Este curso forma parte de Programa Especializado - Introduction to Discrete Mathematics for Computer Science
Combinatorics and Probability
ofrecido por
University of California San Diego
National Research University Higher School of Economics
Programa especializado Introduction to Discrete Mathematics for Computer ScienceUniversity of California San Diego
Acerca de este Curso
4.6
140 calificaciones
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.
Habilidades que obtendrás
Random VariableProbability InterpretationsProbabilityCombinatorics
Programa - Qué aprenderás en este curso
1
Sección
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 (Total: 54 min), 4 readings, 9 quizzes
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
Permutations9m
4 lecturas
Slides1m
Slides1m
Listing All Permutations5m
Slides1m
8 ejercicios de práctica
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
2
Sección
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 (Total: 76 min), 4 readings, 6 quizzes
8 videos
Number of Games in a Tournament10m
Combinations8m
Pascal's Triangle9m
Symmetries4m
Row Sums10m
Binomial Theorem12m
Practice Counting13m
4 lecturas
Generating Combinatorial Objects: Code10m
Slides10m
Slides10m
Slides10m
6 ejercicios de práctica
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
Sección
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 (Total: 36 min), 3 readings, 8 quizzes
8 videos
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
3 lecturas
Salads10m
Slides1m
Slides1m
8 ejercicios de práctica
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
4
Sección
Probability
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 (Total: 126 min), 4 readings, 11 quizzes
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
Independence8m
Monty Hall Paradox8m
`Our Position'6m
4 lecturas
Slidesm
Slidesm
Slidesm
Slidesm
10 ejercicios de práctica
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
4.6
83%
Principales revisiones
por 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.
por 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!
Instructores
Alexander S. Kulikov
Visiting ProfessorDepartment of Computer Science and Engineering
Vladimir Podolskii
Associate ProfessorComputer Science Department
Acerca de 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.
Acerca de 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 communications, IT, mathematics, engineering, and more.
Learn more on www.hse.ru
Acerca del programa especializado 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.
Preguntas Frecuentes
When will I have access to the lectures and assignments?
Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.
What will I get if I subscribe to this Specialization?
When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
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Is financial aid available?
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