Probability

Probability is the math of things you cannot be sure about, which is nearly everything, including whether you will need an umbrella.

By Greg Goodwin · Published October 24, 2026

Cheat Sheet

Scale: every probability is a number from 0 (impossible) to 1 (certain), often written as a percentage.
Two dice: the chance of rolling a total of 7 is 6 out of 36, or 1 in 6.
Independent events: multiply their probabilities, so two heads in a row from a fair coin is 1/2 x 1/2 = 1/4.
At least one: the chance of at least one 6 in four rolls of a die is 1 - (5/6)^4, about 51.8 percent.
Birthday problem: in a group of just 23 people, the chance that two share a birthday is just over 50 percent.
Kolmogorov (1933): modern probability rests on three axioms published in his monograph Grundbegriffe der Wahrscheinlichkeitsrechnung.

The 60-Second Version

Probability is the branch of mathematics that measures how likely things are. Every event is given a number between 0 and 1: 0 means it cannot happen, 1 means it surely will, and a fair coin landing heads sits at one half. To find the probability of an event, you usually count the outcomes you want and divide by the total number of equally likely outcomes. Rules then let you combine events: for independent events you multiply, and for mutually exclusive ones you add. The subject began with gamblers' puzzles and grew into a foundation for statistics, insurance, medicine, physics and machine learning. Its biggest lesson is that human intuition about chance is unreliable, which is why a short list of rules beats a gut feeling. Probability describes what happens across many trials, and it makes no promise about the next single one.

The Long Version

From Dice Games to a Discipline

Probability grew out of games of chance. Sixteenth-century Italians such as Cardano wrote about gambling, but many historians date the birth of the formal study to 1654, when Blaise Pascal and Pierre de Fermat exchanged letters about the problem of points: how to fairly divide the stakes of a game interrupted before its end. Their solutions treated future possibilities as counted cases rather than guesses. Christiaan Huygens learned of the correspondence and published a treatise in 1657, which helped spread the ideas across Europe.

Counting and Combining

The core method is simple. List every outcome of a process, and for equally likely outcomes, the probability of an event is the number of favorable outcomes divided by the total. Two dice have 36 combinations, and six of them sum to 7, so the chance of a 7 is 1 in 6. When events are independent, such as successive coin flips, multiply: two heads is 1/4. For questions like at least one 6 in four rolls, it is easier to find the chance of no sixes, (5/6)^4, and subtract from 1, giving about 51.8 percent. Complements are one of probability's best tricks.

When Intuition Fails

People are poor judges of chance. The birthday problem is the classic demonstration: with only 23 people in a room, the chance that two share a birthday is just over 50 percent, far higher than most people guess. The gambler's fallacy is the belief that after a streak of tails, heads is due; in fact a fair coin has no memory. Conditional probability produces more surprises, as in medical tests where a rare disease and an imperfect test combine to make a positive result less alarming than it feels. Learning the rules is a defense against these traps.

Laws, Axioms, and Foundations

Jakob Bernoulli's Ars Conjectandi, published after his death in 1713, laid out the law of large numbers: as trials accumulate, observed frequencies settle toward the true probability. Laplace's 1812 work organized the field further. Yet the subject lacked a rigorous foundation until Andrey Kolmogorov's 1933 monograph, which grounded probability in measure theory and reduced it to three axioms: probabilities are never negative, the probability of the whole sample space is 1, and probabilities of disjoint events add. Everything else in the theory follows from those rules.

Where It Shows Up

Probability is everywhere in modern life. Insurers price policies by estimating risks, doctors weigh test results, meteorologists issue a 30 percent chance of rain, and engineers calculate failure rates. Statistics uses probability to decide whether a result is meaningful or a fluke, and machine learning systems use it to make predictions under uncertainty. Bayes' theorem, published after Thomas Bayes' death in 1763, lets you update a belief as new evidence arrives, an idea now central to modern data science. Learning the basics gives you a better filter for headlines, risks and sales pitches.

Why People Care

Probability turns uncertainty into numbers you can reason with, which matters for everything from medical decisions to games to news about risk. A little training protects you from classic mistakes, like believing a streak is due to end.

Glossary

Sample space: The set of all possible outcomes of an experiment, such as the six faces of a die.
Event: A set of outcomes you care about, like rolling an even number.
Independence: Two events are independent when knowing one happened tells you nothing about the other.
Conditional probability: The probability of an event given that another has already occurred.
Expected value: The long-run average result of a random process, found by weighting each outcome by its probability.

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