Probability in mathematics. Introduction and simulations

13/04/2026

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

Complementary Event

An event that occurs if and only if the original event does not; the sum of their probabilities is always equal to one.

Conditional Probability

The probability of an event A occurring given that an event B has already occurred, altering the original sample space.

Event

A subset of the sample space representing one or more specific outcomes to which a probability can be assigned.

Independent Event

An event whose probability of occurrence is not affected by the previous outcome of another distinct event.

Laplace’s Rule

A principle that defines the probability of an event as the ratio of the number of favorable cases to the total number of possible cases, assuming all are equally likely.

Law of Large Numbers

A theorem stating that as the number of trials of an experiment increases, the relative frequency of an event tends to stabilize at its theoretical probability.

Mutually Exclusive Events

A pair of events that cannot occur simultaneously in a single trial of a random experiment.

Probability

A numerical measure of the uncertainty associated with the occurrence of an event, expressed as a value between 0 (impossibility) and 1 (absolute certainty).

Random Experiment

A process or action whose exact outcome cannot be predicted with certainty before it occurs, even under the same initial conditions.

Sample Space

The set of all possible outcomes that can result from a given random experiment.

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!

Complementary Event

An event that occurs if and only if the original event does not; the sum of their probabilities is always equal to one.

Conditional Probability

The probability of an event A occurring given that an event B has already occurred, altering the original sample space.

Event

A subset of the sample space representing one or more specific outcomes to which a probability can be assigned.

Independent Event

An event whose probability of occurrence is not affected by the previous outcome of another distinct event.

Laplace’s Rule

A principle that defines the probability of an event as the ratio of the number of favorable cases to the total number of possible cases, assuming all are equally likely.

Law of Large Numbers

A theorem stating that as the number of trials of an experiment increases, the relative frequency of an event tends to stabilize at its theoretical probability.

Mutually Exclusive Events

A pair of events that cannot occur simultaneously in a single trial of a random experiment.

Probability

A numerical measure of the uncertainty associated with the occurrence of an event, expressed as a value between 0 (impossibility) and 1 (absolute certainty).

Random Experiment

A process or action whose exact outcome cannot be predicted with certainty before it occurs, even under the same initial conditions.

Sample Space

The set of all possible outcomes that can result from a given random experiment.

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Probability simulations

Normal
Plinko

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?

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!

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Andrey Kolmogorov

1903

–

1987

Andrey Kolmogorov founded modern probability theory, applying it to statistics, turbulence, and natural phenomena, establishing a solid mathematical framework for uncertainty

“Probability theory is the logic of uncertainty”

Carl Friedrich Gauss

1777

–

1855

Carl Friedrich Gauss developed fundamental methods in algebra, number theory, and statistics, consolidating the foundation of modern mathematics.

“Mathematics is the queen of the sciences and number theory its queen”

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Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

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Kurt Gödel

1906

–

1978

Kurt Gödel revolutionized modern mathematics and logic with his incompleteness theorems, proving that any consistent system contains undecidable propositions

“Only those who understand logic can discover the limits of human reaso”

Pierre-Simon Laplace

1749

–

1827

Pierre-Simon Laplace developed probability theory and mathematical statistics, and made fundamental contributions to astronomy and celestial mechanics.

“Probability theory is the science of analyzing chance”

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Test your knowledge

What is probability, and why is it essential for understanding random phenomena?

Probability is a fundamental branch of mathematics that studies the likelihood of events occurring in situations involving uncertainty. It is based on random experiments—processes whose outcomes cannot be predicted with certainty even under identical conditions. Concepts such as the sample space, which contains all possible outcomes, and events, which are subsets of that space, form the foundation of probability theory. A probability value between 0 and 1 is assigned to each event, indicating how likely it is to occur. Probability is essential in science, engineering, economics, statistics and decision‑making because it allows us to model uncertain processes, estimate risks, predict outcomes and analyze patterns. It provides the mathematical framework for understanding everything from dice rolls to complex natural and social phenomena.

What methods are used to calculate probabilities, and how do they relate to probability distributions?

Several methods exist depending on the situation. Classical probability applies when all outcomes are equally likely, such as in cards or dice. Frequentist probability relies on repeated trials and the proportion of times an event occurs, making it fundamental in empirical statistics. Conditional probability examines the likelihood of an event given that another has already occurred. These methods connect directly to probability distributions, which describe how probabilities are assigned across possible outcomes. Uniform distributions assign equal probability to all values; binomial distributions model the number of successes in repeated independent trials; and the normal distribution describes continuous phenomena concentrated around a mean. These distributions allow for prediction, data modeling and analysis under uncertainty.

What does “probability” really mean in everyday life?

It means measuring how likely something is to happen. When we say “it will probably rain” or “I have a good chance of winning”, we are talking about probability. In math, probability gives us a number between 0 and 1 to express that idea. A probability of 0 means impossible; 1 means certain. Everything else is in between. Probability helps us analyze games, understand natural events, make decisions when we don’t know the outcome and interpret scientific data.

What is the difference between classical, frequentist and conditional probability?

Classical probability is used when all outcomes are equally likely, such as drawing a card or rolling a fair die. Frequentist probability is based on repeating an experiment many times and observing how often an event occurs. If something happens 25 times out of 100, its probability is about 0.25. Conditional probability studies the likelihood of an event given that another event has already happened. For example, if you remove a blue ball from a bag, the probability of drawing a red one changes because the total number of balls has changed.

What are probability distributions used for?

They describe how probabilities are spread across the possible outcomes of an experiment. A uniform distribution is used when all outcomes are equally likely. A binomial distribution counts how many successes occur in repeated trials. The normal distribution appears in many real‑world situations such as heights, measurement errors or natural variations. Knowing these distributions helps us make predictions, analyze data and understand how random processes behave.