Introduction

Probability theory is the mathematical study of random phenomena. The theory develops techniques on how to assign numerical values from $[0,1]$ to the possible outcomes of a random experiment whose result is not known in advance. These numerical values are colloquially known as probabilities. The theory provides rules and develops techniques for computing with those numerical values. The theory rests on three primitives: a sample space of possible outcomes, a collection of events, and a probability measure assigning each event a likelihood consistent with a short list of axioms.

This course has been adapted from the following resources:

• Alon, Noga, and Joel H. Spencer. (2016). The Probabilistic Method (4th ed.). Wiley. ISBN 978-1-119-06195-3.
• Andreescu, Titu, and Zuming Feng. (2003). 102 Combinatorial Problems: From the Training of the USA IMO Team. Birkhäuser Boston. ISBN 978-0-8176-4317-3.
• Art of Problem Solving. Alcumus. https://artofproblemsolving.com/alcumus.
• Art of Problem Solving. AoPS Wiki — Competition Archives. https://artofproblemsolving.com/wiki.
• Blitzstein, Joseph K., and Jessica Hwang. (2019). Introduction to Probability (2nd ed.). Chapman & Hall/CRC. ISBN 978-1-138-36991-7.
• Djukić, Dušan, Vladimir Janković, Ivan Matić, and Nikola Petrović. (2011). The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiad, 1959–2009 (2nd ed.). Springer. ISBN 978-1-4419-9854-5.
• Durrett, Rick. (2019). Probability: Theory and Examples (5th ed.). Cambridge University Press. ISBN 978-1-108-47368-2.
• Engel, Arthur. (1998). Problem-Solving Strategies. Springer-Verlag. ISBN 978-0-387-98219-9.
• Fox, Jacob. MAT 307: Combinatorics, Lecture 13: Graphs of High Girth and High - Chromatic Number. MIT-hosted lecture notes, https://math.mit.edu/~fox/MAT307-lecture13.pdf.
• Grinstead, Charles M., and J. Laurie Snell. Introduction to Probability (2nd ed.). American Mathematical Society. Dartmouth-hosted edition, - https://math.dartmouth.edu/~prob/prob/prob.pdf.
• International Mathematical Olympiad. Problems and Shortlists. https://www.imo-official.org/problems.aspx.
• Mathematical Association of America. AIME: Problems and Solutions. https://maa.org/maa-invitational-competitions/.
• Mathematical Association of America. AMC 10/12: Problems and Solutions. https://maa.org/student-programs/amc/.
• Mathematical Association of America. AMC 8: Problems and Solutions. https://maa.org/student-programs/amc/.
• Mathematical Association of America. USAMO: Problems and Solutions. https://maa.org/maa-invitational-competitions/.
• Melcher, Tai. "Introduction to Probability" MATH 3100, Fall 2026, University of Virginia. Classroom Lecture.
• Patrick, David. (2005). Introduction to Counting & Probability. AoPS Incorporated. ISBN 0-9773045-0-7.
• Patrick, David. (2018). Intermediate Counting & Probability. AoPS Incorporated. ISBN 978-1-934124-06-2.
• Ross, Sheldon M. (2010). A First Course in Probability (8th ed.). Pearson Prentice Hall. ISBN 978-0-13-603313-4.
• Soberón, Pablo. (2013). Problem-Solving Methods in Combinatorics: An Approach to Olympiad Problems. Birkhäuser Basel. DOI 10.1007/978-3-0348-0597-1.
• West, Douglas B. (2001). Introduction to Graph Theory (2nd ed.). Prentice Hall. ISBN 978-0-13-014400-3.