probabilitytheory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. It provides a framework for quantifying uncertainty and making predictions about the likelihood of events. At its core, probability theory deals with experiments that have uncertain outcomes. An experiment is a repeatable procedure whose result is not predictable with certainty. The set of all possible outcomes of an experiment is called the sample space. An event is a subset of the sample space, representing a specific outcome or collection of outcomes.

The probability of an event is a number between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty. There are different interpretations of probability. The classical interpretation defines probability as the ratio of favorable outcomes to the total number of equally likely outcomes. The frequentist interpretation defines probability as the limit of the relative frequency of an event as the number of trials approaches infinity. The subjective interpretation views probability as a degree of belief.

Key concepts in probability theory include random variables, which are variables whose value is a numerical outcome of a random phenomenon. Probability distributions describe the likelihood of different values for a random variable. Common distributions include the binomial distribution, Poisson distribution, and normal distribution. Probability theory is fundamental to many fields, including statistics, physics, finance, computer science, and engineering, enabling informed decision-making in the face of uncertainty.