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lec1 - Lecture 1 Combinatorial analysis Text Based on...

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Lecture 1: Combinatorial analysis. Text: Based on § 1.1-1.5 Tool used in calculating probabilities Combinatorics: theory of counting. Basic facts from combinatorics: Multiplication principle/Basic Principle of counting: If one experiment has m outcomes and another experiment has n outcomes, then there are mn possible outcomes for the two experiments. Example: You are asked to create a password with one alphabet followed by one digit from 0-9. How many combinations of alphabets and digits are there (i.e. how many possible such passwords can be created)? 26 alphabets × 10 digits. Multiplication Principal for More than Two Experiments/Generalized ba- sic principle of counting: (stated as in Casella and Berger) If a job consists of k separate tasks, the i th of which can be done in n i ways, i = 1 , . . . , k , then the entire job can be done in n 1 × n 2 × . . . × n k ways. Example: (Rice) An 8-bit binary word is a sequence of 8 digits, of which each may be either a 0 or a 1. How many different 8-bit words are there? Example: (Rice) A DNA molecule is a sequence of four types of nucleotides, denoted by A,G,C, and T. The molecule can be millions of units long and can thus encode an enormous amount of information. For example, for a molecule 1 million (10 6 ) units long, there are different possible sequences.
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