Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion.

Overview

Probability theory is used to describe phenomena that occur with uncertainty. It provides a means to quantify the likelihood of a set of outcomes. Probability is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The theory of probability is an essential part of various fields including mathematics, statistics, finance, gambling, science, artificial intelligence, and philosophy.

History

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century. The development of the theory was furthered by contributions from many fields, including those of Christiaan Huygens, Jacob Bernoulli, and Pierre-Simon Laplace.

Mathematical Description

Probability theory is generally considered to be founded on the axiomatic approach developed by Andrey Kolmogorov in the 1930s. Kolmogorov's axioms are based on assigning a probability measure to a set of outcomes and require the probability of the entire sample space to be 1.

Random Variables

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables, discrete and continuous. A discrete random variable is one that has a countable number of possible values, while a continuous random variable is one that has an infinite number of possible values.

Probability Distributions

A probability distribution assigns a probability to each measurable subset of the possible outcomes of a random variable. Important examples of probability distributions include the Binomial distribution, the Normal distribution, and the Poisson distribution.

Stochastic Processes

A stochastic process is a collection of random variables representing the evolution of some system of random values over time. Examples include the Random walk, Markov chain, and Brownian motion.

Applications

Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis, and financial regulation.

See Also

• Statistics
• Game theory
• Decision theory