3-1 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

3-2 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistics for Business and Economics Chapter 3 Probability

3-3 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Contents 1.Events, Sample Spaces, and Probability 2.Unions and Intersections 3.Complementary Events 4.The Additive Rule and Mutually Exclusive Events 5.Conditional Probability 6.The Multiplicative Rule and Independent Events 7.Bayes’s Rule

3-4 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives 1.Develop probability as a measure of uncertainty 2.Introduce basic rules for finding probabilities 3.Use probability as a measure of reliability for an inference 4.Provide an advanced rule for finding probabilities

3-5 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge What’s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). So toss a coin twice. Do it! Did you get one head & one tail? What’s it all mean?

3-6 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Many Repetitions!* Number of Tosses Total Heads Number of Tosses

3-7 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.1 Events, Sample Spaces, and Probability

3-8 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Experiments & Sample Spaces 1.Experiment Process of observation that leads to a single outcome that cannot be predicted with certainty 2.Sample point Most basic outcome of an experiment 3.Sample space ( S ) Collection of all sample points Sample Space Depends on Experimenter!

3-9 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Visualizing Sample Space 1.Listing S = {Head, Tail} 2.Venn Diagram H T S

3-10 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Space Examples Toss a Coin, Note Face{Head, Tail} Toss 2 Coins, Note Faces{HH, HT, TH, TT} Select 1 Card, Note Kind {2♥, 2♠,..., A♦} (52) Select 1 Card, Note Color{Red, Black} Play a Football Game{Win, Lose, Tie} Inspect a Part, Note Quality{Defective, Good} Observe Gender{Male, Female} Experiment Sample Space

3-11 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Events 1. Specific collection of sample points 2. Simple Event Contains only one sample point 3. Compound Event Contains two or more sample points

3-12 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. S HH TT TH HT Sample Space S = {HH, HT, TH, TT} Venn Diagram Outcome Experiment: Toss 2 Coins. Note Faces. Compound Event: At least one Tail

3-13 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Event Examples 1 Head & 1 Tail HT, TH Head on 1st Coin HH, HT At Least 1 Head HH, HT, TH Heads on Both HH Experiment: Toss 2 Coins. Note Faces. Sample Space:HH, HT, TH, TT Event Outcomes in Event

3-14 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Probabilities

3-15 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. What is Probability? 1.Numerical measure of the likelihood that event will occur P(Event) P(A) Prob(A) 2.Lies between 0 & 1 3.Sum of sample points is Certain Impossible

3-16 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Probability Rules for Sample Points Let p i represent the probability of sample point i. 1.All sample point probabilities must lie between 0 and 1 (i.e., 0 ≤ p i ≤ 1). 2.The probabilities of all sample points within a sample space must sum to 1 (i.e.,  p i = 1).

3-17 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Equally Likely Probability P(Event) = X / T X = Number of outcomes in the event T = Total number of sample points in Sample Space Each of T sample points is equally likely — P(sample point) = 1/T © T/Maker Co.

3-18 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Steps for Calculating Probability 1.Define the experiment; describe the process used to make an observation and the type of observation that will be recorded 2.List the sample points 3.Assign probabilities to the sample points 4.Determine the collection of sample points contained in the event of interest 5.Sum the sample points probabilities to get the event probability

3-19 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Combinations Rule A sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible is denoted byand is equal to where the factorial symbol (!) means that For example,0! is defined to be 1.

3-20 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.2 Unions and Intersections

3-21 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Compound Events Compound events: Composition of two or more other events. Can be formed in two different ways.

3-22 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Unions & Intersections 1. Union Outcomes in either events A or B or both ‘OR’ statement Denoted by  symbol (i.e., A  B) 2. Intersection Outcomes in both events A and B ‘AND’ statement Denoted by  symbol (i.e., A  B)

3-23 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. S BlackAce Event Union: Venn Diagram Event Ace  Black: A,..., A , 2 ,..., K  Event Black: 2 , 2 , ..., A  Sample Space: 2,  2 , 2 ,..., A  Event Ace: A, A , A , A  Experiment: Draw 1 Card. Note Kind, Color & Suit.

3-24 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Event Ace  Black: A,..., A ,  2 ,..., K  Event Union: Two–Way Table Sample Space ( S ): 2, 2 , 2 ,..., A  Simple Event Ace: A, A , A , A  Simple Event Black: 2 ,..., A  Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Type RedBlack Total AceAce & Red Ace & Black Ace Non & Red Non & Black Non- Ace TotalRedBlack S Non-Ace

3-25 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. S BlackAce Event Intersection: Venn Diagram Event Ace  Black: A , A  Event Black: 2 ,..., A  Sample Space: 2, 2 , 2 ,..., A  Experiment: Draw 1 Card. Note Kind, Color & Suit. Event Ace: A, A , A , A 

3-26 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Space (S): 2, 2 , 2 ,..., A  Event Intersection: Two–Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Event Ace  Black: A , A  Simple Event Ace: A, A , A , A  Simple Event Black: 2 ,..., A  Color Type RedBlack Total AceAce & Red Ace & Black Ace Non & Red Non & Black Non- Ace TotalRedBlack S Non-Ace

3-27 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Compound Event Probability 1.Numerical measure of likelihood that compound event will occur 2.Can often use two–way table Two variables only

3-28 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Event B 1 B 2 Total A 1 P(AP(A 1  B 1 ) P(AP(A 1  B 2 ) P(AP(A 1 ) A 2 P(AP(A 2  B 1 ) P(AP(A 2  B 2 ) P(AP(A 2 ) P(BP(B 1 ) P(BP(B 2 )1 Event Probability Using Two–Way Table Joint ProbabilityMarginal (Simple) Probability Total

3-29 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Color Type RedBlack Total Ace 2/52 4/52 Non-Ace 24/52 48/52 Total 26/52 52/52 Two–Way Table Example Experiment: Draw 1 Card. Note Kind & Color. P(Ace) P(Ace  Red) P(Red)

3-30 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge 1. P(A) = 2. P(D) = 3. P(C  B) = 4. P(A  D) = 5. P(B  D) = Event CDTotal A 426 B What’s the Probability?

3-31 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Solution* The Probabilities Are: 1. P(A) = 6/10 2. P(D) = 5/10 3. P(C  B) = 1/10 4. P(A  D) = 9/10 5. P(B  D) = 3/10 Event CDTotal A 426 B

3-32 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.3 Complementary Events

3-33 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Complementary Events Complement of Event A The event that A does not occur All events not in A Denote complement of A by A C S ACAC A

3-34 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Rule of Complements The sum of the probabilities of complementary events equals 1: P(A) + P(A C ) = 1 S ACAC A

3-35 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. S Black Complement of Event Example Event Black: 2 , 2 ,..., A  Complement of Event Black, Black C : 2, 2 ,..., A, A  Sample Space: 2, 2 , 2 ,..., A  Experiment: Draw 1 Card. Note Color.

3-36 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.4 The Additive Rule and Mutually Exclusive Events

3-37 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Mutually Exclusive Events Events do not occur simultaneously A  does not contain any sample points   Mutually Exclusive Events

3-38 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. S  Mutually Exclusive Events Example Events  and are Mutually Exclusive Experiment: Draw 1 Card. Note Kind & Suit. Outcomes in Event Heart: 2, 3, 4,..., A Sample Space: 2, 2 , 2 ,..., A  Event Spade: 2 , 3 , 4 ,..., A 

3-39 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Additive Rule 1.Used to get compound probabilities for union of events 2. P(A OR B) = P(A  B) = P(A) + P(B) – P(A  B) 3.For mutually exclusive events: P(A OR B) = P(A  B) = P(A) + P(B)

3-40 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Additive Rule Example Experiment: Draw 1 Card. Note Kind & Color. P(Ace  Black) = P(Ace)+ P(Black)– P(Ace Black)  Color Type RedBlack Total Ace 224 Non-Ace Total = + – =

3-41 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge 1. P(A  D) = 2. P(B  C) = Event CDTotal A 426 B Using the additive rule, what is the probability?

3-42 Copyright © 2014, 2011, and 2008 Pearson Education, Inc Solution* Using the additive rule, the probabilities are: P(A  D) = P(A) + P(D) – P(A  D) P(B  C) = P(B) + P(C) – P(B  C) = + – =

3-43 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.5 Conditional Probability

3-44 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditional Probability 1. Event probability given that another event occurred 2. Revise original sample space to account for new information Eliminates certain outcomes 3. P(A | B) = P(A and B) = P(A  B  P(B) P(B)

3-45 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. S BlackAce Conditional Probability Using Venn Diagram Black ‘Happens’: Eliminates All Other Outcomes Event (Ace  Black) (S)(S) Black

3-46 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Conditional Probability Using Two–Way Table Experiment: Draw 1 Card. Note Kind & Color. Revised Sample Space Color Type RedBlack Total Ace 224 Non-Ace Total 26 52

3-47 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Using the table then the formula, what’s the probability? Thinking Challenge 1. P(A|D) = 2. P(C|B) = Event CDTotal A 426 B

3-48 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Solution* Using the formula, the probabilities are:

3-49 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.6 The Multiplicative Rule and Independent Events

3-50 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Multiplicative Rule 1.Used to get compound probabilities for intersection of events 2. P(A and B) = P(A  B) = P(A)  P(B|A) = P(B)  P(A|B) 3. For Independent Events: P(A and B) = P(A  B) = P(A)  P(B)

3-51 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Multiplicative Rule Example Experiment: Draw 1 Card. Note Kind & Color. Color Type RedBlack Total Ace 224 Non-Ace Total P(Ace  Black) = P(Ace)∙P(Black | Ace)

3-52 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistical Independence 1. Event occurrence does not affect probability of another event Toss 1 coin twice 2. Causality not implied 3.Tests for independence P(A | B) = P(A) P(B | A) = P(B) P(A  B) = P(A)  P(B)

3-53 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge 1. P(C  B) = 2. P(B  D) = 3. P(A  B) = Event CDTotal A 426 B Using the multiplicative rule, what’s the probability?

3-54 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Solution* Using the multiplicative rule, the probabilities are:

3-55 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Tree Diagram Experiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don’t replace. Dependent! B R B R B R 6/20 5/19 14/19 14/20 6/19 13/19 P(R  R)=(6/20)(5/19) =3/38 P(R  B)=(6/20)(14/19) =21/95 P(B  R)=(14/20)(6/19) =21/95 P(B  B)=(14/20)(13/19) =91/190

3-56 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 3.7 Bayes’s Rule

3-57 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Bayes’s Rule Given k mutually exclusive and exhaustive events B 1, B 1,... B k, such that P(B 1 ) + P(B 2 ) + … + P(B k ) = 1, and an observed event A, then

3-58 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Bayes’s Rule Example A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are defective, while 1% of Factory II’s are defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?

3-59 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Bayes’s Rule Example Factory II Factory I Defective Defective Good Good

3-60 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Probability Rules for k Sample Points, S 1, S 2, S 3,..., S k 1. 0 ≤ P(S i ) ≤ 1 2.

3-61 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Random Sample All possible such samples have equal probability of being selected.

3-62 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Combinations Rule Counting number of samples of n elements selected from N elements

3-63 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Bayes’s Rule