The Relation Between One and Two Dimensional Random Walks to Pascal’s Triangle and Analogous Higher-Dimensional Structures

Abstract

This study aims to explore the use of Pascal's triangle and other analogous structures in higher dimensions to predict the position of particles inside a random walk after a certain number of steps. Pascal's triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it, and explores higher dimensional structures like Pascal’s pyramid (the three-dimensional equivalent of Pascal’s triangle). The first part of the study delves into the concept of Pascal's triangle and its fundamental properties. After that, Pascal's triangle was used in predicting the position of particles inside a random walk. A random walk is a mathematical model used to describe the movement of particles in a system where the direction and magnitude of each step are random. The probability of a particle ending up at a particular point after a certain number of steps in a random walk can be calculated using Pascal's triangle. The numerator of the fraction representing the probability is the value at the point in Pascal's triangle, while the denominator is the sum of the numbers in the corresponding row or layer of the triangle. The paper demonstrates how these concepts can be used to predict the position of particles inside a random walk and provide a real-life example of their application in the simulation of Brownian motion.