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## Probability theory(probabilitytheory)
Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failureto occur is random. Probabilities
The probability that an event Two crucial concepts in the theory of probability are those of a random variable and of the probability distribution of a random variable; see those articles for more information. ## A somewhat more abstract view of probability
"Pure" mathematicians usually take probability theory to be the study of probability spaces and random variables — anapproach introduced by
Andrey NikolaevichKolmogorov
in the
1930s
. A
probability space
is a triple (Ω, - Ω is a non-empty set, sometimes called the "sample space", each of whose members is thought of as a potential outcome ofa random experiment. For example, if 100 voters are to be drawn randomly from among all voters in California and asked whom theywill vote for governor, then the set of all sequences of 100 Californian voters would be the sample space Ω.
*F*is a sigma-algebra of subsets of Ω whose members arecalled "events". For example the set of all sequences of 100 Californian voters in which at least 60 will vote for Schwarzeneggeris identified with the "event" that at least 60 of the 100 chosen voters will so vote. To say that*F*is a sigma-algebranecessarily implies that the complement of any event is an event, and the union of any (finite or countably infinite) sequence ofevents is an event.
- P is a probability measure on
*F*, i.e., a measure such that P(Ω) = 1.
It is important to note that A random variable is a measurable function on Ω. For example, the number of voters who will vote for Schwarzenegger inthe aforementioned sample of 100 is a random variable.
If ## Philosophy of application of probability
Some statisticians will assign probabilities only to events that they think of as random, according to their relativefrequencies of occurrence, or to subsets of populations as proportions of the whole; those are ## See also- expectation
- likelihood
- probability
- probability axioms
- probability distribution
- random variable
- statistical independence
- variance
- List of publications instatistics
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