Probability theory aims to provide a mathematical framework to describe, model, analyze, and solve problems involving random phenomena and complex systems. While its original motivation was the study of gambling problems, probability has become successful in applications in finance, computer science, engineering, statistical mechanics, and biology. In the mathematical sciences, probability is fundamental for the analysis of statistical procedures, and the “probabilistic method” is an important tool for proving existence theorems in discrete mathematics.
Stochastic processes are probabilistic models for random quantities evolving in time or space. The evolution is governed by some dependence relationship between the random quantities at different times or locations. Major classes of stochastic processes are random walks, Markov processes, branching processes, renewal processes, martingales, and Brownian motion. Important application areas are mathematical finance, queuing processes, analysis of computer algorithms, economic time series, image analysis, social networks, and modeling biomedical phenomena. Stochastic process models are used extensively in operations research applications.