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Untertitel: Englisch, Griechisch, Chinesisch (vereinfacht)

Kompetenzen, die Sie erwerben

Random VariableProbability InterpretationsProbabilityCombinatorics

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Ca. 21 Stunde zum Abschließen

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


Untertitel: Englisch, Griechisch, Chinesisch (vereinfacht)

Lehrplan - Was Sie in diesem Kurs lernen werden

3 Stunden zum Abschließen

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 Videos (Gesamt 54 min), 4 Lektüren, 9 Quiz
12 Videos
Rule of Sum3m
How Not to Use the Rule of Sum3m
Convenient Language: Sets4m
Generalized Rule of Sum3m
Number of Paths4m
Rule of Product3m
Back to Recursive Counting3m
Number of Tuples5m
Licence Plates3m
Tuples with Restrictions5m
4 Lektüren
Listing All Permutations5m
8 praktische Übungen
Rule of Sum in Programming4m
Numbers Divisible by 2 or 38m
Operations with Sets10m
Generalized Rule of Sum18m
Rule of Product in Programming10m
Applications of the Rule of Product12m
Counting with Restrictions20m
3 Stunden zum Abschließen

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 Videos (Gesamt 76 min), 4 Lektüren, 6 Quiz
8 Videos
Number of Games in a Tournament10m
Pascal's Triangle9m
Row Sums10m
Binomial Theorem12m
Practice Counting13m
4 Lektüren
Generating Combinatorial Objects: Code10m
6 praktische Übungen
Number of Segments and Diagonals20m
Forming Sport Teams15m
Number of Iterations of Nested For Loops4m
Sum of the First Six Rows of Pascal's Triangle2m
Expanding (3a-2b)^k20m
Practice Counting10m
3 Stunden zum Abschließen

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 Videos (Gesamt 36 min), 3 Lektüren, 8 Quiz
8 Videos
Combinations with Repetitions7m
Distributing Assignments Among People3m
Distributing Candies Among Kids3m
Numbers with Fixed Sum of Digits4m
Numbers with Non-increasing Digits2m
Splitting into Working Groups4m
3 Lektüren
8 praktische Übungen
Combinations with Repetitions10m
Distributing Assignments Among People10m
Distributing Candies Among Kids15m
Numbers with Fixed Sum of Digits15m
Numbers with Non-increasing Digits7m
Splitting into Working Groups10m
Problems in Combinatorics45m
5 Stunden zum Abschließen


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 Videos (Gesamt 126 min), 4 Lektüren, 11 Quiz
17 Videos
Galton Board6m
Natural Sciences and Mathematics6m
Rolling Dice7m
More Probability Spaces10m
Not Equiprobable Outcomes4m
More About Finite Spaces6m
Mathematics for Prisoners7m
Not All Questions Make Sense10m
What Is Conditional Probability?7m
How Reliable Is The Test?8m
Bayes' Theorem8m
Conditional Probability: A Paradox7m
Past and Future8m
Monty Hall Paradox8m
`Our Position'6m
4 Lektüren
10 praktische Übungen
Concentration for Galton Board10m
Computing Probabilities for Two Dice12m
Computing Probabilities: More Examples12m
Fair Decisions and Imperfect Coins20m
Inclusion-Exclusion Formula10m
Computing Conditional Probabilities16m
Prisoner, King and Conditional Probabilities10m
More Conditional Probabilities8m
More About Independence20m
Monty Hall Gone Crazy20m
3 Stunden zum Abschließen

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 Videos (Gesamt 54 min), 6 Lektüren, 9 Quiz
9 Videos
Linearity of Expectation7m
Birthday Problem10m
Expectation is Not All4m
From Expectation to Probability2m
Markov’s Inequality7m
Application to Algorithms4m
6 Lektüren
Average Value of a Dice Throw: Experiment10m
Dice Game Experiment10m
9 praktische Übungen
Random Variables8m
Linearity of Expectation10m
Bob’s Party5m
More Linearity10m
Average Income8m
Bob’s Party Revisited5m
Alice’s tests5m
2 Stunden zum Abschließen

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 Videos (Gesamt 17 min), 3 Lektüren, 1 Quiz
3 Videos
Playing the Game8m
Project Description4m
3 Lektüren
Experiment: Dice Game10m
1 praktische Übung
Final Project: Dice Game
47 BewertungenChevron Right


ziehen Sie für Ihren Beruf greifbaren Nutzen aus diesem Kurs


erhalten Sie eine Gehaltserhöhung oder Beförderung


von 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.

von 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

Über University of California 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....

Über 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 www.hse.ru...

Über die Spezialisierung 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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