Pravdpodobnost

Pravdpodobnost is a branch of mathematics that studies the likelihood of events and provides a formal framework for reasoning about uncertainty arising from random phenomena, data, and models.

A probabilistic model consists of a sample space, a set of events, and a probability measure that assigns numbers between 0 and 1 to events, satisfying principles such as non-negativity, normalization (the probability of the entire space is 1), and additivity over disjoint events. In practice, probabilities are described using discrete distributions with probability mass functions or continuous distributions with probability density functions; a cumulative distribution function summarizes the probability that a random variable does not exceed a given value.

Basic rules include the addition rule P(A or B) = P(A) + P(B) − P(A and B), and the multiplication rule P(A and B) = P(A)P(B) when A and B are independent. Conditional probability P(A|B) = P(A and B)/P(B) captures how probability changes with new information, and Bayes’ theorem links prior beliefs and evidence to posterior probabilities.

Random variables map outcomes to numbers, enabling the computation of expected value (mean), variance, and other moments that describe a distribution. Common distributions include the binomial, normal, Poisson, and exponential families, each with characteristic shapes and parameters.

Two central results are the law of large numbers, which describes convergence of averages to the true mean, and the central limit theorem, which explains why sums of independent variables tend to follow a normal distribution under broad conditions.

Interpretations vary: the frequentist perspective treats probability as a long-run frequency, while the Bayesian view regards probability as a degree of belief updated by data via Bayes’ rule. Pravdpodobnost has wide applications in science, engineering, finance, statistics, medicine, and machine learning, where uncertainty must be quantified and managed.