09-07 Axioms - Counting

# 09-07 Axioms - Counting - BTRY 4080 STSCI 4080 Fall 2010 70...

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BTRY 4080 / STSCI 4080 Fall 2010 70 Equally Likely Outcomes & Combinatorial Analysis Ross: Chapter 1, Sections 1.1 - 1.5 Ross: Chapter 2, Section 2.5

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BTRY 4080 / STSCI 4080 Fall 2010 71 Sample Spaces With Equally Likely Outcomes (Part 1) Assumption: For an experiment with a sample space S that has a fnite number o± possible outcomes, all outcomes are considered equally likely to occur. Generic example: S = { s 1 ,s 2 3 ,...,s N } ±or N< . “Equally likely” outcomes simply means that P ( { s i } )= 1 N ,i =1 to N For any event E S : P ( E Number o± outcomes in E Number o± outcomes in S = N i =1 I { s i E } N
BTRY 4080 / STSCI 4080 Fall 2010 72 Example: If two dice are rolled, what is the probability of the event E = { sum of upturned faces equals 7 } ? S = { ?? } E = { ?? } P ( E )?

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BTRY 4080 / STSCI 4080 Fall 2010 73 Example: Suppose we toss 4 fair dice. Let A = { all dice show a different face } How should we determine P ( A ) ? Proceeding as before, one could: (1) Enumerate all 1296 ways in which 4 dice can land “face up” (i.e., ±nd S ); (2) Determine that there are 360 outcomes in S that also correspond to A ; (3) Compute P ( A ) = 360 / 1296 . =0 . 278 (i.e., 28% ).
BTRY 4080 / STSCI 4080 Fall 2010 74 Basic Principle of Counting, Section 1.2 1. Basic principle: Suppose that completion of a “job” involves two tasks, say A and B .The rea re m ways to complete task A ; there are n ways to complete taks B . Suppose both tasks must be completed in order to complete the job. Then, there are m × n ways in which one can complete the job. 2. Generalized principle: Suppose a job consists of k separate tasks, and the i th task can be done in n i ways. If completion of the job requires completion of all k tasks, then there are ± k i =1 n i possible ways in which to complete the entire job.

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BTRY 4080 / STSCI 4080 Fall 2010 75 The Multiplicative Principle 2 A B Destination Origin 1 m 2 1 n Task A can be completed in one of m ways. For each possible choice, task B can be completed in one of n ways. Hence, total number of ways to complete both tasks (hence, the job) is m × n .
BTRY 4080 / STSCI 4080 Fall 2010 76 Example: Suppose we toss 4 fair dice. Let A = { all dice show a different face } If I toss 4 dice: (1) in how many total ways can the 4 faces turn up? (2) in how many ways can the 4 faces turn up so that each face is distinct?

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BTRY 4080 / STSCI 4080 Fall 2010 77 Example : A college planning committee consists of 3 freshmen, 4 sophomores, 5 juniors, and 2 seniors. A subcommittee of 4 , consisting of 1 person from each class, is to be chosen. Q : How many different subcommittees are possible?
BTRY 4080 / STSCI 4080 Fall 2010 78 Example : In a 7-place license plate, the frst 3 places are to be occupied by letters and the fnal 4 by numbers, e.g., ITH-0827 . Q : How many di±±erent 7-place license plates are possible i± there can be no repetition o± letters? o± numbers? o± both?

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BTRY 4080 / STSCI 4080 Fall 2010 79 An alternative view of (certain) counting problems: Theorem 5 Suppose we have an urn containing n distinguishable balls Suppose we take a sample of r balls. DeFne an outcome of this experiment as the ordered sequence of r labels that results from the selection process. Then: If the balls are sampled with replacement, there are n r possible outcomes.
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09-07 Axioms - Counting - BTRY 4080 STSCI 4080 Fall 2010 70...

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