Zufallsversuch

Zufallsversuch is the German term for a random experiment. It refers to a procedure that can be performed in such a way that its outcome is uncertain, but the set of possible results is fixed. After carrying out the procedure, exactly one outcome from a predefined sample space Ω is observed. The concept is fundamental in probability theory and is used to model single trials of stochastic processes, games of chance, measurements, or other uncertain phenomena.

Mathematically, a Zufallsversuch is modeled within a probability space (Ω, F, P). Ω is the sample space containing all possible outcomes; F is a σ-algebra of events, i.e., subsets of Ω to which probabilities are assigned; P is a probability measure with 0 ≤ P(A) ≤ 1 for A ∈ F and P(Ω) = 1. A single trial yields an outcome ω ∈ Ω, and an event A occurs if ω ∈ A, with probability P(A). Random variables are functions X: Ω → R that summarize outcomes by assigning numeric values to ω.

Common examples include tossing a fair coin (Ω = {Kopf, Zahl}, each with probability 0.5), rolling a six-sided die (Ω = {1,2,3,4,5,6}, P(k) = 1/6), and drawing a card from a standard 52-card deck (with or without replacement, which affects independence). Repeating identical Zufallsversuche yields a sequence of outcomes, analyzed as a stochastic process, with notions such as independence, dependence, and distributions.

In German-language texts, Zufallsversuch is often used interchangeably with terms like Zufallsexperiment or Experiment to denote a single trial whose outcome is random; the formal treatment relies on the probability space framework.