Probability Overview - The Mathematics of Chance

Probability is the branch of mathematics that gives us a way to measure and reason about uncertainty. Every day we make decisions based on what we think is likely to happen. Probability puts that everyday intuition on a firm mathematical footing.

Gambling drove the earliest serious mathematics of chance: Italian mathematician Gerolamo Cardano wrote a manuscript on games of chance as early as the 1560s, but it was not published until a century after his death, so credit for founding the field usually goes to Pascal and Fermat's 1654 letters. Their work stayed focused on gambling for decades, until the Swiss mathematician Jacob Bernoulli published the Ars Conjectandi in 1713, proving what is now called the Law of Large Numbers and showing that probability could be applied far beyond the gaming table – to insurance, population statistics, and scientific measurement. That shift, from a gambler's toy to a serious scientific tool, is exactly why probability now underpins everything from vaccine trials to spacecraft reliability testing.

What Is Probability?

Probability is a number that measures how likely an event is to occur. It always lies between 0 and 1 inclusive. A probability of 0 means the event cannot happen. A probability of 1 means it is certain to happen. Anything in between represents some degree of likelihood.

Where Is Probability Used?

Probability is not just a classroom topic. The table below shows six fields where it plays a real, practical role.

Field	How probability is used
Weather forecasting	A 70% chance of rain means P(rain) = 0.7.
Medicine	Clinical trials estimate the probability that a treatment works.
Insurance	Premiums are priced using the probability of a claim.
Finance	Risk models estimate the probability of a market move.
Games and sport	Odds on a team winning reflect probability.
Science	Quantum mechanics describes particles using probability.

A Brief History

Formal probability theory began in the 17th century. French mathematicians Blaise Pascal and Pierre de Fermat exchanged letters about gambling problems. Their correspondence laid the groundwork for the entire field. Later, Jacob Bernoulli, Abraham de Moivre, and Pierre-Simon Laplace built these early ideas into a rigorous branch of mathematics.

Two Interpretations of Probability

Type	Meaning	Example
Theoretical probability	Based on equally likely outcomes and logical reasoning.	P(heads) = 1/2 for a fair coin.
Experimental probability	Based on observed results from actual trials.	Flipping a coin 100 times and recording the results.

As the number of trials increases, experimental probability gets closer to theoretical probability. This is known as the Law of Large Numbers.

Topics in This Section

This section is split into eight short lessons. Together, they build a complete picture of how probability works.

Experiments – what a probability experiment is.
Outcomes – the possible results of an experiment.
Events – collections of outcomes we care about.
Sample Space – the full set of all possible outcomes.
Probability Scale – measuring likelihood from 0 to 1.
Independent Events – when one event does not affect another.
Conditional Probability – probability given that something has already happened.
Tree Diagrams – a visual tool for multi-step probability.

Key Takeaways

Probability measures likelihood on a scale from 0 (impossible) to 1 (certain).
Theoretical probability is calculated from reasoning; experimental probability comes from observed data.
The Law of Large Numbers tells us experimental results approach theoretical values over many trials.
Probability is used in science, medicine, finance, weather, and everyday decisions.

Practice: Theoretical vs Experimental

Experimental Probability