# probability

The probability is a classification of statements and judgments according to the degree of certainty (certainty). The certainty of predictions is of particular importance . In mathematics, probability theory has developed its own specialty that describes probabilities as mathematical objects whose formal properties are also transferred to statements and judgments in everyday life and philosophy.

The probability that a certain result will occur in an individual case in a repeatable test or process (a random experiment ) is the numerical ratio of these "favorable" results to the possible results at all. The different definitions of probability ( conceptions of probability) differ in how one gets the numerical value of the probability.

### Principle of symmetry - classic, Laplace view

In the case of an ideal, "fair" dice (that is, no result is preferred due to asymmetrical mass distribution or the like), because of the symmetry, each of the six sides has the same "chance" from the outset ( a priori probability ) of being up after the throw lie. So, for example, the probability of throwing an odd number is 3/6 = 0.5 because there are three favorable outcomes (1, 3, 5) but six possible outcomes.

This is the so-called classical definition as it was developed by Christiaan Huygens and Jakob I Bernoulli and formulated by Laplace . It is the basis of classical probability theory . The elementary events have this same a priori probability .

### Objectivistic concept of probability

#### Frequency principle - statistical conception of probability

The experiment is repeated many times, then the relative frequencies of the respective elementary events are calculated. The probability of an event is now the limit to which the relative frequency tends towards an infinite number of repetitions. This is the so-called “Limes definition” according to von Mises . The law of large numbers plays a central role here. Prerequisites are that the experiment can be repeated as required and that the individual runs are independent of one another. Another name for this concept is Frequentistic Probability . This concept of probability is meant, for example, in physics for the decay probability of a radionuclide ; the experiments here are the individual, mutually independent decays of the atomic nuclei.

Example: You roll the die 1000 times and get the following distribution: The 1 falls 100 times (this corresponds to a relative frequency of 10%), the 2 falls 150 times (15%), the 3 also 150 times (15%) , the 4 in 20%, the 5 in 30% and the 6 in 10% of the cases. The suspicion arises that the dice is not fair. If after 10,000 runs the numbers have stabilized at the given values, then one can say with some certainty that, for example, the probability of rolling a 3 is 15%.

#### Propensity theory

The propensity theory interprets probability as a measure of the tendency of a process to a certain result.

#### Quantum mechanical probability concept

In non-relativistic quantum mechanics , the wave function of a particle is used as its fundamental description. The integral of the square of the magnitude of the wave function over a spatial area corresponds to the probability of encountering the particle there. So it is not a mere statistical, but a non-determined probability.

### Subjectivist conception of probability

In the case of one-off random events, the probability of their occurrence can only be estimated , not calculated. Central aspects are here experts know experience and intuition . Therefore one speaks of a subjectivistic conception of probability, see also Bayesian concept of probability .

Example: After someone has owned different cars, he rates the probability as high (for example “I am 80% sure”) that he will be satisfied with the XY brand the next time he buys a car. This prediction value can be changed up or down, for example by a test report.

However, this intuitive probability assessment harbors a multitude of "stumbling blocks" B. in the subjective perception (risks, such as having an accident due to increased speed , tend to be lower and chances, such as winning the lottery , are estimated to be higher than the actual probability of occurrence) or the asymmetrical information (the subjective assessment of accident risks corresponds the frequency of mentions in the media rather than the actual accident statistics, classic examples are the overestimated probabilities of occurrence of a shark attack or an aircraft accident ).

### Axiomatic definition of probability

The axiomatic (based on axioms) definition of probability according to Kolmogorow is the defining definition for mathematics today, see Axioms von Kolmogorow .

## Stochastics

Lotto probability
Number of
correct ones
Probability
[%]
0

43.5965
1

41.30195
2

13.2378
3

1.76504
4th

0.09686
5

0.00184
6th

0.00001

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

The mathematical term of probability is often used: The calculation of probability or probability theory (part of stochastics) takes care of the mathematical systematization of probabilities. A distinction is made here between probability distribution, probability function, conditional probability and many other terms.

Probabilities are numbers between 0 and 1, with zero and one being acceptable values. An impossible event is assigned the probability 0, and a certain event the probability 1. However, the reverse of this only applies if the number of all events is at most countably infinite. In “uncountably infinite” probability spaces, an event can occur with probability 0, it is then called almost impossible, an event with probability 1 does not have to occur, it is then called almost certain.

## Psychology - assessing probabilities

It is often said that people have a bad feeling for probability; in this context one speaks of "probability idiots" (see also numerical illiteracy ). The following examples:

• The birthday paradox : There are 23 people on a football pitch (two times eleven players and one referee). The probability that at least two people have their birthday on the same day is greater than 50%.
• You have participated in a preventive medical examination and received a positive result. You also know that you have no special risk factors for the diagnosed disease compared to the general population: With the calculation methods of the conditional probability , you can estimate the actual risk that the diagnosis made by the test actually applies. Two items of information are of particular importance in determining the risk of a false positive result: the reliability ( sensitivity and specificity ) of the test and the observed basic frequency of the disease in question in the general population. Knowing this actual risk can help to weigh the sense of further (possibly more serious) treatments. In such cases, the representation of the absolute frequency on the complete decision tree and a consultation with the doctor based on it give a more comprehensible impression than the mere interpretation of percentages based on the test result viewed in isolation.

## Philosophy - Understandings of Probability

While there is broad agreement about the mathematical handling of probabilities (see probability theory ), there is disagreement about what the calculation rules of mathematical theory may be applied to. This leads to the question of how to interpret the term “probability”.

"Probability" is often used in two different contexts:

1. Aleatoric probability (also: ontic / objective / statistical probability) describes the relative frequency of future events that are determined by a random physical process. A more precise distinction is made between deterministic physical processes, which in principle would be predictable with sufficiently precise information (throwing the dice, weather forecast), and nondeterministic processes, which in principle cannot be predicted (radioactive decay).
2. Epistemic probability (also: subjective / personal probability) describes the uncertainty about statements for which causal relationships and backgrounds are only incompletely known. These statements could relate to past or future events. For example, natural laws are occasionally assigned epistemic probabilities, as are statements in politics (“The tax cut comes with a 60% probability.”), Economics or jurisprudence.

Aleatoric and epistemic probability are loosely associated with the frequentist and Bayesian concepts of probability.

It is an open question whether aleatoric probability can be reduced to epistemic probability (or vice versa): Does the world appear random to us because we don't know enough about it, or are there fundamentally random processes, such as the objective interpretation of quantum mechanics assumes? Although the same mathematical rules for dealing with probabilities apply to both points of view, the respective point of view has important consequences for which mathematical models are considered valid.