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Probability ⚪｜Definition｜1st｜20251122123847-00-⌔
Probability
Probability concerns events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.123 This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of either “heads” or “tails” is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.4
Printed 2026-06-28.
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Strictly speaking, a probability of 0 indicates that an event almost never takes place, whereas a probability of 1 indicates than an event almost certainly takes place. This is an important distinction when the sample space is infinite. For example, for the continuous uniform distribution on the real interval [5, 10], there are an infinite number of possible outcomes, and the probability of any given outcome being observed – for instance, exactly 7 – is 0. This means that an observation will almost surely not be exactly 7. However, it does not mean that exactly 7 is impossible. Ultimately some specific outcome (with probability 0) will be observed, and one possibility for that specific outcome is exactly 7. ↩
“Kendall’s Advanced Theory of Statistics, Volume 1: Distribution Theory”, Alan Stuart and Keith Ord, 6th ed., (2009), ISBN 978-0-534-24312-8. ↩
William Feller, An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed., (1968), Wiley, ISBN 0-471-25708-7. ↩
Probability Theory. The Britannica website. ↩
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