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step if a blue ball is withdrawn, based on the goal of eventually withdrawing a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