{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

09-16 Equally Likely Outcomes &amp; Counting (1)

09-16 Equally Likely Outcomes &amp; Counting (1) - BTRY...

This preview shows pages 1–11. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 finite number of possible outcomes, all outcomes are considered equally likely to occur. Generic example: S = { s 1 ,s 2 ,s 3 ,...,s N } for N < . “Equally likely” outcomes simply means that P ( { s i } ) = 1 N , i = 1 to N For any event E S : P ( E ) = Number of outcomes in E Number of 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 )?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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., find 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 . There are 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 producttext k i =1 n i possible ways in which to complete the entire job.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 first 3 places are to be occupied by letters and the final 4 by numbers, e.g., ITH-0827 . Q : How many different 7-place license plates are possible if there can be no repetition of letters? of numbers? of both?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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. Define 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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}