이 강좌는 Introduction to Discrete Mathematics for Computer Science 전문 분야의 일부입니다.

제공자:

캘리포니아 샌디에고 대학교

국립 연구 고등 경제 대학

About this Course

최근 조회 22,704회

Counting is one of the basic mathematically related tasks we encounter on a day to day basis. The main question here is the following. If we need to count something, can we do anything better than just counting all objects one by one? Do we need to create a list of all phone numbers to ensure that there are enough phone numbers for everyone? Is there a way to tell that 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 course we discuss most standard combinatorial settings that can help to answer questions of this type. We will especially concentrate on developing the ability to distinguish these settings in real life and algorithmic problems. This will help the learner to actually implement new knowledge. Apart from that we will discuss recursive technique for counting that is important for algorithmic implementations.
One of the main `consumers’ of Combinatorics is Probability Theory. This area is connected with numerous sides of life, on one hand being an important concept in everyday life and on the other hand being an indispensable tool in such modern and important fields as Statistics and Machine Learning. In this course we will concentrate on providing the working knowledge of basics of probability and a good intuition in this area. The practice shows that such an intuition is not easy to develop.
In the end of the course we will create a program that successfully plays a tricky and very counterintuitive dice game.
As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.

지금 바로 시작해 나만의 일정에 따라 학습을 진행하세요.

일정에 따라 마감일을 재설정합니다.

권장: 6 weeks, 3-5 hours/week ...

자막: 영어, 그리스어, 중국어 (간체자)

Random VariableProbability InterpretationsProbabilityCombinatorics

지금 바로 시작해 나만의 일정에 따라 학습을 진행하세요.

일정에 따라 마감일을 재설정합니다.

권장: 6 weeks, 3-5 hours/week ...

자막: 영어, 그리스어, 중국어 (간체자)

주

1Suppose 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 (Total 54 min), 4 readings, 9 quizzes

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

Permutations9m

Slides1m

Slides1m

Listing All Permutations5m

Slides1m

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

Tuples5m

Counting with Restrictions20m

주

2In 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 (Total 76 min), 4 readings, 6 quizzes

Number of Games in a Tournament10m

Combinations8m

Pascal's Triangle9m

Symmetries4m

Row Sums10m

Binomial Theorem12m

Practice Counting13m

Generating Combinatorial Objects: Code10m

Slides10m

Slides10m

Slides10m

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

주

3We 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 (Total 36 min), 3 readings, 8 quizzes

Review3m

Salad5m

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

Salads10m

Slides1m

Slides1m

Salads10m

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

주

4The 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 (Total 126 min), 4 readings, 11 quizzes

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

Independence8m

Monty Hall Paradox8m

`Our Position'6m

Slides

Slides

Slides

Slides

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

주

5In 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 (Total 54 min), 6 readings, 9 quizzes

Average5m

Expectation9m

Linearity of Expectation7m

Birthday Problem10m

Expectation is Not All4m

From Expectation to Probability2m

Markov’s Inequality7m

Application to Algorithms4m

Average Value of a Dice Throw: Experiment10m

Slides1m

Slides1m

Dice Game Experiment10m

Slides1m

Slides1m

Random Variables8m

Average12m

Expectations10m

Linearity of Expectation10m

Bob’s Party5m

More Linearity10m

Average Income8m

Bob’s Party Revisited5m

Alice’s tests5m

주

6In 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 (Total 17 min), 3 readings, 1 quiz

Experiment: Dice Game10m

Slides1m

Slides1m

Final Project: Dice Game

4.6

47개의 리뷰이 강좌를 통해 확실한 경력상 이점 얻기

급여 인상 또는 승진하기

대학: ZB•Oct 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.

대학: CZ•Sep 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!

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....

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...

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....

강의 및 과제를 언제 이용할 수 있게 되나요?

강좌에 등록하면 바로 모든 비디오, 테스트 및 프로그래밍 과제(해당하는 경우)에 접근할 수 있습니다. 상호 첨삭 과제는 이 세션이 시작된 경우에만 제출하고 검토할 수 있습니다. 강좌를 구매하지 않고 살펴보기만 하면 특정 과제에 접근하지 못할 수 있습니다.

이 전문 분야를 구독하면 무엇을 이용할 수 있나요?

강좌를 등록하면 전문 분야의 모든 강좌에 접근할 수 있고 강좌를 완료하면 수료증을 취득할 수 있습니다. 전자 수료증이 성취도 페이지에 추가되며 해당 페이지에서 수료증을 인쇄하거나 LinkedIn 프로필에 수료증을 추가할 수 있습니다. 강좌 내용만 읽고 살펴보려면 해당 강좌를 무료로 청강할 수 있습니다.

환불 규정은 어떻게 되나요?

재정 지원을 받을 수 있나요?

궁금한 점이 더 있으신가요? 학습자 도움말 센터를 방문해 보세요.