Probability theory

Probability theory is a branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables and stochastic processes: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. As a mathematical foundation for statistics, probability theory is essential to most human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to description of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at microscopic scales, described in quantum mechanics.

The mathematical theory of probability has its roots in attempts to analyse games of chance by Pierre de Fermat and Blaise Pascal in the seventeenth century. Although an individual coin toss or the roll of a die is random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.

Initially, probability theory mainly considered discrete events, and its methods were very combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, the foundations of which were laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of probability space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory 1933. Fairly quickly this became the undisputed axiomatic basis for modern probability theory.

Representation and interpretation of probability values

The probability of an event is generally represented as a real number between 0 and 1, inclusive. The closer an event's probability is to 1, the more likely it is to occur. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. (The subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".)

For example, if two mutually exclusive events are assumed equally probable, such as a flipped or spun coin landing heads-up or tails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2". A formal and more accurate representation would be p=.5.

Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events. The odds of heads-up, for the tossed/spun coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1".

Odds a:b for some event are equivalent to probability a/(a+b). For example, 1:1 odds are equivalent to probability 1/2, and 3:2 odds are equivalent to probability 3/5.

There remains the question of exactly what can be assigned probability, and how the numbers so assigned can be used; this is the question of probability interpretations. There are some who claim that probability can be assigned to any kind of an uncertain logical proposition; this is the Bayesian interpretation. There are others who argue that probability is properly applied only to random events as outcomes of some specified random experiment, for example sampling from a population; this is the frequentist interpretation. There are several other interpretations which are variations on one or the other of those, or which have less acceptance at present.

Distributions

A probability distribution is a function that assigns probabilities to events or propositions. For any set of events or propositions there are many ways to assign probabilities, so the choice of one distribution or another is equivalent to making different assumptions about the events or propositions in question.

There are several equivalent ways to specify a probability distribution. Perhaps the most common is to specify a probability density function. Then the probability of an event or proposition is obtained by integrating the density function. The distribution function may also be specified directly. In one dimension, the distribution function is called the cumulative distribution function. Probability distributions can also be specified via moments or the characteristic function, or in still other ways.

A distribution is called a discrete distribution if it is defined on a countable, discrete set, such as a subset of the integers. A distribution is called a continuous distribution if it has a continuous distribution function, such as a polynomial or exponential function. Most distributions of practical importance are either discrete or continuous, but there are examples of distributions which are neither.

Important discrete distributions include the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution, and the Maxwell-Boltzmann distribution.

Important continuous distributions include the normal distribution, the gamma distribution, the Student's t-distribution, and the exponential distribution.

Probability in mathematics

To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor. Certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible, for example, to flip 10 heads in a row. What then does the number "50%" mean in this context?

One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent—that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let N_H be the number of times the coin lands heads, then we can, for any N, consider the ratio $N_{H} \over N$.

As N gets larger and larger, we expect that in our example the ratio $N_{H} \over N$ will get closer and closer to 1/2. This allows us to "define" the probability $\Pr(H)$ of flipping heads as the limit, as N approaches infinity, of this sequence of ratios:

$\Pr(H)=\lim _{N\to \infty }{N_{H} \over N}$

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,

$\left|\Pr(H)-{N_{H} \over N}\right|<\epsilon$

In other words, by saying that "the probability of heads is 1/2", we mean that if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitrarily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.

Note that a proper definition requires measure theory, which provides means to cancel out those cases where the above limit does not provide the "right" result (or is even undefined) by showing that those cases have a measure of zero.

The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz & Guildenstern Are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event—after all, it is possible (although unlikely) that a fair coin would give this result—or whether his assumption that the coin is fair is at fault.

See also

• glossary of probability and statistics
• list of probability topics
• list of statistical topics
• list of publications in statistics
• probabilistic logic
• predictive modelling
• fuzzy measure theory
• expected value
• likelihood function
• variance
• statistical independence
• notation in probability

Bibliography

• Pierre Simon de Laplace (1812) Analytical Theory of Probability

The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.

• Andrei Nikolajevich Kolmogorov (1950) Foundations of the Theory of Probability

The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933.

• Harold Jeffreys (1939) The Theory of Probability

An empiricist, Bayesian approach to the foundations of probability theory.

• Edward Nelson (1987) Radically Elementary Probability Theory

Discrete foundations of probability theory, based on nonstandard analysis and internal set theory. downloadable. http://www.math.princeton.edu/~nelson/books.html

• Patrick Billingsley: Probability and Measure, John Wiley and Sons, New York, Toronto, London, 1979.

• Henk Tijms (2004) Understanding Probability

A lively introduction to probability theory for the beginner, Cambridge Univ. Press.

• Gut, Allan (2005). Probability: A Graduate Course. Springer. ISBN 0387228330.

External links

• Analytical argumentations of probability and statistics.
• Probability Surveys Expertly written surveys of nearly all branches of contemporary probability theory are available online, free of charge.