Combinatorics and Probability

Go to Course: https://www.coursera.org/learn/combinatorics

Introduction

**Course Review: Combinatorics and Probability on Coursera** If you are looking for an engaging and intellectually stimulating course that combines theoretical mathematics with practical application, then "Combinatorics and Probability" on Coursera is definitely worth your time. This course offers a solid introduction to combinatorial mathematics and probability theory, making it an excellent choice for students, professionals, and anyone with a curious mind interested in enhancing their analytical skills. ### Overview Counting is a fundamental task that transcends everyday situations, from determining the number of combinations in a game of chance to evaluating complex algorithms in computer science. This course answers critical questions like: Can we count without listing every object? How do we assess the efficiency of algorithms before running them? Through these inquiries, the course delves into the fascinating realms of combinatorics and probability. ### Course Syllabus Breakdown #### 1. Basic Counting In this module, participants are introduced to the essential concepts of combinatorics. The focus is on effective counting techniques that allow us to quantify scenarios without exhaustive listing. The inclusion of Python code snippets not only aids comprehension but also enhances programming skills. Understanding basic counting principles lays a strong foundation for tackling more complex problems in the remainder of the course. #### 2. Binomial Coefficients Here, students explore binomial coefficients and their applications. Through practical examples—like selecting teams and traversing grids—learners discover the interconnectedness of different combinatorial problems. This hands-on approach helps to crystallize understanding while revealing the beauty of mathematical patterns. #### 3. Advanced Counting Moving beyond the foundational principles, this module introduces advanced counting techniques, including combinations with repetitions. It emphasizes the importance of gaining experience through practice, providing learners with challenging problems to solve. This section is invaluable for those looking to deepen their understanding and apply their knowledge to real-world scenarios. #### 4. Probability This module addresses the mathematical models underlying probability, introducing concepts like probability spaces and conditional probability. The clear, practical examples help demystify often-misunderstood probability principles, including the application of Bayes’ theorem. This knowledge is crucial for making informed decisions in uncertain situations. #### 5. Random Variables The course progresses to examine the quantitative aspects of outcomes in various scenarios. By introducing the concept of random variables, students learn to analyze and interpret uncertainty. This module is particularly beneficial for anyone interested in data-driven decision-making and statistical modeling. #### 6. Project: Dice Games The culmination of the course manifests in an exciting hands-on project that involves optimizing strategies for a simple dice game. This practical application ties together all the theoretical concepts covered in previous modules. It challenges students to think critically and creatively, ensuring that the learning experience is not only educational but also enjoyable. ### Recommendations "Combinatorics and Probability" is highly recommended for: - **Students in Mathematics, Data Science, or Computer Science**: This course provides essential knowledge that will benefit your studies and potential career. - **Professionals in Analytics or Risk Assessment**: Understanding combinatorial methods and probability is crucial for making data-driven decisions. - **Curious Learners**: If you have a genuine interest in mathematics and want to strengthen your analytical thinking, this course is ideal. ### Conclusion Overall, "Combinatorics and Probability" on Coursera stands out for its comprehensive syllabus, clear instruction, and practical applications. Whether you are looking to enhance your professional skill set or simply enjoy the beauty of mathematics, this course equips you with valuable tools for understanding complex problems and making informed decisions. With engaging content and a hands-on project, it ensures a fulfilling and intellectually rewarding experience. Enroll today and unlock the power of combinatorics and probability!

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.

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!

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.

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.

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.

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.