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© 2011 Pearson Education, Inc

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Statistics for Business and Economics Chapter 3 Probability

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© 2011 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.Random Sampling 8.Baye’s Rule

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© 2011 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

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© 2011 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?

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© 2011 Pearson Education, Inc Many Repetitions!* Number of Tosses Total Heads Number of Tosses

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© 2011 Pearson Education, Inc 3.1 Events, Sample Spaces, and Probability

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© 2011 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 possible outcomes Sample Space Depends on Experimenter!

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© 2011 Pearson Education, Inc Sample Space Properties Experiment: Observe Gender © T/Maker Co. 1.Mutually Exclusive 2 outcomes can not occur at the same time —Male & Female in same person 2.Collectively Exhaustive One outcome in sample space must occur. —Male or Female

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© 2011 Pearson Education, Inc Visualizing Sample Space 1.Listing S = {Head, Tail} 2.Venn Diagram H T S

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© 2011 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

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© 2011 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

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© 2011 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

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© 2011 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

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© 2011 Pearson Education, Inc Probabilities

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

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© 2011 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).

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© 2011 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.

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© 2011 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

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© 2011 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.

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© 2011 Pearson Education, Inc 3.2 Unions and Intersections

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© 2011 Pearson Education, Inc Compound Events Compound events: Composition of two or more other events. Can be formed in two different ways.

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© 2011 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)

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© 2011 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.

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© 2011 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

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© 2011 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

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© 2011 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

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© 2011 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

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© 2011 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

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© 2011 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)

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© 2011 Pearson Education, Inc 1. P(A) = 2. P(D) = 3. P(C B) = 4. P(A D) = 5. P(B D) = Thinking Challenge Event CDTotal A 426 B What’s the Probability?

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© 2011 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

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© 2011 Pearson Education, Inc 3.3 Complementary Events

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© 2011 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

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© 2011 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

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© 2011 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.

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© 2011 Pearson Education, Inc 3.4 The Additive Rule and Mutually Exclusive Events

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© 2011 Pearson Education, Inc Mutually Exclusive Events Events do not occur simultaneously A does not contain any sample points Mutually Exclusive Events

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© 2011 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

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© 2011 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)

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© 2011 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 = + – =

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© 2011 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?

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© 2011 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) = + – =

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© 2011 Pearson Education, Inc 3.5 Conditional Probability

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© 2011 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)

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© 2011 Pearson Education, Inc S BlackAce Conditional Probability Using Venn Diagram Black ‘Happens’: Eliminates All Other Outcomes Event (Ace Black) (S) Black

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© 2011 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

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© 2011 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

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© 2011 Pearson Education, Inc Solution* Using the formula, the probabilities are:

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© 2011 Pearson Education, Inc 3.6 The Multiplicative Rule and Independent Events

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© 2011 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)

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© 2011 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)

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© 2011 Pearson Education, Inc 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) Statistical Independence

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© 2011 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?

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© 2011 Pearson Education, Inc Solution* Using the multiplicative rule, the probabilities are:

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© 2011 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

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© 2011 Pearson Education, Inc 3.7 Random Sampling

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© 2011 Pearson Education, Inc Importance of Selection How a sample is selected from a population is of vital importance in statistical inference because the probability of an observed sample will be used to infer the characteristics of the sampled population.

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© 2011 Pearson Education, Inc Random Sample If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a random sample.

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© 2011 Pearson Education, Inc Random Number Generators Most researchers rely on random number generators to automatically generate the random sample. Random number generators are available in table form, and they are built into most statistical software packages.

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© 2011 Pearson Education, Inc 3.8 Bayes’s Rule

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© 2011 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

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© 2011 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?

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© 2011 Pearson Education, Inc Bayes’s Rule Example Factory II Factory I Defective Defective Good Good

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© 2011 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.

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© 2011 Pearson Education, Inc Key Ideas Random Sample All possible such samples have equal probability of being selected.

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© 2011 Pearson Education, Inc Key Ideas Combinations Rule Counting number of samples of n elements selected from N elements

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© 2011 Pearson Education, Inc Key Ideas Bayes’s Rule

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