probabability

Probability is a branch of mathematics that studies the likelihood of events. It formalizes uncertainty and provides tools to quantify how likely different outcomes are. A probability model starts with a sample space Ω, the set of all possible outcomes of a random process, and events, which are subsets of Ω. A probability measure P assigns to each event a number in the interval [0,1] such that P(Ω)=1 and, for any countable collection of disjoint events, the probability of their union is the sum of their probabilities (countable additivity).

Models are classified as discrete or continuous. In discrete models, outcomes are countable and described by probability mass functions. In continuous models, outcomes form intervals or spaces with a probability density function, and probabilities are computed by integration. Common rules include the addition rule P(A∪B)=P(A)+P(B)−P(A∩B) and the conditional rule P(A|B)=P(A∩B)/P(B) when P(B)>0. The multiplication rule P(A∩B)=P(A|B)P(B) expresses joint probability, and Bayes' theorem updates beliefs in light of new evidence: P(A|B)=P(B|A)P(A)/P(B).

Fundamental objects include random variables, which map outcomes to real numbers, and derived quantities such as expectation and variance. Important distributions include the binomial, Poisson, geometric (discrete), and normal, exponential, and uniform (continuous). The law of large numbers and the central limit theorem describe how probabilities behave in the long run.

Applications span statistics, finance, science, engineering, and data science, where probability models underpin inference, decision making, and risk assessment.

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