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Stat400Lec4(Ch1.3)_ans

# Stat400Lec4(Ch1.3)_ans - Chapter 1.3 Methods of Enumeration...

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Chapter 1.3 Methods of Enumeration Multiplication principle. Permutation and combination. Sampling with/without replacement. Ordered/unordered sample. Distinguishable permutation.

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Multiplication Principle In general, if there are p experiments and the first has N1 possible outcomes, the second N2, …, and the p th Np possible outcomes, then there are a total of N1 × N2 × Np possible outcomes for the p experiments. Example: A DNA molecule is a sequence of four types of nucleotides, denoted by A; G;C and T. For a molecule 1 million (10^6) units long, how many different possible sequences? 4 × 4 × 4... × 4 = 4^(1000,000)
Permutation In general, there are n! ways to fill in n positions with n different objects, where n! = n × (n - 1) × ... × 2 × 1; 0! = 1. Definition: Each of the n! arrangements of n different objects is called a permutation of the n objects. Example: Suppose that 3 positions are to be filled with 3 different objects (O1,O2,O3). How many possible arrangements are there? 3 × 2 × 1 =6

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Each of the n P r arrangements is called a permutation of the n objects taken r at a time. r n P
Example There are 5 seats in a classroom and 3 students registered for the class. How many seating arrangements do we have? Using Multiplication rule: E1: S1 has 5 choices , E2: S2 has 4 choices, E3: S3 has 3 choices # of arrangements for composite events E1E2E3 is: 5 × 4 × 3=5!/2!=5!/(5-3)!= 60.

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With or without replacement Sampling with replacement: an object is selected and then replaced (put it back) before the next object is selected.
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Stat400Lec4(Ch1.3)_ans - Chapter 1.3 Methods of Enumeration...

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