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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 ?
"Chance created the situation; genius utilized it," says history.
Thoughts of the hideous pack which tenanted the ship induced cold tremors along the spine of the cowardly prowler; but life itself depended upon the success of his venture, and so he was enabled to steel himself to the frightful chances which lay before him.
We do not know whether he lived long enough for a chance of that promotion whose way was so arduous.
His own life depended upon the chance of his eluding, or outdistancing Achmet Zek, when that worthy should have discovered that he had escaped.
"Eighty chances!" replied the passenger, turning his back on him.
He has hardly had time to throw away that chance you gave him before this letter comes, and puts the ball at his foot for the second time.
Captain Ebsworth says, and I say, let chance decide it.
For such a pretty maid as 'tis, this is a fine chance!"
Cirripedes long appeared to me to present a case of very great difficulty under this point of view; but I have been enabled, by a fortunate chance, elsewhere to prove that two individuals, though both are self-fertilising hermaphrodites, do sometimes cross.
If he can't, we shall have another chance of catching them in the shrubbery, before many more nights are over our heads."
Every one breakfasted at a different hour in the Red House, and the Squire was always the latest, giving a long chance to a rather feeble morning appetite before he tried it.
"We can't all be Stanleys and Burtons," said I; "besides, we don't get the chance,--at least, I never had the chance.