Combinatorics and Probability from Coursera

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 online 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.

What's inside

Syllabus

Basic Counting

Counting is one of the basic mathematically related tasks we encounter on a day to day basis. The main question here is the following: can we count the number of objects without listing all of them? This question arises naturally in various scenarios both in real life and in Computer Science. What is the number of different phone numbers or license plates? What is the number of different combinations one needs to brute force in order to crack a password? Is there a way to tell that an 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 consider the basic building blocks of combinatorics. All of them are easy to understand and at the same time are powerful enough to handle various non-trivial questions. To help you to develop an intuition, we consider short Python code snippets for generating the objects to be counted.

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 most 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!

Advanced Counting

We have already considered most of the 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.

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.

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.

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.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Introduces foundational concepts in Combinatorics necessary for careers involving IT

Offers a practical approach to counting and probability, important concepts in computer science and data science

Provides an accessible introduction to Probability Theory and its applications in real life

Suitable for individuals with basic math and programming knowledge who seek to develop their skills in counting and probability

Teaches recursive techniques for counting, which are essential for efficient algorithms

Provides opportunities to apply knowledge through a hands-on project

Reviews summary

Challenging but rewarding course in combinatorics and probability

Learners say this course starts with engaging assignments about combinatorics, then delves deeply into probability. Many find the material to be difficult but rewarding, especially those who are familiar with high school mathematics. Instructors are described as knowledgeable and have a clear teaching style, however pronunciation issues make understanding difficult at times. Some assignments and exercises are said to be tough and the explanations in the final project are said to be unclear. Overall, learners recommend this course to those who have a strong foundation in math and are willing to put in effort.

Fun and Thought-Provoking

"Learners say this course starts with engaging assignments about combinatorics, then delves deeply into probability."

"Many find the material to be difficult but rewarding, especially those who are familiar with high school mathematics."

Challenging but Rewarding

"Learners say this course starts with engaging assignments about combinatorics, then delves deeply into probability."

"Many find the material to be difficult but rewarding, especially those who are familiar with high school mathematics."

"Some assignments and exercises are said to be tough and the explanations in the final project are said to be unclear."

Final Project Instructions Lacking

"Some assignments and exercises are said to be tough and the explanations in the final project are said to be unclear."

Knowledgeable but Difficult to Understand

"Instructors are described as knowledgeable and have a clear teaching style, however pronunciation issues make understanding difficult at times."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Combinatorics and Probability with these activities:

Review basic probability and statistics

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Reviewing basic probability and statistics will provide a solid foundation for understanding the concepts covered in this course.

Browse courses on Probability

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Read through your notes or textbook chapters on probability and statistics.
Solve practice problems to test your understanding.

Solve combinatorial problems

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Solving combinatorial problems will help you develop the skills necessary to apply the concepts learned in this course to real-world scenarios.

Browse courses on Combinatorics

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Find a set of practice problems online or in a textbook.
Set a timer for yourself and try to solve as many problems as you can within the time limit.
Check your answers and identify any areas where you need improvement.

Create a cheat sheet on probability distributions

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Creating a cheat sheet on probability distributions will help you organize and retain the information you learn in this course.

Browse courses on Probability

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Gather information on different probability distributions, such as their formulas, properties, and applications.
Create a table or diagram that summarizes the key information for each distribution.

Three other activities

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Watch video tutorials on advanced combinatorial techniques

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Watching video tutorials on advanced combinatorial techniques will expose you to new ideas and approaches that can enhance your understanding of the subject.

Browse courses on Combinatorics

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Search for video tutorials on specific combinatorial topics that you are interested in.
Take notes while watching the tutorials and make sure to pause and rewind when necessary.
Practice the techniques you learn by solving problems or completing exercises.

Attend a workshop on probability and statistics

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Attending a workshop on probability and statistics will provide you with an opportunity to learn from experts and engage in hands-on activities.

Browse courses on Probability

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Find a relevant workshop in your area or online.
Register for the workshop and make sure to attend all sessions.
Participate actively in the discussions and activities.

Develop a program to simulate a dice game

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Developing a program to simulate a dice game will help you apply the concepts of probability and statistics to a real-world scenario.

Browse courses on Probability

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Design the game rules and determine the probabilities of different outcomes.
Choose a programming language and write the code for the simulation.
Test the simulation by running it multiple times and analyzing the results.

Career center

Learners who complete Combinatorics and Probability will develop knowledge and skills that may be useful to these careers:

Data Scientist

Data Scientists use probability to build mathematical models that can help organizations make better decisions. The course covers the basics of probability, such as how to calculate probabilities, conditional probability, and Bayes' theorem. This knowledge can help Data Scientists develop models that are more accurate and reliable. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including data science.

See salaries and explore the career path for Data Scientist

Machine Learning Engineer

Machine Learning Engineers use probability and statistics to develop and implement machine learning algorithms. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Machine Learning Engineers develop models that can learn from data and make predictions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including machine learning.

See salaries and explore the career path for Machine Learning Engineer

Business Analyst

Business Analysts use probability and statistics to analyze data and make business decisions. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Business Analysts develop models that can predict future trends and make informed business decisions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including business analysis.

See salaries and explore the career path for Business Analyst

Operations Research Analyst

Operations Research Analysts use probability and statistics to solve problems in a variety of fields, including logistics, manufacturing, and healthcare. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Operations Research Analysts develop models that can optimize processes and improve efficiency. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including operations research.

See salaries and explore the career path for Operations Research Analyst

Statistician

Statisticians use probability and statistics to collect, analyze, and interpret data. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Statisticians develop models that can be used to make informed decisions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including statistics.

See salaries and explore the career path for Statistician

Software Engineer

Software Engineers use combinatorics to design and implement efficient algorithms. The course covers the basics of combinatorics, such as how to count the number of objects in a set, and how to generate permutations and combinations. This knowledge can help Software Engineers develop algorithms that run faster and use less memory. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including software engineering.

See salaries and explore the career path for Software Engineer

Data Analyst

Data Analysts use probability and statistics to analyze data and make decisions. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Data Analysts develop models that can identify trends and make informed decisions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including data analysis.

See salaries and explore the career path for Data Analyst

Financial Analyst

Financial Analysts use probability and statistics to analyze financial data and make investment decisions. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Financial Analysts develop models that can predict future financial performance and make informed investment decisions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including financial analysis.

See salaries and explore the career path for Financial Analyst

Quantitative Analyst

Quantitative Analysts use probability and statistics to make financial decisions. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Quantitative Analysts develop models that can predict future financial performance. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including quantitative finance.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries use probability and statistics to assess risk. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Actuaries develop models that can predict the likelihood of future events. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including actuarial science.

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Risk Manager

Risk Managers use probability and statistics to assess and manage risk. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Risk Managers develop models that can predict the likelihood of future events and assess the potential impact of those events. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including risk management.

See salaries and explore the career path for Risk Manager

Epidemiologist

Epidemiologists use probability and statistics to study the distribution and determinants of health-related states or events (including disease), and the application of this study to the control of diseases and other health problems. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Epidemiologists develop models that can be used to understand the causes of disease and develop strategies to prevent and control it. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including epidemiology.

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Biostatistician

Biostatisticians use probability and statistics to analyze data in the field of biology. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Biostatisticians develop models that can be used to understand biological processes and make informed decisions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including biostatistics.

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Market Researcher

Market Researchers use probability and statistics to collect and analyze data about consumer behavior. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Market Researchers develop models that can predict consumer behavior and make informed marketing decisions. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including market research.

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Insurance Underwriter

Insurance Underwriters use probability and statistics to assess risk and determine insurance premiums. The course covers the basics of probability and statistics, such as how to calculate probabilities, expected values, and standard deviations. This knowledge can help Insurance Underwriters develop models that can predict the likelihood of future events and assess the potential impact of those events. Additionally, the course covers advanced counting techniques, which can be used to solve problems in a variety of fields, including insurance underwriting.

See salaries and explore the career path for Insurance Underwriter