Probability

The likelihood ( probability ) is a classification of statements and judgments on the degree of certainty ( security). Special importance is the certainty of predictions. In mathematics, probability theory has come up with its own field right, the probabilities as mathematical objects describes their formal qualities in everyday life and philosophy be applied to statements and judgments.

Probability concepts

We distinguish different conceptions of probability (probability terms ).

Symmetry principle - classical or Laplace considers

The probability of an event is the ratio of the beneficial results to the total amount of the results. For example, the probability of a six dice throw an odd number, 0.5. This corresponds to a relative frequency of 50 %, because there are six possible outcomes, three of which have the mentioned property.

This is the so -called classical definition, as developed by Christiaan Huygens and Jacob I. Bernoulli and formulated by Laplace. It is the basis of classical probability theory. The elementary events have the same probability of occurrence. Prerequisite is a finite result set and knowledge of a priori probabilities.

Example: In a "fair" dice ( ie no result is due to asymmetric mass distribution or similar preferred) thinks to oneself that every number has an equal chance and will therefore appear in 1/ 6 of all attempts. The probability of the event " even number " is calculated as follows: There are three beneficial results (2, 4, 6), but six possible outcomes, therefore, gives 3 /6 = 0.5 as a result.

Objectivist concept of probability

Frequency principle - Statistical probability of conception

A random experiment is repeated as many times as possible, then the relative frequencies of the respective elementary events are calculated. The probability of an event is now the limit of its relative frequency in (theoretically) infinite number of repetitions. This is the so -called limit - definition ' according to von Mises. The law of large numbers plays a central role here. Prerequisite is the random repeatability of the experiment; the individual passages must be independent of each other. Another name for this concept is the frequentist concept of probability. This concept of probability is meant, for example, in physics at the probability of decay of a radionuclide; the experiments here are the individual, independent decays of atomic nuclei.

Example: You roll the dice 1,000 times, with the following distribution: The one falls 100 times (this corresponds to a relative frequency of 10%) that falls 2 150 times ( 15%), the 3 is also 150 times (15%), 4 20 %, the 5 in 30% and 6 to 10 % of cases. The suspicion is that the die is not fair. After 10,000 rounds, the numbers in the specified values ​​have stabilized, so that one can say with some certainty that, for example, the probability of rolling a 3 is 15%.

Propensitätstheorie

The Propensitätstheorie interprets probability as a measure of the tendency of a process to a particular result.

Quantum mechanical probability of conception

In nonrelativistic quantum mechanics, the wave function of a particle is used as its fundamental description. The integral of the absolute square of the wave function over a region of space there corresponds the probability of finding the particle in it. So this is not a merely statistical, but a non- deterministic probability.

Subjectivist conception probability

With unique random events one can only estimate their probability of occurrence, not calculated. Central factors are here expertise, experience and intuition. Therefore, one speaks of a subjectivist conception probability, see also Bayesian concept of probability.

Example: After someone has owned several cars, he estimates the probability as high (for example, "I am 80% confident " ) to be with the brand XY also next car again satisfied. This predictive value can be changed, for example, a review up or down.

Axiomatic definition of probability

• Axiomatic definition of probability by Kolmogorov - the now authoritative for mathematics definition, see axioms of Kolmogorov.

Stochastics

Stochastics as a branch of mathematics is the study of the frequency and probability. It is a relatively young branch of mathematics to which in a broader sense combinatorics, probability theory and mathematical statistics include.

Often the mathematical concept of probability is used: the probability theory or probability theory ( branch of stochastics ) cares about the systematization of mathematical probabilities. Here probability distribution probability function, conditional probability, and many other terms are distinguished.

Probabilities are numbers between 0 and 1, with zero and one values ​​permissible. An impossible event is 0, the probability assigned to a certain event, the probability 1 The reversal it is true, but only if the total number of events is at most countably infinite. In " uncountably infinite" probability spaces, an event with probability 0 occur, it is then called almost impossible, an event with probability 1 fail to occur, then it means almost certain.

Psychology - Estimating probabilities

It is often asserted that man possesses a bad feeling for the probability, one speaks in this context of the " probability of idiots " (see also Zahlenanalphabetismus ). The following examples:

• The birthday paradox: on a football field are 23 people (twice eleven players and a referee ). The probability that this includes at least two people have the same birthday is greater than 50%.
• You have attended a screening and get a positive result. You know also that you have no special risk factors for the diagnosed disease compared to the total population: Using the methods of calculation of the conditional probability can be estimated the actual risk that the diagnosis created by the test actually true. Here are two items of particular importance to determine the risk of a false positive result: the reliability (selectivity and specificity) of the test and the observed fundamental frequency of that disease in the general population. Can help to balance the sense of further (perhaps more consequential ) treatments to know this actual risk. In such cases, the representation of the absolute frequency at the full decision tree and a subsequent anabolic consultation with the doctor gives a better impression comprehensive union than the mere interpretation of percentages due to the observed isolated test result.

Philosophy - understanding of probability

While rules the mathematical handling of probabilities broad consensus (see probability theory ), there is disagreement as to what the rules of calculation of the mathematical theory may be applied. This leads to the question of the interpretation of the term " probability ".

Frequently " probability " in two different contexts is needed:

Aleatory and epistemic probability are loosely associated with the frequentist and the Bayesian concept of probability.

It is an open question whether aleatory probability can be reduced to epistemic probability ( or vice versa): Appears the world to us by chance, because we do not know enough about them, or is there fundamentally random processes, such as the objective interpretation of quantum mechanics takes? Although the same mathematical rules apply to both points of view to deal with probabilities that each view has important consequences for which mathematical models are considered valid.