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probability, in mathematics, assignment of a number as a measure of the “chance” that a given event will occur. There are certain important restrictions on such a probability measure. In any experiment there are certain possible outcomes; the set of all possible outcomes is called the sample space of the experiment. To each element of the sample space (i.e., to each possible outcome) is assigned a probability measure between 0 and 1 inclusive (0 is sometimes described as corresponding to impossibility, 1 to certainty). Furthermore, the sum of the probability measures in the sample space must be 1.

Probability of Simple and Compound Events

A simple illustration of probability is given by the experiment of tossing a coin. The sample space consists of one of two outcomes—heads or tails. For a perfectly symmetrical coin, the likely assignment would be 1-2 for heads, 1-2 for tails. The probability measure of an event is sometimes defined as the ratio of the number of outcomes. Thus if weather records for July 1 over a period of 40 years show that the sun shone 32 out of 40 times on July 1, then one might assign a probability measure of 32/40 to the event that the sun shines on July 1.

Probability computed in this way is the basis of insurance calculations. If, out of a certain group of 1,000 persons who were 25 years old in 1900, 150 of them lived to be 65, then the ratio 150/1,000 is assigned as the probability that a 25-year-old person will live to be 65 (the probability of such a person's not living to be 65 is 850/1,000, since the sum of these two measures must be 1). Such a probability statement is of course true only for a group of people very similar to the original group. However, by basing such life-expectation figures on very large groups of people and by constantly revising the figures as new data are obtained, values can be found that will be valid for most large groups of people and under most conditions of life.

In addition to the probability of simple events, probabilities of compound events can be computed. If, for example, A and B represent two independent events, the probability that both A and B will occur is given by the product of their separate probabilities. The probability that either of the two events A and B will occur is given by the sum of their separate probabilities minus the probability that they will both occur. Thus if the probability that a certain man will live to be 70 is 0.5, and the probability that his wife will live to be 70 is 0.6, the probability that they will both live to be 70 is 0.5×0.6=0.3, and the probability that either the man or his wife will reach 70 is 0.5+0.6−0.3=0.8.

Permutations and Combinations

In many probability problems, sophisticated counting techniques must be used; usually this involves determining the number of permutations or combinations. The number of permutations of a set is the number of different ways in which the elements of the set can be arranged (or ordered). A set of 5 books in a row can be arranged in 120 ways, or 5×4×3×2×1=5!=120 (the symbol 5!, denoting the product of the integers from 1 to 5, is called factorial 5). If, from the five books, only three at a time are used, then the number of permutations is 60, or In general the number of permutations of n things taken r at a time is given by On the other hand, the number of combinations of 3 books that can be selected from 5 books refers simply to the number of different selections without regard to order. The number in this case is 10: In general, the number of combinations of n things taken r at a time is

Statistical Inference

The application of probability is fundamental to the building of statistical forms out of data derived from samples (see statistics). Such samples are chosen by predetermined and arbitrary selection of related variables and arbitrary selection of intervals for sampling; these establish the degree of freedom. Many courses are given in statistical method. Elementary probability considers only finite sample spaces; advanced probability by use of calculus studies infinite sample spaces. The theory of probability was first developed (c.1654) by Blaise Pascal, and its history since then involves the contributions of many of the world's great mathematicians.


See P. Billingsley, Probability and Measure (1979); I. Hacking, The Emergence of Probability (1984, rev. ed. 2006); J. T. Baskin, Probability (1986); P. Bremaud, Introduction to Probability (1988); S. M. Ross, Introduction to Probability Theory (1989).

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  1. the PROBABILITYof an event, such as the occurrence of heads or tails on the toss of a coin.
  2. social or physical outcomes which are unforseen and perhaps inherently unpredictable.
Chance arises from the existence of physical or social processes which involve random events, a multiplicity of interacting variables in ‘open systems’ (including the changeability of actors' choices), and because actors’ intentions often have UNANTICIPATED CONSEQUENCES. While an inherent CONTINGENCY in social events is seen by some theorists as ruling out general theories, this neglects the availability of generalized ‘probabilistic’ accounts and the fact that it is the goal of general theories to abstract from particular events (and provide EXPLANATION or analytical frameworks), not necessarily to predict or control events.

Coping with chance in social life is a source of MAGIC and RELIGION and the basis of important leisure forms, including games of chance and GAMBLING. See also RISK SOCIETY.

Collins Dictionary of Sociology, 3rd ed. © HarperCollins Publishers 2000


See also Fate.
Charity (See GENEROSITY.)
Bridoison, Taiel de
judge who casts dice to decide cases. [Fr. Lit.: Pantagruel]
Fata Morgana
lake-dwelling sorceress and personification of chance. [Ital. Lit.: Orlando Innamorato]
goddess of chance. [Rom. Myth.: Kravitz, 58]
Jimmy the Greek
renowned American oddsmaker. [Am. Culture: Wallechinsky, 468]
Russian roulette
suicidal gamble involving a six-shooter, loaded with one bullet. [Folklore: Payton, 590]
god of chance. [Rom. Myth.: Espy, 42–43]
Three Princes of Serendip
always make discoveries by accident. [Br. Lit.: Three Princes of Serendip]
Urim and Thummin
oracular gems used for casting lots, set in Aaron’s breastplate. [O.T.: Exodus 28:30; Leviticus 8:8]
Allusions—Cultural, Literary, Biblical, and Historical: A Thematic Dictionary. Copyright 2008 The Gale Group, Inc. All rights reserved.
References in classic literature ?
Even if they do not know for what purpose they are fattened, they will at least know that all that happened to the ram did not happen accidentally, and will no longer need the conceptions of chance or genius.
Now for my own chance." He cast the dice in his turn.
As Paulvitch forged ahead with the current he asked himself these questions, and many more beside, not the least disquieting of which were those which related to his future should it chance that the Kincaid had already steamed away, leaving him to the merciless horrors of the savage wilderness.
On the above principle, nurserymen always prefer getting seed from a large body of plants of the same variety, as the chance of intercrossing with other varieties is thus lessened.
Franklin and Miss Rachel took in each other's society, and noting what a pretty match they were in all personal respects, we naturally speculated on the chance of their putting their heads together with other objects in view besides the ornamenting of a door.
From a chance remark of the Abyssinian, Werper discovered the purpose of the expedition, and when he realized that these men were the enemies of Achmet Zek, he took heart, and immediately blamed his predicament upon the Arab.
"No matter," replied Forster; "I think that by putting on the very highest speed we might have a chance of getting over."
"You said to give him a chance. Well, give it to him.
I wish I could feel sure that you are right also in determining to try that last chance with the Count.
It was at the end of a week of this sort of waiting, roaming about in the bitter winds or loafing in saloons, that Jurgis stumbled on a chance in one of the cellars of Jones's big packing plant.
There we were all four tied by the leg and with precious little chance of ever getting out again, while we each held a secret which might have put each of us in a palace if we could only have made use of it.
"You don't mean that Frank has got another chance?" exclaimed Mr.