Probability in mathematics. Introduction and simulations

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!