Stochastic ( from Ancient Greek στοχαστικὴ τέχνη stochastikē techne, Latin ars conjectandi, so art of conjecture ',' rate 'art' ) is a branch of mathematics and understood as a generic term, the fields of Probability theory and statistics together.

The historical aspects are presented in the article history of probability theory.


Mathematical Stochastics is concerned with the description and analysis of random experiments such as the throwing of thumbtacks, dice or coin toss and influenced by random temporal trends and spatial structures.

Such events, developments and structures are often documented by data statistics provides appropriate methods for their analysis. With the help of stochastics can be approximately calculated the probability of lottery winnings or the size of the uncertainty in opinion polls determine. The stochastics is also for the financial mathematics of interest and helps with their methodology, for example, in the pricing of options.

Probabilities and random experiments

Under a forecast refers to:

  • A measure of the uncertainty of future events,
  • A measure of the degree of personal conviction ( Bayesian concept of probability ), and ultimately an extension of propositional logic.

Specification of probabilities

Probabilities with the letter (of French probabilité, introduced by Laplace ) or shown. They do not carry unit, but are numbers between zero and one, where zero and one also permissible probabilities are. Therefore, they can as percentages (20%), decimals (), fractures ( ), quotas ( 2 of 10 and 1 out of 5) or relative numbers (1 to 4 ) can be specified (all figures describe the same probability).

Are frequent misunderstandings, if not properly distinguish between " to" and "from ": "1 to 4" means that the one desired event 4 undesired events are facing. Thus there are 5 events, one of which is the desired, ie " 1 of 5".

When you run an random experiment several times in succession, the relative frequency of an event can be calculated by dividing the absolute frequency, ie the number of successful attempts by the number of attempts made. An infinite number of attempts, this relative frequency goes to the probability. In practice, the number of necessary for an acceptable match of relative frequency and probability experiments is often underestimated.

Probabilities of zero and one ↔ impossible and safe events

The fact that an event the probability is assigned to zero, ie only that its admission is fundamentally impossible if there are only finitely many different experimental outputs.

This is illustrated by the following example: In a random experiment any real number between 0 and 1 is drawn. It is assumed that each number is equal to occur - it is thus provided that the uniform distribution on the interval. Then, since there are infinitely many integers in the interval, for each individual number from the interval, the probability is equal to zero, yet from each number draw result as possible.

An impossible event is in the context of this example, as the contraction of the two, ie the elementary event.

An event is called safe if it has a probability of 1. The probability that an impossible event does not occur, is 1 and is a secure event. An example of a certain event in a dice game with a six-sided die is the event " there will be no seven cubed ".

Constraints, axioms

Basic assumptions of the stochastics are described in the Kolmogorov axioms by Andrei Kolmogorov. For these and their implications can be concluded that:

The probability of the event that includes all possible test outputs is:

The probability of an impossible event is:

All probabilities are between zero and one, including:

The probability of occurrence of an event and its non-occurrence add up to one:

In a complete system of events ( this must all be pairwise disjoint and their union be equal to) the sum of the probabilities is equal to:

Laplace experiments

As Laplace experiments, named after the mathematician Pierre -Simon Laplace, random experiments are referred to, for the following two points are met:

  • There are only finitely many possible experimental outputs.
  • All possible outcomes are equally likely.

Simple examples of Laplace experiments are dicing a cube, tossing a coin ( leaving aside the fact that they stand on the edge can stay ) and the lottery numbers.

The probability of Laplace experiment is calculated according to

Probability Theory

  • Random variable
  • Probability distributions ( uniform distribution, normal distribution, exponential distribution, binomial distribution, Bernoulli distribution, hypergeometric distribution, Poisson distribution, mixing distribution)
  • Probability density distribution function


Combinatorics is a branch of mathematics, which of themselves with determining the number of possible arrangements or selections

  • Distinguishable or indistinguishable objects
  • With or without regard to the order

Employed. In the modern Combinatorics, these problems are reformulated as pictures, so that the task of combinatorics may substantially limit itself to include these pictures.

Game theory

Game theory is a branch of mathematics which is concerned with systems with multiple stakeholders ( players, agents ) to analyze. Game theory attempts inter alia, derive the rational choice behavior in situations of social conflict.


Statistics are based on mathematical methodology for analyzing quantitative data. It combines empirical data with theoretical models.

Other terms from the stochastics, examples

  • Accident
  • Law of large numbers
  • Law of small numbers
  • Probability
  • Conditional probability
  • Expected value
  • Stochastic independence
  • Stochastic process
  • Markov chain

See also, application examples

  • Division problem
  • Goat problem, known as the "three -door problem."
  • Formulary Stochastics