Undergraduate Probability

with Professor Roman Vershynin

A free video course


This is a free video course in probability. It corresponds to a standard, semester-long undergraduate probability class in many North American universities. If you are an ambitious high school student, you may be well prepared for this course. It is completely free, publicly available, and no subscription is needed.

The only prerequisite is calculus.You need to know how to differentiate and integrate. To fully enjoy this class, it would be great if you know limits and series as well. You should also be familiar with the basic operations on sets. To refresh your memory, review the algebra of sets.

The course is based on examples. Every episode starts from intuitive examples to motivate the theory, then goes back to examples to reinforce the skills.

Got a nice problem? If you have a cute problem to illustrate one of the episodes, Email it to me along with the solution, I can post it here. Let's make a nice bank of problems for everyone to enjoy.

Season 1: Combinatorics

  • Episode 1: Do you have a birthmate? Watch Notes

    Want to learn more about Euler's number and Euler's limit? Check out Section 3 of this workbook.
  • Episode 2: Permutations and factorials Watch Notes

    Multiplication principle. Permutations. Factorial.
  • Episode 3: Factorials, befriended Watch Notes

    How many of your friends have the same birthday? Stirling's approximation of the factorial.
  • Episode 4: Binomial coefficients Watch Notes

    Permutations with repetitions. Combinations. Binomial coefficients.
  • Episode 5: How many solutions does an equation have? Watch Notes

    Why is 0!=1? Symmetry of binomial coefficients. The number of integer solutions to an equation. Want to learn more? Check out the partition function.
  • Episode 6: What is the probability to win a lottery? Watch Notes

    Here we practice with binomial coefficients. We work on three examples: splitting the class into two teams, winning the lottery, and selecting a representative sample of lab rats.
  • Episode 7: Binomial theorem and Pascal's triangle Watch Notes

    We rediscover the binomial theorem. It implies that the sum of all binomial coefficients equals 2^n. We then give an alternative, combinatorial proof of this identity. Another cute identity is Pascal's rule. We prove it in a combinatorial way and organize it visually as Pascal's triangle. This rule allows us to quickly compute all binomial coefficients, and thus write down binomial formula of any power.

Season 2: The Axioms of Probability

  • Episode 1: What is an event? Watch Notes

    We define three major building blocks of probability theory: sample spaces, outcomes, and events.
  • Episode 2: Operations on events Watch Notes

    We define three major operations on events: union, intersection, and complement. These are the same as basic operations on sets: Chapter 1 is a good review. De Morgan's laws say that the complement operation turns intersections into unions, and vice versa.
  • Episode 3: What is probability? Watch Notes

    We give a formal, axiomatic definition of probability. We then explore a special kind of probability -- the uniform probability, where all outcomes are equally likely.
  • Episode 4: The probability to detect fake news Watch Notes

    We give two consequences of the axioms of probability: the probability of the complement, and the monotonicity property. We compute the probability to expose the producer of fake news in 100 days, if we check 1 news story per day.
  • Episode 5: Inclusion-exclusion principle Watch Notes

    The inclusion-exclusion principle allows us to compute the probability of the union of any number of events. We illustrate it by solving the "matching problem" where we compute the probability of a derangement.

Season 3: Conditional Probability and Independence

  • Episode 1: Conditional probability Watch Notes

    Motivated by a cancer & smoking study, we introduce the notion of conditional probability. We illustrate this notion with examples on survival probabilities and others.
  • Episode 2: The law of total probability Watch Notes

    From definition of conditional probability, we derive a "chain rule" and then the law of total probability. This allows to compute probabilities by "conditioning". We illustrate this trick by computing the probability of a successful blood transfusion.
  • Episode 3: Bayes formula Watch Notes

    Bayes formula allows us to update measure of belief in light of new evidence -- from prior to posterior probabilities.
  • Episode 4: Monty Hall problem Watch Notes

    One of the famous paradoxes in probability is Monty Hall problem, or equivalently Three Prisoners problem. We analyze this paradox using Bayes formula, and then give an intuitive explanation.
  • Episode 5: What is independence? Watch Notes

    We define independence, a central notion of probability theory, and a strictly weaker notion of pairwise independence. We note that independence property is stable with respect to set operations.
  • Episode 6: Finding your birthmate with 50-50 chance Watch Notes

    In the rest of this season, we practice with problems about independence. This episode revisits the birthmate problem from S1:E1 but this time we solve it using independence.
  • Episode 7: Problems on independence: airplane failure, network connectivity Watch Notes

    We solve two problems related to independence: find the probability that the plane can fly if its engines fail at random, and find the probability that towns stay connected in a snowstorm. Conditioning trick simplifies the second problem. That problem is a particular the Erdos-Renyi model of a random graph.
  • Episode 8: Problem of points Watch Notes

    We find the probability that the player who first flips a head wins. Conditioning on the first flip "resets" the game and yields an equation, solving which we get the answer to this problem. Similarly, we solve a version of the famous problem of points, an important problem in the early days of the development of probability theory.
  • Episode 9: Random walk and gambler's ruin Watch Notes

    We introduce a simple random walk, a fascinating object of study in probability theory. We solve gambler's ruin problem: what is the probability that a random walk reaches n (payoff) before reaching 0 (bankruptcy)?
  • Episode 10: Secretary problem Watch Notes

    The secretary problem, a.k.a. "best prize problem", is a classical problem of probability theory. In this episode we design a stopping strategy that allows us to select the best among N prizes, presented to us in a sequence, with probability approximately 1/e = 0.37...

Season 4: Random variables

  • Episode 1: What is a random variable? Watch Notes

    We introduce the notions of a random variable, distribution, and the probability mass function of a discrete random variable.
  • Episode 2: Expectation of a discrete random variable Watch Notes

    We define the expected value of a discrete random variable, and compute the expected winnings in the "6 out of 49" lottery.
  • Episode 3: Linearity of expectation. St. Petersburg paradox Watch Notes

    We study basic properties of expectation, and learn about St. Petersburg paradox - a game where the expected winnings are infinite.
  • Episode 4: Group testing Watch Notes

    We illustrate the linearity of expectation by analyzing group testing, a technological method where one wants to detect abnormalities with as few tests as possible.
  • Episode 5: Expected number of cycles in a random permutation Watch Notes

    We prove that the expected number of cycles in a random permutation of n objects equals the harmonic number, so it is logarithmic in n.
  • Episode 6: What is variance? Watch Notes

    We introduce the concepts of variance and standard deviation of a random variable, and we practice with them by comparing two investment options.
  • Episode 7: Properties of variance Watch Notes

    We cover basic properties of variance. Then we revisit the matching problem from S2:E5. We compute the expectation and variance of the number of students who receive their own exams, if the exams are handed out at random.

Season 5: Will be released in Summer 2023, stay tuned