Probabilitet

Probabilitet, or probability, is a branch of mathematics that studies the likelihood of events in the presence of uncertainty. It provides a quantitative measure between 0 and 1 that describes how probable an outcome is in a specified random process. In formal terms, probability is defined on a probability space (Ω, F, P), where Ω is the set of possible outcomes, F is a σ-algebra of events, and P assigns a probability to each event.

Probabilities obey axioms: non-negativity, P(Ω) = 1, and additivity over disjoint events. The basic rules include the sum rule P(A ∪ B) = P(A) + P(B) for disjoint A and B, the product rule for independent events P(A ∩ B) = P(A)P(B), and conditional probability P(A|B) = P(A ∩ B) / P(B). Bayes' theorem relates P(A|B) and P(B|A).

Two main interpretations exist: the frequentist view ties probability to long-run relative frequencies, while the Bayesian view treats probability as a degree of belief that can be updated with data.

A random variable maps outcomes to numbers. Discrete variables take countable values; continuous variables take values in intervals. Key summaries include the expectation (mean), variance and standard deviation. Common distributions include the binomial, normal, Poisson, exponential, and uniform distributions.

Important results include the Law of Large Numbers, which connects observed frequencies to true probabilities, and the Central Limit Theorem, which explains why many distributions approximate normality under aggregation.

Applications span statistics, science, engineering, finance and risk assessment. Probabilitet provides the foundation for inference, decision making under uncertainty, and stochastic modeling of systems.