probabilities

Probability is a branch of mathematics that studies and quantifies the likelihood of events under uncertainty. It provides a mathematical framework for describing randomness in experiments, games, measurements, and real-world processes.

The basic objects are the sample space S, consisting of all possible outcomes, and events, which are subsets of S. A probability measure P assigns to each event a number between 0 and 1, such that P(S)=1 and, for disjoint events A and B, P(A ∪ B)=P(A)+P(B). These Kolmogorov axioms underpin modern probability theory and, in measure-theoretic form, extend to infinite families of events.

Probability interpretations include classical probability (equally likely outcomes: P(A)=|A|/|S|), empirical (relative frequency from observed data), and subjective probability (degree of belief).

Key rules include conditional probability P(A|B)=P(A∩B)/P(B) when P(B)>0, the multiplication rule P(A∩B)=P(B)P(A|B), and the law of total probability P(A)=∑ P(A|Bi)P(Bi) for a partition {Bi}. Bayes' theorem updates beliefs: P(A|B)=P(B|A)P(A)/P(B).

A random variable X maps outcomes to real numbers. Discrete variables have probability mass functions, continuous variables have probability density functions, with cumulative distribution functions F(x)=P(X≤x). The expectation E[X] and variance Var(X) summarize central tendency and dispersion. Independence means joint probabilities factorize.

Distributions such as the uniform, binomial, and normal model common phenomena. Probability theory underpins statistics, finance, science, and risk assessment by providing tools for inference, hypothesis testing, and decision making under uncertainty.