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UT Dallas CS 6313 - ch02

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Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 2ProbabilityApplied Statistics and Probability for EngineersSixth EditionDouglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Random Experiment• An experiment is a procedure that is– carried out under controlled conditions, and– executed to discover an unknown result.• An experiment that results in different outcomes even when repeated in the same manner every time is a random experiment.Sec 2-1.1 Random Experiments 2Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Sample Spaces• The set of all possible outcomes of a random experiment is called the sample space, S.• S is discrete if it consists of a finite or countable infinite set of outcomes.• S is continuous if it contains an interval of real numbers.Sec 2-1.2 Sample Spaces 3Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 2-1: Defining Sample Spaces• Randomly select a camera and record the recycle time of a flash. S = R+= {x | x > 0}, the positive real numbers. • Suppose it is known that all recycle times are between 1.5 and 5 seconds. Then S = {x | 1.5 < x < 5} is continuous.• It is known that the recycle time has only three values(low, medium or high). Then S= {low, medium, high} is discrete.• Does the camera conform to minimum recycle time specifications? S = {yes, no} is discrete.Sec 2-1.2 Sample Spaces 4Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Sample Space Defined By A Tree DiagramSec 2-1.2 Sample Spaces 5Example 2-2: Messages are classified as on-time(o) or late(l). Classify the next 3 messages. S = {ooo, ool, olo, oll, loo, lol, llo, lll}Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Events are Sets of Outcomes• An event (E) is a subset of the sample space of a random experiment.• Event combinations– The Union of two events consists of all outcomes that are contained in one event or the other, denoted as E1E2. – The Intersection of two events consists of all outcomes that are contained in one event and the other, denoted as E1E2.– The Complement of an event is the set of outcomes in the sample space that are not contained in the event, denoted as E.Sec 2-1.3 Events 6Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 2-3 Discrete EventsSuppose that the recycle times of two cameras are recorded. Consider only whether or not the cameras conform to the manufacturing specifications. We abbreviate yes and no as y and n. The sample space is S = {yy, yn, ny, nn}.Suppose, E1denotes an event that at least one camera conforms to specifications, then E1= {yy, yn, ny}Suppose, E2denotes an event that no camera conforms to specifications, then E2= {nn}Suppose, E3denotes an event that at least one camera does notconform, then E3= {yn, ny, nn},– Then E1E3= S– Then E1E3= {yn, ny}– Then E1 = {nn}Sec 2-1.3 Events 7Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 2-4 Continuous EventsMeasurements of the thickness of a part are modeled with the sample space: S = R+.Let E1= {x | 10 ≤ x < 12},Let E2= {x | 11 < x < 15}– Then E1E2= {x | 10 ≤ x < 15}– Then E1E2= {x | 11 < x < 12}– Then E1 = {x | 0 < x < 10 or x ≥ 12}– Then E1 E2= {x | 12 ≤ x < 15}Sec 2-1.3 Events 8Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Venn DiagramsSec 2-1.3 Events 9Events A & B contain their respective outcomes. The shaded regions indicate the event relation of each diagram.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Mutually Exclusive EventsSec 2-1.3 Events 10• Events A and B are mutually exclusive if they share no common outcomes.• The occurrence of one event precludes the occurrence of the other.• Symbolically, A B = ØCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Mutually Exclusive Events - Laws• Commutative law (event order is unimportant): – A B = B A and A B = B A• Distributive law (like in algebra):– (A B) C = (A C) (B C)– (A B) C = (A C) (B C)• Associative law (like in algebra):– (A B) C = A (B C)– (A B) C = A (B C)Sec 2-1.3 Events 11Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Mutually Exclusive Events - Laws• DeMorgan’s law:– (A B) = A B The complement of the union is the intersection of the complements.– (A B) = A B The complement of the intersection is the union of the complements.• Complement law: (A) = A.Sec 2-1.3 Events 12Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Counting Techniques• There are three special rules, or counting techniques, used to determine the number of outcomes in events.• They are :1. Multiplication rule2. Permutation rule3. Combination rule• Each has its special purpose that must be applied properly – the right tool for the right job.Sec 2-1.4 Counting Techniques 13Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Counting – Multiplication Rule• Multiplication rule:– Let an operation consist of k steps and there are • n1ways of completing step 1,• n2ways of completing step 2, … and• nkways of completing step k.– Then, the total number of ways to perform ksteps is:• n1 · n2· … · nkSec 2-1.4 Counting Techniques 14Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 2-5 - Web Site Design• In the design for a website, we can choose to use among:– 4 colors,– 3 fonts, and– 3 positions for an image.How many designs are possible?• Answer via the multiplication rule: 4 · 3 · 3 = 36Sec 2-1.4 Counting Techniques 15Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Counting – Permutation Rule• A permutation is a unique sequence of distinct items.• If S = {a, b, c}, then there are 6 permutations– Namely: abc, acb, bac, bca, cab, cba (order matters)• Number of permutations for a set of n items is n!• n! = n·(n-1)·(n-2)·…·2·1• 7! = 7·6·5·4·3·2·1 = 5,040 = FACT(7) in Excel• By definition: 0! = 1Sec 2-1.4 Counting Techniques 16Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Counting–Subset Permutations and an example• For a sequence of r items from a set of n items:• Example 2-6: Printed Circuit Board• A printed circuit board has eight different locations in which a component can be placed. If four different components are to be placed on the board, how many designs are


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UT Dallas CS 6313 - ch02

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