À propos de ce cours
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47 avis

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Approx. 21 heures pour terminer

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


Sous-titres : Anglais, Grec, Chinois (simplifié)

Compétences que vous acquerrez

Random VariableProbability InterpretationsProbabilityCombinatorics

100 % en ligne

Commencez dès maintenant et apprenez aux horaires qui vous conviennent.

Dates limites flexibles

Réinitialisez les dates limites selon votre disponibilité.

Niveau débutant

Approx. 21 heures pour terminer

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


Sous-titres : Anglais, Grec, Chinois (simplifié)

Programme du cours : ce que vous apprendrez dans ce cours

3 heures pour terminer

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 vidéos (Total 54 min), 4 lectures, 9 quiz
12 vidéos
Rule of Sum3 min
How Not to Use the Rule of Sum3 min
Convenient Language: Sets4 min
Generalized Rule of Sum3 min
Number of Paths4 min
Rule of Product3 min
Back to Recursive Counting3 min
Number of Tuples5 min
Licence Plates3 min
Tuples with Restrictions5 min
Permutations9 min
4 lectures
Slides1 min
Slides1 min
Listing All Permutations5 min
Slides1 min
8 exercices pour s'entraîner
Rule of Sum in Programming4 min
Numbers Divisible by 2 or 38 min
Operations with Sets10 min
Generalized Rule of Sum18 min
Rule of Product in Programming10 min
Applications of the Rule of Product12 min
Tuples5 min
Counting with Restrictions20 min
3 heures pour terminer

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 vidéos (Total 76 min), 4 lectures, 6 quiz
8 vidéos
Number of Games in a Tournament10 min
Combinations8 min
Pascal's Triangle9 min
Symmetries4 min
Row Sums10 min
Binomial Theorem12 min
Practice Counting13 min
4 lectures
Generating Combinatorial Objects: Code10 min
Slides10 min
Slides10 min
Slides10 min
6 exercices pour s'entraîner
Number of Segments and Diagonals20 min
Forming Sport Teams15 min
Number of Iterations of Nested For Loops4 min
Sum of the First Six Rows of Pascal's Triangle2 min
Expanding (3a-2b)^k20 min
Practice Counting10 min
3 heures pour terminer

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 vidéos (Total 36 min), 3 lectures, 8 quiz
8 vidéos
Review3 min
Salad5 min
Combinations with Repetitions7 min
Distributing Assignments Among People3 min
Distributing Candies Among Kids3 min
Numbers with Fixed Sum of Digits4 min
Numbers with Non-increasing Digits2 min
Splitting into Working Groups4 min
3 lectures
Salads10 min
Slides1 min
Slides1 min
8 exercices pour s'entraîner
Salads10 min
Combinations with Repetitions10 min
Distributing Assignments Among People10 min
Distributing Candies Among Kids15 min
Numbers with Fixed Sum of Digits15 min
Numbers with Non-increasing Digits7 min
Splitting into Working Groups10 min
Problems in Combinatorics45 min
5 heures pour terminer


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 vidéos (Total 126 min), 4 lectures, 11 quiz
17 vidéos
Galton Board6 min
Natural Sciences and Mathematics6 min
Rolling Dice7 min
More Probability Spaces10 min
Not Equiprobable Outcomes4 min
More About Finite Spaces6 min
Mathematics for Prisoners7 min
Not All Questions Make Sense10 min
What Is Conditional Probability?7 min
How Reliable Is The Test?8 min
Bayes' Theorem8 min
Conditional Probability: A Paradox7 min
Past and Future8 min
Independence8 min
Monty Hall Paradox8 min
`Our Position'6 min
4 lectures
10 exercices pour s'entraîner
Concentration for Galton Board10 min
Computing Probabilities for Two Dice12 min
Computing Probabilities: More Examples12 min
Fair Decisions and Imperfect Coins20 min
Inclusion-Exclusion Formula10 min
Computing Conditional Probabilities16 min
Prisoner, King and Conditional Probabilities10 min
More Conditional Probabilities8 min
More About Independence20 min
Monty Hall Gone Crazy20 min
3 heures pour terminer

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 vidéos (Total 54 min), 6 lectures, 9 quiz
9 vidéos
Average5 min
Expectation9 min
Linearity of Expectation7 min
Birthday Problem10 min
Expectation is Not All4 min
From Expectation to Probability2 min
Markov’s Inequality7 min
Application to Algorithms4 min
6 lectures
Average Value of a Dice Throw: Experiment10 min
Slides1 min
Slides1 min
Dice Game Experiment10 min
Slides1 min
Slides1 min
9 exercices pour s'entraîner
Random Variables8 min
Average12 min
Expectations10 min
Linearity of Expectation10 min
Bob’s Party5 min
More Linearity10 min
Average Income8 min
Bob’s Party Revisited5 min
Alice’s tests5 min
2 heures pour terminer

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 vidéos (Total 17 min), 3 lectures, 1 quiz
3 vidéos
Playing the Game8 min
Project Description4 min
3 lectures
Experiment: Dice Game10 min
Slides1 min
Slides1 min
1 exercice pour s'entraîner
Final Project: Dice Games
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Meilleurs avis

par 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.

par 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

À propos de Université de Californie à 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....

À propos de Université nationale de recherche, École des hautes études en sciences économiques

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 www.hse.ru...

À propos de la Spécialisation 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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