Todennäköisyyksiä

Todennäköisyyksiä is the plural form of todennäköisyys, a concept central to statistics and mathematics that measures how likely it is that a given event will occur. In its formal sense, a probability is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probabilities are governed by basic rules that enable quantitative reasoning about random phenomena.

The mathematical foundation rests on Kolmogorov’s axioms: non-negativity, normalization (the probability of the sure event is 1), and additivity for disjoint events. Events can be modeled as discrete or continuous. Discrete probabilities are described by a probability mass function, while continuous probabilities use a probability density function; both yield probabilities through sums or integrals. The cumulative distribution function summarizes the likelihood that a variable does not exceed a given value.

Key rules include the addition rule for mutually exclusive events, P(A or B) = P(A) + P(B), and the multiplication rule for independent events, P(A and B) = P(A)P(B). Conditional probability, P(A|B) = P(A∩B)/P(B), measures the likelihood of an event given that another has occurred. Bayes’ theorem relates posterior, likelihood, and prior probabilities: P(A|B) = P(B|A)P(A)/P(B).

Interpretations of probability differ: frequentists view probabilities as long-run frequencies, while Bayesians interpret them as degrees of belief that update with evidence. Common probability distributions, such as binomial, normal, exponential, and uniform, are used to model diverse phenomena. Applications span research, engineering, finance, risk assessment, and data analysis. In Finnish usage, todennäköisyydet describe these measures of likelihood in various contexts.