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Probability in mathematics. Introduction and simulations

What do you know about mathematical probability, its calculation and its applications?

The online probability simulations on this page will help you to better understand what is probability in mathematics. We will learn some basic concepts such as random experiments or the sample space and we will be introduced to the calculation of events or the different types of probability distributions

What is probability in mathematics

Probability in mathematics is a fundamental branch of mathematics that is used to study and measure the possibility of a particular event occurring. It is based on the analysis of random situations and helps us to make informed decisions and predict outcomes.

Random experiments, sample space and events

A random experiment is one whose outcome cannot be predicted with certainty, even if repeated under the same conditions. The set of all possible outcomes of a random experiment is known as the sample space, and each of these possible outcomes is called an elementary event. An event, in probabilistic terms, is any subset of the sample space; for example, when rolling a die, obtaining an even number is an event that groups the results 2, 4 and 6. Understanding these concepts is key to analyze situations of uncertainty, calculate probabilities and establish mathematical models that allow us to make informed decisions based on the behavior of random phenomena.

Calculating the probability of an event

Probability is expressed numerically between 0 and 1, where 0 means that the event is impossible and 1 means that it is certain to occur. For example, if we flip a coin, the probability of getting heads is 1/2, since there are two possible outcomes (heads or tails) and only one of them is the one we are looking for. There are different methods to calculate the probability of an event, depending on the type of experiment or situation:

Classical probability

It is applied when all possible outcomes are equally probable. For example, in a standard deck of cards, the probability of drawing an ace is 4/52, since there are 4 aces in a total of 52 cards.

Frequential probability

It is based on repeated observation of an experiment. For example, if we throw a die 100 times and get a 3 on 20 occasions, the estimated probability of getting a 3 is 20/100, i.e. 1/5.

Conditional probability

It is used when the probability of an event depends on another event having already occurred. It is expressed as P(A|B), the probability of A occurring given that B has already occurred. For example, if there is a bag with 5 red balls and 3 blue balls, the probability of drawing a red ball given that a blue ball has already been drawn is 5/8.

Probability distributions

Probability distributions describe how the probabilities of the different possible outcomes of a random experiment are distributed. In other words, a probability distribution assigns each value or set of values the probability of occurrence. The most important types of distributions include the following

Uniform distribution

All outcomes have the same probability.

Binomial distribution

Models the number of successes in a series of independent trials with two possible outcomes.

Normal or Gaussian bell distribution

Fundamental in statistics and appears frequently in natural and social phenomena because of its symmetric shape and concentration around the mean.

These distributions allow analysis and prediction of the behavior of data under conditions of uncertainty.

Probability applications

Probability is applied in many areas of life, such as gambling, finance, statistics, decision making and science in general. It allows us to assess risks, analyze data, predict outcomes and make informed decisions based on available information.

The online probability simulations on this page are a great help to master this important part of mathematics – use them and you won’t regret it!

Probability simulations

Normal distribution


The normal distribution, also known as Gaussian bell, is one of the most important probability distributions in statistics. It is characterized by its symmetric, bell shape, where most of the data are clustered around the mean and the probability decreases as we move away from it. This distribution appears naturally in many phenomena, such as the height of people or measurement errors, and is fundamental for data analysis and decision making in contexts of uncertainty. This simulation is a practical example of the normal distribution. There are 10 black stones and 10 white stones in your pocket. What is the distribution of the number of black stones when I take out 10 randomly?
Click here to start the simulation

Plinko probability


Suelta pelotas a través de una malla triangular con estacas y ve cómo se acumulan en contenedores. Cambia a una vista de histograma y compara la distribución de pelotas para una distribución binomial perfecta. ¡Ajusta la probabilidad binomial y desarrolla tu conocimiento en estadística!


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