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Probability theory
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Probability theory

Probability theory is the mathematical study of probability.

Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occur is random. Probabilities are assigned to events according to the probability axioms.

The probability that an event occurs given the known occurrence of an event is the conditional probability of given ; its numerical value is (as long as is nonzero). If the conditional probability of given is the same as the ("unconditional") probability of , then and are said to be independent events. That this relation between and is symmetric may be seen more readily by realizing that it is the same as saying .

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.

Table of contents
1 A somewhat more abstract view of probability
2 Philosophy of application of probability
3 See also

A somewhat more abstract view of probability

"Pure" mathematicians usually take probability theory to be the study of probability spaces and random variables — an approach introduced by Andrey Nikolaevich Kolmogorov in the 1930s. A probability space is a triple (Ω, F, P), where

It is important to note that P is defined on F and not on Ω. With Ω denumerable we can define F := powerset(Ω) which is trivially a sigma-algebra and the biggest one we can create using Ω. In a discrete space we can therefore omit F and just write (Ω, P) to define it. If on the other hand Ω is non-denumerable and we use F = powerset(Ω) we get into trouble defining our probability measure P because F is too 'huge'. So we have to use a smaller sigma-algebra F (eg. the Borel algebra of Ω). We call this sort of probability space a continuous probability space and are led to questions in measure theory when we try to define P.

A random variable is a measurable function on Ω. For example, the number of voters who will vote for Schwarzenegger in the aforementioned sample of 100 is a random variable.

If X is any random variable, the notation P(X ≥ 60) is shorthand for P({ ω in Ω : X(ω) ≥ 60 }), so that "X ≥ 60" is an "event".

Philosophy of application of probability

Some statisticians will assign probabilities only to events that they think of as random, according to their relative frequencies of occurrence, or to subsets of populations as proportions of the whole; those are frequentists. Others assign probabilities to propositions that are uncertain according either to subjective degrees of belief in their truth, or to logically justifiable degrees of belief in their truth. Such persons are Bayesians. A Bayesian may assign a probability to the proposition that there was life on Mars a billion years ago, since that is uncertain; a frequentist would not assign such a probability, since it is not a random event that has a long-run relative frequency of occurrence.

See also