Math/Stat 431 Introduction to the Theory of Probability

Fall 2023 Lecture

Meetings: MWF 11:00am-11:50am in 5231 Sewell Social Sciences
Instructor: Benedek Valků
Office: 409 Van Vleck.
Office Hours: Tu 3:45pm-4:45pm, Fr 2:30pm-3:30pm or by appointment
E-mail: valko at math dot wisc dot edu

This is the course homepage. Part of this information is repeated in the course syllabus that you find on Canvas.  Important deadlines from the Registrar's page.

Course description

Math 431 is an introduction to the theory of probability, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.

Math 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.

From a broad intellectual perspective, probability is one of the core areas of mathematics with its own distinct style of reasoning. Among the other core areas are analysis, algebra, geometry/topology, logic and computation.

To go beyond 431 in probability you should take next Math 521 - Analysis, and after that one or both of these: Math 632 - Introduction to Stochastic Processes and Math 635 - Introduction to Brownian Motion and Stochastic Calculus. Those who would like a proof based introduction to probability could consider taking  Math 531 - Probability Theory (531 requires a proof based course as a prerequisite).

Where is probability used?

Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. Aside from being a beautiful subject in and of itself, is used throughout the sciences and industry.  For example, in biology many models of cellular phenomena are now modeled probabilistically as opposed to deterministically. As for industry, many models used by insurance and financial companies are probabilistic in nature. Thus, those wishing to go into actuarial science or finance need to have a solid understanding of probability. Probabilistic models show up in the study of networks, making probability theory useful for those interested in computer science and information technology.


To be technically prepared for Math 431 one needs to be comfortable with the language of sets and calculus, including multivariable calculus, and be ready for abstract reasoning. Basic techniques of counting is also useful, but we will review these along the way. Probability theory can seem very hard in the beginning, even after success in past math courses.


Anderson, Seppšlšinen, Valků: Introduction to Probability, Cambridge University Press, 2017. 


We will use the Canvas site of the course to post quizzes, homework assignments and solutions.


We will be using Piazza for class discussion.  The system is catered to getting you help fast and efficiently from classmates and myself.  Rather than emailing math questions to me, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email Our class Piazza page can be accessed from the Canvas page of the course.


Course grades will be based on homework and  quizzes (17%), two midterm exams (2x25%), and a comprehensive final exam (33%).
Midterm exams will be in the evenings of the following dates.
No calculators, cell phones, or other gadgets will be permitted in exams, only pencil and paper. The final grades will be determined according to the following scale:

A: [100,89), AB: [89,87), B: [87,76), BC: [76,74), C: [74,62), D: [62,50), F: [50,0].

Final letter grades are not curved but the grade lines above may be lowered at the end. Class
attendance is not part of the grading


You will have a Mastery Check Quiz on basic set operations, and basic counting techniques, counting for 2% of your course grade. There will also be short weekly online quizzes on Canvas counting for 5% of your course grade.


Homework assignments are due roughly every Friday. Assignments will be posted on Canvas, and they will have to be submitted in Canvas as a single pdf file.

Challenge Problems

We will also post challenge problems during the semester. These will be graded separately from the regular homework assignments. The total score from the challenge problems will be normalized at the end of the semester, usually the best problem solvers get a couple of extra points towards their final grade. These problems are not graded like the rest of the homework, with ordinary standards of partial credit. The scale for the challenge problems is 0-3, with 0 unless you make significant progress towards the answer. Note that you do not need to work on these problems if you do not want to, they are there for the students who enjoy working on harder problems.

Weekly schedule

Here is a tentative weekly schedule, to be adjusted as we go.

Weeks 1-2     Axioms of probability, sampling, consequences of the rules of probability
Weeks 3-4     Conditional probability and independence
Weeks 5-6     Random variables (distribution, expectation, variance)
Weeks 6-7     Normal and Poisson approximation of binomial distribution
Week 8          Moment generating function and transformation of random variables
Weeks 9-12   Joint behavior of multiple random variables (joint distribution, expectation, covariance)
Week 13        Estimating tail probabilities, the Law of large numbers and the Central limit theorem
Weeks 14-15 Conditional distribution and conditional expectation

The Math Club provides interesting lectures and other math-related events. Everybody is welcome.