Probability

Introduction A random walk is one of the most fundamental stochastic processes in probability theory. At its core, it describes a path that consists of a succession of random steps. Despite its deceptively simple definition, the random walk model underpins a surprisingly wide range of phenomena—from the diffusion of particles in physics to stock‑price dynamics in finance, from the spread of diseases in epidemiology to algorithmic techniques in computer science. ...

Introduction Monte Carlo methods are a family of computational algorithms that rely on repeated random sampling to obtain numerical results. From estimating the value of π to pricing complex financial derivatives, Monte Carlo techniques have become indispensable across scientific research, engineering, finance, and data science. Their power lies in the ability to solve problems that are analytically intractable by turning them into stochastic experiments that computers can execute millions—or even billions—of times. ...

Table of Contents Introduction Probability Fundamentals Conditional Probability and the Chain Rule Probability Distributions How LLMs Use Probability From Theory to Practice Common Misconceptions Conclusion Resources Introduction If you’ve ever wondered how ChatGPT, Claude, or other large language models generate coherent text that seems almost human-like, the answer lies in mathematics—specifically, probability theory. While the internal mechanics of these models involve complex neural networks and billions of parameters, at their core, they operate on a surprisingly elegant principle: predicting the next word by calculating probabilities. ...