Chapter 2 Probability

Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter2 Probability

Uncertainty • Life is not deterministic. • We do not knoweverydetailfor sure andhavetogivedecisionseveryday. • Shouldwetakeourumbrellawith us?

Decisionmakingunderuncertainty • Consider a newsboy, he willhaveexactly 3 customerswhowantsto buy newspaper. • Howmanynewspapersshould he buy? • If 50% chance of 4 customersand 50% chance of 6 customers, howmanynewspapersshould he buy? Sec 2-

Decisionmakingunderuncertainty • Whatiftheprofit of thetable is uncertain? • Whatiftheresourcesareuncertain? • How do wedecide? Sec 2-

Let’srolltwodice • Whatarethepossibleoutcomes? • Howmanydifferentoutcomes in terms of playingbackgammon? Sec 2-1.1 Random Experiments

Sum of outcomes of twodice Probability Possiblevalues

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. • Examples… Sec 2-1.1 Random Experiments

SampleSpaces • 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

Exampleexperiments • Roll a die • Rolltwodice • Rollthreedice • Tosstwocoins • Drawtwocardsfrom a deck - Givethesamplespaceforeachexperiment. Sec 2-1.2 Sample Spaces

Sample Space Defined By A Tree Diagram Messages are classified as on-time(o) or late(l). Classify the next 3 messages. S = {ooo, ool, olo, oll, loo, lol, llo, lll} Sec 2-1.2 Sample Spaces

Events are Sets of Outcomes • An event (E) is a subset of the sample space of a random experiment. • Sum of twodice is 10 • Twocards has thesametype • Firsttwomessagesare on-time Sec 2-1.3 Events

Events are Sets of Outcomes • TheUnion of two events consists of all outcomes that are contained in one event or the other, denoted as E1E2. • TheIntersection of two events consists of all outcomes that are contained in one event and the other, denoted as E1E2. • TheComplement 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

Discrete Events Consider twocamerasandeachcamerawhetheror not conform to the manufacturing specifications. • E1 denotes an event that at least one camera conforms to specifications • E2 denotes an event that no camera conforms to specifications • E3 denotes an event that at least one camera does not conform. • E1E3 • E1E3 • E1’ Sec 2-1.3 Events

Continuous Events Measurements of the thickness of a part, x E1 = {x | 10 ≤ x < 12}, E2 = {x | 11 < x < 15} • E1 E2 • E1E2 • E1 • E1E2 Sec 2-1.3 Events

Venn Diagrams Events A & B contain their respective outcomes. The shaded regions indicate the event relation of each diagram. Sec 2-1.3 Events

Mutually Exclusive Events • Events Aand B are mutually exclusive because they share no common outcomes. • The occurrence of one event precludes the occurrence of the other. • Symbolically, AB = Ø Sec 2-1.3 Events

Mutually Exclusive Events - Example • Experiment = Rolling twodice. • A: Thesum is 2, B: Thesum is 5. Sec 2-1.3 Events

Mutually Exclusive Events • Events Aand B are mutually exclusive because they share no common outcomes. • The occurrence of one event precludes the occurrence of the other. • Symbolically, AB = Ø Sec 2-1.3 Events

Mutually Exclusive Events • A sample of twoprintedcircuitboards is selectedfrom a • batch: 90 nondefectiveboards, 8 boardswithminor • defects, 2 boardswithmajordefects. • A: one of them has minorother has majordefects • B: both of themarenondefective Sec 2-1.3 Events

Laws • Commutative law (event order is not important): • 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

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

Counting Techniques • There are three counting techniques, used to determine the number of outcomes in an event. • Multiplication • Permutation • Combination • Each has its special purpose that must be applied properly – the right tool for the right job. Sec 2-1.4 Counting Techniques

Multiplication • Multiplication rule: • Let an operation consist of m steps and there are • nk ways of completing step k. • Then, the total number of ways to perform m steps is: • n1 · n2 · … ·nm Sec 2-1.4 Counting Techniques

Web Site Design • In the design for a website, we can choose among: • 4 colors, • 3 fonts, • 3 positions for an image. How many designs are possible? Sec 2-1.4 Counting Techniques

Counting – Permutation Rule • A permutation is a unique sequence of distinct items. (order is important) • Howmanywaystoordertheelements of the set {a, b, c} • Number of permutations for a set of n items is n! • n! = n×(n-1) ×(n-2) ×…2×1 • By definition: 0! = 1 Sec 2-1.4 Counting Techniques

Counting–Subset Permutations and an example • 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 possible? Sec 2-1.4 Counting Techniques

Counting–Subset Permutations and an example • For a sequence of r items from a set of n items: • Choose r from n andthenorderthem. Sec 2-1.4 Counting Techniques

Counting - Similar Item Permutations • In a hospital, a operating room needs to schedule three knee surgeries and two hip surgeries in a day. • Weconsiderallkneesurgeriesidentical as well as allhipsurgeries. • What is the set of allsequences? Sec 2-1.4 Counting Techniques

Counting - Similar Item Permutations • Used for counting the sequences when some items are identical. • The number of permutations of r types of items • nk is thenumber of type k items • n = n1 + n2 + … + nritems Sec 2-1.4 Counting Techniques

Counting – Combination Rule • A combination is a selection of r items from a set of n where order does not matter. • If S = {a, b, c} • Howmanyways of choosing 1 item? • Howmanyways of choosing 2 items? • Howmanyways of choosing 3 items? Sec 2-1.4 Counting Techniques

Sampling without Replacement • A bin of 50 parts contains 3 defectives and 47 non-defective parts. • A sample of 6 parts is selected from the 50 without replacement. • Howmanyways of choosing 6 out of 50? • How many of themcontains exactly 2 defective parts? Sec 2-1.4 Counting Techniques

Sampling without Replacementcontd. Sec 2-1.4 Counting Techniques

Chapter 2 Probability