Об этом курсе
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Начальный уровень


Субтитры: Английский, Греческий, Китайский (упрощенное письмо)

Приобретаемые навыки

Random VariableProbability InterpretationsProbabilityCombinatorics

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Субтитры: Английский, Греческий, Китайский (упрощенное письмо)

Программа курса: что вы изучите

3 ч. на завершение

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 видео ((всего 54 мин.)), 4 материалов для самостоятельного изучения, 9 тестов
12 видео
Rule of Sum3мин
How Not to Use the Rule of Sum3мин
Convenient Language: Sets4мин
Generalized Rule of Sum3мин
Number of Paths4мин
Rule of Product3мин
Back to Recursive Counting3мин
Number of Tuples5мин
Licence Plates3мин
Tuples with Restrictions5мин
4 материала для самостоятельного изучения
Listing All Permutations5мин
8 практического упражнения
Rule of Sum in Programming4мин
Numbers Divisible by 2 or 38мин
Operations with Sets10мин
Generalized Rule of Sum18мин
Rule of Product in Programming10мин
Applications of the Rule of Product12мин
Counting with Restrictions20мин
3 ч. на завершение

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 видео ((всего 76 мин.)), 4 материалов для самостоятельного изучения, 6 тестов
8 видео
Number of Games in a Tournament10мин
Pascal's Triangle9мин
Row Sums10мин
Binomial Theorem12мин
Practice Counting13мин
4 материала для самостоятельного изучения
Generating Combinatorial Objects: Code10мин
6 практического упражнения
Number of Segments and Diagonals20мин
Forming Sport Teams15мин
Number of Iterations of Nested For Loops4мин
Sum of the First Six Rows of Pascal's Triangle2мин
Expanding (3a-2b)^k20мин
Practice Counting10мин
3 ч. на завершение

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 видео ((всего 36 мин.)), 3 материалов для самостоятельного изучения, 8 тестов
8 видео
Combinations with Repetitions7мин
Distributing Assignments Among People3мин
Distributing Candies Among Kids3мин
Numbers with Fixed Sum of Digits4мин
Numbers with Non-increasing Digits2мин
Splitting into Working Groups4мин
3 материала для самостоятельного изучения
8 практического упражнения
Combinations with Repetitions10мин
Distributing Assignments Among People10мин
Distributing Candies Among Kids15мин
Numbers with Fixed Sum of Digits15мин
Numbers with Non-increasing Digits7мин
Splitting into Working Groups10мин
Problems in Combinatorics45мин
5 ч. на завершение


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 видео ((всего 126 мин.)), 4 материалов для самостоятельного изучения, 11 тестов
17 видео
Galton Board6мин
Natural Sciences and Mathematics6мин
Rolling Dice7мин
More Probability Spaces10мин
Not Equiprobable Outcomes4мин
More About Finite Spaces6мин
Mathematics for Prisoners7мин
Not All Questions Make Sense10мин
What Is Conditional Probability?7мин
How Reliable Is The Test?8мин
Bayes' Theorem8мин
Conditional Probability: A Paradox7мин
Past and Future8мин
Monty Hall Paradox8мин
`Our Position'6мин
4 материала для самостоятельного изучения
10 практического упражнения
Concentration for Galton Board10мин
Computing Probabilities for Two Dice12мин
Computing Probabilities: More Examples12мин
Fair Decisions and Imperfect Coins20мин
Inclusion-Exclusion Formula10мин
Computing Conditional Probabilities16мин
Prisoner, King and Conditional Probabilities10мин
More Conditional Probabilities8мин
More About Independence20мин
Monty Hall Gone Crazy20мин
3 ч. на завершение

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 видео ((всего 54 мин.)), 6 материалов для самостоятельного изучения, 9 тестов
9 видео
Linearity of Expectation7мин
Birthday Problem10мин
Expectation is Not All4мин
From Expectation to Probability2мин
Markov’s Inequality7мин
Application to Algorithms4мин
6 материала для самостоятельного изучения
Average Value of a Dice Throw: Experiment10мин
Dice Game Experiment10мин
9 практического упражнения
Random Variables8мин
Linearity of Expectation10мин
Bob’s Party5мин
More Linearity10мин
Average Income8мин
Bob’s Party Revisited5мин
Alice’s tests5мин
2 ч. на завершение

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 видео ((всего 17 мин.)), 3 материалов для самостоятельного изучения, 1 тест
3 видео
Dice Game3мин
Playing the Game8мин
Project Description4мин
3 материала для самостоятельного изучения
Experiment: Dice Game10мин
1 практическое упражнение
Final Project: Dice Game
Рецензии: 48Chevron Right


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Лучшие отзывы о курсе Combinatorics and Probability

автор: 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.

автор: 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

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