#### Keywords

Urn Models, Infinite Products, Infinite Sums, Inventing Identities

#### Abstract

This paper considers an urn and its evolution in discrete time steps. The

urn initially has two different colored balls(blue and red). We discuss different

cases where k blue balls (k = 1, 2, 3, ... ) will be added (or removed) at every

step if a blue ball is withdrawn, based on the goal of eventually withdrawing a

red ball P(R eventually). We compute the probability of eventually withdrawing

a red ball with two different methods–one using infinite sums and other using

infinite products. One advantage of this is that we can obtain P(R eventually) in

a complex but nicely patterned form using one method, and a simple form using

the other method. Since the results must be equal, we obtain some interesting

identities. We also present a general result and invent new identities, illustrated

by an example using the Fibonacci numbers. Additionally, we transform Wallis

Product(a) and Wallis Product(b) into urn models. Finally, we illustrate some

results by simulating the urn processes in R.

#### Primary Advisor

Dr. Myron Hlynka

#### Co-Advisor

Dr. Percy Brill

#### Program Reader

Dr. Abdulkadir Hussein

#### Degree Name

Master of Science

#### Department

Mathematics and Statistics

#### Document Type

Major Research Paper

#### Convocation Year

2019