2. Combinatorics PDF

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Summary of Ch2 of Introduction to Probability, Statistics, and Random Processes by Pishro-Nik...

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2. Combinatorics Tuesday, August 28, 2018

4:24 PM

2.1 Counting

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"At least" means union When you are dealing with unions, you should do the complement to work with the intersections instead Sampling: choosing an element from that set. With or without replacement: self explanatory Ordered or unordered: self explanatory 4 possibilities of sampling from sets: ○ Ordered sampling with replacement (simple multiplying) ○ Ordered sampling without replacement (permutations) ○ Unordered sampling without replacement (combinations) ○ Unordered sampling with replacement (stars and bars) Summary of what is to come:

2.1.1 Ordered Sampling with Replacement

2.1.2 Ordered Sampling without Replacement: Permutations

* Note: when dealing with a "at least", do complements (as stated above in 2.1) *

2.1.3 Unordered Sampling without Replacement: Combinations

*Good note*

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Stated as "n choose k" For probability, don't forget to divide by the total number of possibilities

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Has to do with successes and failures, where one event is considered a success and all others are considered failures. There is also the multinomial coefficients case where a group of k wants to be divided into r groups where ni is the size of group i.

2.1.3 Unordered Sampling with Replacement - Basically the total number of distinct k samples from n-element set is trying to see how may solutions there are to the equation:

Which is equal to:

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Basically think of this as stars and bars...