Probabiliteten

Probabilitäten (probabilities) are a mathematical concept used to quantify the likelihood that a given event occurs. They take values between 0 and 1, where 0 means impossibility and 1 means certainty. They are central to the study of uncertainty, statistics, and decision making.

In modern probability theory, probabilities are formalized as measures on a sigma-algebra over a sample space. The Kolmogorov axioms require non-negativity, P(Ω) = 1, and countable additivity, ensuring consistency across events.

There are several interpretations of probability: classical (based on symmetry and counting), frequentist (long-run frequencies), and Bayesian (degrees of belief updated with evidence).

Basic rules include the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B), the multiplication rule P(A ∩ B) = P(A)P(B|A), and conditional probability P(A|B) = P(A ∩ B)/P(B).

Random variables map outcomes to numbers, and distributions describe their behavior. Discrete variables have probability mass functions, continuous variables have probability density functions, and the cumulative distribution function links these descriptions. The expectation and variance summarize central tendency and spread.

Key results such as the law of large numbers and the central limit theorem explain convergence properties and normal approximations in many practical problems.

Probabilities underpin statistics, finance, risk assessment, science, and machine learning, providing a rigorous framework for inference, decision making, and uncertainty quantification.