Probability in mathematics. Introduction and simulations
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
This Thematic Unit is part of our Mathematics collection
STEM OnLine mini dictionary
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!
STEM OnLine mini dictionary
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 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
–
Joseph Fourier
–
Become a giant
Probability and Statistics in Data Science using Python
Introduction to Probability
Fundamentals of Statistics
Fat Chance: Probability from the Ground Up
Introduction to Algebra
Pre-University Calculus
MathTrackX: Special Functions
Polynomials, Functions and Graphs
Professional development for Educators
Reimagining higher education teaching in the age of AI
An Introduction to Evidence-Based Undergraduate STEM Teaching
Teach computing: Physical computing with Raspberry Pi and Python
Innovating Instruction: Learning Design in the STEM Classroom
Giants of science
“If I have seen further, it is by standing on the shoulders of giants”
Isaac Newton
Andrey Kolmogorov
–
Carl Friedrich Gauss
–
Become a giant
Probability and Statistics in Data Science using Python
Introduction to Probability
Fundamentals of Statistics
Fat Chance: Probability from the Ground Up
MathTrackX: Special Functions
Introduction to Algebra
How to Learn Math: For Students
Maths Essentials
Professional development for Educators
Teaching Science and Engineering
Teach computing: Introducing physical computing
BlendedX: Blended Learning with edX
Advancing Learning Through Evidence-Based STEM Teaching
Test your knowledge
What is probability, and why is it essential for understanding random phenomena?
What methods are used to calculate probabilities, and how do they relate to probability distributions?
What does “probability” really mean in everyday life?
What is the difference between classical, frequentist and conditional probability?
What are probability distributions used for?
You may also be interested
2026 Best Online STEM Education Platform
Last simulations
To learn and experience
Take your knowledge to the next level with science kits and hands-on tools that connect theory with experimentation
Desk Galton Board
Teachers essentials
Notion
Organize your notes, tasks, and projects in one place. Notion combines notes, lists, and calendars in one flexible space.
Canva
Design presentations, diagrams, and infographics with easy-to-use templates and professional results
Genially
Create interactive content, presentations, and educational games in a simple and visual way
Desmos
The most intuitive graphing calculator to explore mathematics and create dynamic classroom activities
Tinkercad
Design in 3D and simulate electronic circuits for free; it’s the perfect tool for simple robotics projects
Your Infinite Library
Enjoy the freedom to explore millions of titles and a curated selection of magazines on any device
Join Prime for students and young adults
The best series. Millions of songs. All included with your subscription.
Science made fun
Learn with the best courses
edX
Online courses from leading universities and companies. Learn practical skills and earn professional certifications
Coursera
Courses from leading universities and companies. Learn practical skills and earn professional certifications
Udemy
Thousands of online courses tailored to your level. Learn at your own pace with expert instructors