunities, optimize pricing, and manage inventory, among other things.

2. Equipo editorial, Etecé. (2025, 10 marzo). Probabilidad - Concepto, tipos, fórmula, usos y ejemplos. Concepto. https://concepto.de/probabilidad/

3. Equipo editorial, Etecé. (2024b, noviembre 15). Estadística - Concepto, niveles, historia, importancia y ramas. Concepto. https://concepto.de/estadistica/

5. Anderson, R, D., Williams, A, T., Sweeney, & J, D. (2025, 27 marzo). Statistics | Definition, Types, & Importance. Encyclopedia Britannica. https://www.britannica.com/science/statistics