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Aprox. 21 horas para completar

Sugerido: 6 weeks, 3-5 hours/week ...


Legendas: Inglês, Grego, Chinês (simplificado)

Habilidades que você terá

Random VariableProbability InterpretationsProbabilityCombinatorics

100% online

Comece imediatamente e aprenda em seu próprio cronograma.

Prazos flexíveis

Redefinir os prazos de acordo com sua programação.

Nível iniciante

Aprox. 21 horas para completar

Sugerido: 6 weeks, 3-5 hours/week ...


Legendas: Inglês, Grego, Chinês (simplificado)

Programa - O que você aprenderá com este curso

3 horas para concluir

Basic Counting

Suppose we need to count certain objects. Can we do anything better than just list all the objects? Do we need to create a list all phone numbers to check whether there are enough phone numbers for everyone? Is there a way to tell whether our algorithm will run in a reasonable time before implementing and actually running it? All these questions are addressed by a mathematical field called Combinatorics. In this module we will give an introduction to this field that will help us to answer basic versions of the above questions.

12 vídeos ((Total 54 mín.)), 4 leituras, 9 testes
12 videos
Rule of Sum3min
How Not to Use the Rule of Sum3min
Convenient Language: Sets4min
Generalized Rule of Sum3min
Number of Paths4min
Rule of Product3min
Back to Recursive Counting3min
Number of Tuples5min
Licence Plates3min
Tuples with Restrictions5min
4 leituras
Listing All Permutations5min
8 exercícios práticos
Rule of Sum in Programming4min
Numbers Divisible by 2 or 38min
Operations with Sets10min
Generalized Rule of Sum18min
Rule of Product in Programming10min
Applications of the Rule of Product12min
Counting with Restrictions20min
3 horas para concluir

Binomial Coefficients

In how many ways one can select a team of five students out of ten students? What is the number of non-negative integers with at five digits whose digits are decreasing? In how many ways one can get from the bottom left cell to the top right cell of a 5x5 grid, each time going either up or to the right? And why all these three numbers are equal? We'll figure this out in this module!

8 vídeos ((Total 76 mín.)), 4 leituras, 6 testes
8 videos
Number of Games in a Tournament10min
Pascal's Triangle9min
Row Sums10min
Binomial Theorem12min
Practice Counting13min
4 leituras
Generating Combinatorial Objects: Code10min
6 exercícios práticos
Number of Segments and Diagonals20min
Forming Sport Teams15min
Number of Iterations of Nested For Loops4min
Sum of the First Six Rows of Pascal's Triangle2min
Expanding (3a-2b)^k20min
Practice Counting10min
3 horas para concluir

Advanced Counting

We have already considered most of the most standard settings in Combinatorics, that allow us to address many counting problems. However, successful application of this knowledge on practice requires considerable experience in this kind of problems. In this module we will address the final standard setting in our course, combinations with repetitions, and then we will gain some experience by discussing various problems in Combinatorics.

8 vídeos ((Total 36 mín.)), 3 leituras, 8 testes
8 videos
Combinations with Repetitions7min
Distributing Assignments Among People3min
Distributing Candies Among Kids3min
Numbers with Fixed Sum of Digits4min
Numbers with Non-increasing Digits2min
Splitting into Working Groups4min
3 leituras
8 exercícios práticos
Combinations with Repetitions10min
Distributing Assignments Among People10min
Distributing Candies Among Kids15min
Numbers with Fixed Sum of Digits15min
Numbers with Non-increasing Digits7min
Splitting into Working Groups10min
Problems in Combinatorics45min
5 horas para concluir


The word "probability" is used quite often in the everyday life. However, not always we can speak about probability as some number: for that a mathematical model is needed. What is this mathematical model (probability space)? How to compute probabilities (if the model is given)? How to judge whether the model is adequate? What is conditional probability and Bayes' theorem? How our plausible reasoning can be interpreted in terms of Bayes' theorem? In this module we cover these questions using some simple examples of probability spaces and real life sutiations.

17 vídeos ((Total 126 mín.)), 4 leituras, 11 testes
17 videos
Galton Board6min
Natural Sciences and Mathematics6min
Rolling Dice7min
More Probability Spaces10min
Not Equiprobable Outcomes4min
More About Finite Spaces6min
Mathematics for Prisoners7min
Not All Questions Make Sense10min
What Is Conditional Probability?7min
How Reliable Is The Test?8min
Bayes' Theorem8min
Conditional Probability: A Paradox7min
Past and Future8min
Monty Hall Paradox8min
`Our Position'6min
4 leituras
10 exercícios práticos
Concentration for Galton Board10min
Computing Probabilities for Two Dice12min
Computing Probabilities: More Examples12min
Fair Decisions and Imperfect Coins20min
Inclusion-Exclusion Formula10min
Computing Conditional Probabilities16min
Prisoner, King and Conditional Probabilities10min
More Conditional Probabilities8min
More About Independence20min
Monty Hall Gone Crazy20min
3 horas para concluir

Random Variables

In the previous module we discussed how to compute probabilities of random events. But in many practical situation we are interested not only in positive or negative outcome, but also in some quantitative characteristics of an outcome. Among these cases are number of steps of an algorithms, number of points that one can win in the games involving any kind of randomness, all quantitative characteristics of a random person in some group of people. Basically settings of this kind arise in all situations when (a) any kind of uncertainty is presented (b) we are interested in quantitative characteristics. The mathematical model for this is called random variables. And we will discuss them in this module.

9 vídeos ((Total 54 mín.)), 6 leituras, 9 testes
9 videos
Linearity of Expectation7min
Birthday Problem10min
Expectation is Not All4min
From Expectation to Probability2min
Markov’s Inequality7min
Application to Algorithms4min
6 leituras
Average Value of a Dice Throw: Experiment10min
Dice Game Experiment10min
9 exercícios práticos
Random Variables8min
Linearity of Expectation10min
Bob’s Party5min
More Linearity10min
Average Income8min
Bob’s Party Revisited5min
Alice’s tests5min
2 horas para concluir

Project: Dice Games

In this module, we will apply accumulated knowledge to create a project solving a certain dice game. The game is very simple: two players pick a dice each from a given pool of dices with various numbers on their sides. Then each player throws his dice and the one with the greater number on his dice wins. The game looks very simple and it seems that it is very easy to play this game optimally once we know our pool of dices. Yet it turns out that this intuition is overwhelmingly wrong: the game turns out to be very counterintuitive. In this module we will discuss the game in detail and create a program that finds an optimal strategy to play the game on a given pool of dices.

3 vídeos ((Total 17 mín.)), 3 leituras, 1 teste
3 videos
Playing the Game8min
Project Description4min
3 leituras
Experiment: Dice Game10min
1 exercício prático
Final Project: Dice Game2h
47 avaliaçõesChevron Right


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Principais avaliações do Combinatorics and Probability

por ZBOct 13th 2018

I really enjoyed taking this course. The teaching was pretty good and some of the quiz questions will challenge you if you haven't done Combinatorics before.

por CZSep 11th 2018

The final project is hard for me cuz I don't have Python experience. and the logic is a little bit complicated. That's not for absolutely beginners!



Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

Vladimir Podolskii

Associate Professor
Computer Science Department

Sobre Universidade da Califórnia, San Diego

UC San Diego is an academic powerhouse and economic engine, recognized as one of the top 10 public universities by U.S. News and World Report. Innovation is central to who we are and what we do. Here, students learn that knowledge isn't just acquired in the classroom—life is their laboratory....

Sobre National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more. Learn more on

Sobre o Programa de cursos integrados Introduction to Discrete Mathematics for Computer Science

Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and financial analysts (it is not a coincidence that math puzzles are often used for interviews). We cover the basic notions and results (combinatorics, graphs, probability, number theory) that are universally needed. To deliver techniques and ideas in discrete mathematics to the learner we extensively use interactive puzzles specially created for this specialization. To bring the learners experience closer to IT-applications we incorporate programming examples, problems and projects in our courses....
Introduction to Discrete Mathematics for Computer Science

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