Syllabus


Course title:

Theory of Probability (Fall 2026)

Sections

MATH-UA.0333-005 and MA-UY 3014-C

Time:

Tuesday Thursday  09:30 AM - 10:45 AM

Room:

Courant Institute / Warren Weaver Hall (251 Mercer St), Room 201

Instructor:

Professor Yuri Bakhtin  (he/him/his)

Communication

Most communication will be done via Brightspace. The best way to contact me is via Brightspace or email yb22@nyu.edu. For efficient email communication and etiquette, I recommend reading materials linked from https://cims.nyu.edu/~bakhtin/email-etiquette.html

Office hours:

[Tentative] Tuesday 2:30-3:30pm, Thursday 1:30-2:30pm. My office is 729 at Courant Institute.

Recitation:

[Tentative] Fridays 9:30-10:45 AM at CI / WWH Room 201. 

Recitation Leader: TBA

Course description:

An introduction to the mathematical treatment of random phenomena occurring in the natural and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains, applications.

Textbook:

The textbook we will be using for the course is Introduction to probability, Second Edition  by Joseph K.Blitzstein and Jessica Hwang (The first edition is OK, too). Paper copies of the textbook can be bought at  at this link or on Amazon. Free online access (no download though) is provided by the authors at Blitzstein Hwang Probability.pdf. Other forms of electronic access are available at this link through NYU.

The book is accompanied by various video materials, problem solutions, etc., available at https://stat110.hsites.harvard.edu/

We will cover or touch the material from almost every chapter in this book but often we will not follow the book literally.

Prerequisites:

This course is intended for math majors and other students with a strong interest in mathematics. It requires fluency in calculus topics such as limits, derivatives, series, (multi-variable) integration. It makes sense to look through the textbook in advance to know what to expect.

Homework

Homework will account for 10% of the final score.  Homework problem sets along with instructions and deadlines will be posted on Brightspace (usually, weekly) and you will submit your work electronically. It can be typed or handwritten but it has to be in the PDF format. Late homework will not be accepted. Two worst homework scores will be dropped for each student.

You may work on problems in groups and use AI/LLM as described below but each student must write down their own solutions.

You are allowed to use Large Language Models (LLMs), such as ChatGPT or Claude, as tutoring aids for mathematical problem solving. Your own problem-solving efforts are paramount though. In addition, while LLM models can be valuable resources, they are subject to both obvious and subtle errors. For maximum growth and test preparation, I recommend that you solve homework on your own. If you only read/copy solutions and don’t practice solving problems on your own for real, your performance on exams will be poor. The best way to use AI is actually to treat it as a tutor. AI may help you with a study plan if you need to catch up, give you more exercises on a topic, identify your weak spots, etc.

Regrade policy: if you think there is a grading mistake for one of your solutions, you may request a regrade but only within two weeks after the grade is published.

Quizzes

Throughout the course, 6 (six) quizzes will be given at the beginning of randomly chosen lectures. Each quiz will be closed book and will be entirely based on one of the most recent homework problems, after the assignment is due. It will be assumed that you already know how to solve the problem and all you need to do is to reproduce the solution you have submitted as a part of the homework assignment. For each student, their best 5 quizzes out of 6 will count for 10% of the course grade (i.e., the worst quiz score will be dropped).

Exams:

Two midterm exams ((Tuesday October 13 and Thursday November 12) ) and the final exam are in-class and closed book. You are only allowed a pen/pencil, and an eraser. I will provide paper and you will not be allowed to use any other paper.  No electronics (including but not limited to computers, phones, watches, AI glasses), books, notes, or any other aids will be allowed, and a  zero-tolerance policy will be strictly enforced. In addition, oral spot-checks may be applied after an in-class exam to ensure that  the student is the sole author of their submitted work. These checks are a part of the exam and count toward the grade. A student's unsatisfactory explanation of their own work will result in a zero score.

 

Your total midterm score will be determined as follows: if your second midterm grade is higher than the first one, then the second midterm grade will be your midterm score; otherwise, the average of the two midterms will be used.

Grading:

Two worse homework scores will be dropped.

The worst quiz score will be dropped.

[Please do not rely on this too much and try to do your best on all the homework and quizzes. These policies take care of all the unexpected situations during the semester (illness, family/personal issues, computer or network issues, tardiness, absent-mindedness, etc)]

The total homework score counts for 10% of the course grade.

The total quiz score counts for 10% of the course grade.

The final grade will be calculated based on the comparison between your final exam grade and your midterm grade as follows:

1. If the final exam grade is higher than the midterm grade:

Final Grade =  10% × Homework + 10% × Quizzes                    

                       + 20% × Midterms + 60% × Final Exam

2. If the final exam grade is lower than the midterm grade:

Final Grade = 10% × Homework + 10% × Quizzes
                     + 40% × Midterms + 40% × Final Exam

 

The final grades will be assigned based on the following thresholds:

90% guarantees an "A"

80% guarantees a "B"

70% guarantees a "C"

60% guarantees a "D".

Various intermediate grades such as “A-” or “C+” will also be given, at my discretion.

Integrity policy

Using any unpermitted aid at a quiz or exam will result in a penalty. The minimum penalty is a zero score on the entire quiz / exam. For example, a mere presence of an electronic device within reach or any unauthorized communication will immediately result in a zero score.

Oral spot-checks may be applied after an in-class exam to ensure that  the student is the sole author of their submitted work. These checks are a part of the exam and count toward the grade. A student's unsatisfactory explanation of their own work will result in a zero score.

 

Tentative schedule:

Sep 3

Introduction. Classical def of probability. Some combinatorics. Sections 1.1-1.3

Sep 8

More combinatorics. Sections 1.4,1.5

Sep 10

Sample space and events. Sections 1.6, 1.7

Sep 15

Conditional probability. Bayes’ rule and the formula of total probability. Sections 2.1-2.4

Sep 15

Add/Drop Deadline

Sep 17

Independence of events. Section 2.5

Sep 22

Applications of conditioning. Sections 2.6-2.9

Sep 24

Discrete random variables and their distributions. Sections 3.1-3.5

Sep 29

Some useful distributions. Independent random variables. Sections 3.6-3.10

Oct 1

Discrete random variables, their variance and examples. Sections 4.5, 4.6

Oct 6

Expectations and variances of random variables. Sections 4.1-4.6

Oct 8

Poisson distribution and Poisson approximation. Sections 4.7, 4.8, 4.10.

Oct 13

Midterm exam

Oct 15

Continuous random variables, distribution and density, expectation, some useful continuous distributions. Sections 5.1-5.3

Oct 20

Transformations of random variables. Some useful distributions. Sections 5.4, 5.5, partially 8.1.

Oct 22

Poisson process. Section 5.6

Oct 27

Joint distributions. Conditional probability. Sections 7.1,7.2

Oct 29

Independence. Covariance, correlation. Section 7.3

Nov 3

Transformations of random  variables. Jacobian. Section 8.1

Nov 5

Sums of independent random variables. Convolutions. Section 8.2

Nov 10

Markov’s inequality.  Chebyshev’s inequality.

Convergence in probability. Law of large numbers I.  Sections 10.1, 10.2

 

Nov 12

Midterm exam

Nov 17

Convergence in distribution. Central Limit Theorem. Section 10.3 Moment generating functions. Sections 6.1, 6.2.

Nov 19

Moments and Moment generating functions. Sections 6.4, 6.5, 6.6, 6.8

Nov 24

Central Limit Theorem. Section 10.3

Nov 26

Thanksgiving Recess

Dec 1

Central Limit Theorem and applications in statistics. Sections 10.4, 10.5

Dec 3

Markov Chains. Sections 11.1, 11.2

Dec 8

Markov Chains, stationary distributions, reversibility. Sections 11.3, 11.4.

Dec 10

Review

TBA

Final exam