Ereignissen

Ereignissen is the plural form of the German noun Ereignis, which translates to event or occurrence. In mathematics and statistics, Ereignis is used to denote a possible outcome or a set of outcomes of a random experiment, typically considered as a subset of a sample space Ω. The collection of all such events is described within a structure called a sigma-algebra, and a probability measure P assigns probabilities to these events. An event is said to occur when the realized outcome lies in the corresponding subset.

Common operations on events correspond to standard set operations: the union of events A and B (A ∪ B) represents “A or B”; the intersection (A ∩ B) represents “A and B”; the complement (A^c) represents “not A.” The probability of a union of disjoint events equals the sum of their probabilities (the additivity property). Two events A and B may be independent if P(A ∩ B) = P(A)P(B). Conditional probability P(A|B) = P(A ∩ B)/P(B) describes the probability of A given that B has occurred.

Examples help illustrate the concept. In a fair six-sided die, the event E1 = “the outcome is even” corresponds to {2, 4, 6}, and the event E2 = “the outcome is 1” corresponds to {1}. The probability P(E1) is 1/2 and P(E2) is 1/6. Such events form the basis of probabilistic modeling, hypothesis testing, and various applications in science and everyday reasoning.

Beyond mathematics, Ereignissen also appears in general German text to refer to occurrences or incidents. The singular Ereignis is often used in philosophy, literature, and everyday language.