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.
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Combinatorics and Probability
캘리포니아 샌디에고 대학교이 강좌에 대하여
귀하가 습득할 기술
- Random Variable
- Probability Interpretations
- Probability
- Combinatorics
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캘리포니아 샌디에고 대학교
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강의 계획표 - 이 강좌에서 배울 내용
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.
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COMBINATORICS AND PROBABILITY의 최상위 리뷰
It was pretty interesting. Although, the requirement to know the python programming in the last course task was really unexpected to me.
informative material presented clearly and simply. I had studied bayes before and it was nice to get a concise review.
I've covered these topics before but there were a lot of great problems and extensions of concepts in this course. Worth doing.
Special thanks to Prof. Vladimir Podolskii and Prof. Alexander S. Kulikov for their amazing explanations and diligent visuals of the concepts as well as problem sets. You Rock!
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