valószínségtér

Valószínűségtér is a fundamental concept in probability theory, providing a mathematical framework for describing random phenomena. It consists of three components: a sample space, a set of events, and a probability measure. The sample space, often denoted by $\Omega$, is the set of all possible outcomes of a random experiment. For example, when flipping a coin, the sample space is {Heads, Tails}.

The set of events, typically denoted by $\mathcal{F}$, is a collection of subsets of the sample space. These subsets represent the specific outcomes or combinations of outcomes we are interested in. For instance, in a dice roll, the event "rolling an even number" would be the subset {2, 4, 6} of the sample space {1, 2, 3, 4, 5, 6}. The collection of events must satisfy certain mathematical properties to be considered a valid sigma-algebra.

The probability measure, denoted by $P$, is a function that assigns a numerical probability to each event in the event set. This measure must satisfy three axioms: the probability of any event must be non-negative, the probability of the entire sample space must be 1, and for any sequence of mutually exclusive events, the probability of their union is the sum of their individual probabilities. Together, these three components – sample space, event space, and probability measure – form the probability space, which is the rigorous foundation for the study of probability.