3 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Probability Chapter 3

3 - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Define Experiment, Outcome, Event, Sample Space, & Probability 2.Explain How to Assign Probabilities 3.Use a Contingency Table, Venn Diagram, or Tree to Find Probabilities 4.Describe & Use Probability Rules

3 - 3 © 2000 Prentice-Hall, 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 - 4 © 2000 Prentice-Hall, Inc. Many Repetitions!* Number of Tosses Total Heads / Number of Tosses

3 - 5 © 2000 Prentice-Hall, Inc. Experiments, Outcomes, & Events

3 - 6 © 2000 Prentice-Hall, Inc. Experiments & Outcomes 1.Experiment Process of Obtaining an Observation, Outcome or Simple Event Process of Obtaining an Observation, Outcome or Simple Event 2.Sample Point Most Basic Outcome of an Experiment Most Basic Outcome of an Experiment 3.Sample Space (S) Collection of All Possible Outcomes Collection of All Possible Outcomes Sample Space Depends on Experimenter!

3 - 7 © 2000 Prentice-Hall, Inc. Outcome Examples Toss a Coin, Note FaceHead, Tail Toss 2 Coins, Note FacesHH, HT, TH, TT Select 1 Card, Note Kind 2, 2 ,..., A  (52) Select 1 Card, Note ColorRed, Black Play a Football GameWin, Lose, Tie Inspect a Part, Note QualityDefective, OK Observe GenderMale, Female ExperimentSample Space

3 - 8 © 2000 Prentice-Hall, Inc. Outcome Properties 1.Mutually Exclusive 2 Outcomes Can Not Occur at the Same Time 2 Outcomes Can Not Occur at the Same Time Both Male & Female in Same Person Both Male & Female in Same Person 2.Collectively Exhaustive 1 Outcome in Sample Space Must Occur 1 Outcome in Sample Space Must Occur Male or Female Male or Female Experiment: Observe Gender © T/Maker Co.

3 - 9 © 2000 Prentice-Hall, Inc. Events 1.Any Collection of Sample Points 2.Simple Event Outcome With 1 Characteristic Outcome With 1 Characteristic 3. Compound Event Collection of Outcomes or Simple Events Collection of Outcomes or Simple Events 2 or More Characteristics 2 or More Characteristics Joint Event Is a Special Case Joint Event Is a Special Case 2 Events Occurring Simultaneously 2 Events Occurring Simultaneously

© 2000 Prentice-Hall, Inc. Event Examples Sample SpaceHH, HT, TH, TT 1 Head & 1 TailHT, TH Heads on 1st CoinHH, HT At Least 1 HeadHH, HT, TH Heads on BothHH Experiment: Toss 2 Coins. Note Faces. EventOutcomes in Event

© 2000 Prentice-Hall, Inc. Sample Space

© 2000 Prentice-Hall, Inc. Visualizing Sample Space 1.Listing S = {Head, Tail} S = {Head, Tail} 2.Venn Diagram 3.Contingency Table 4.Decision Tree Diagram

© 2000 Prentice-Hall, Inc. S Tail HH TT TH HT Sample Space S = {HH, HT, TH, TT} Venn Diagram Outcome Experiment: Toss 2 Coins. Note Faces. Compound Event

© 2000 Prentice-Hall, Inc. 2 nd Coin Coin 1 st HeadTail Total HeadHHHT HH, HT TailTHTT TH, TT TotalHH, THHT, TTS Contingency Table Experiment: Toss 2 Coins. Note Faces. S = {HH, HT, TH, TT} Sample Space Outcome (Count, Total % Shown Usually) Simple Event (Head on 1st Coin)

© 2000 Prentice-Hall, Inc. Tree Diagram Outcome S = {HH, HT, TH, TT} Sample Space Experiment: Toss 2 Coins. Note Faces. T H T H T HH HT TH TT H

© 2000 Prentice-Hall, Inc. Compound Events

© 2000 Prentice-Hall, Inc. Forming Compound Events 1.Intersection Outcomes in Both Events A and B Outcomes in Both Events A and B ‘AND’ Statement ‘AND’ Statement  Symbol (i.e., A  B)  Symbol (i.e., A  B) 2.Union Outcomes in Either Events A or B or Both Outcomes in Either Events A or B or Both ‘OR’ Statement ‘OR’ Statement  Symbol (i.e., A  B)  Symbol (i.e., A  B)

© 2000 Prentice-Hall, Inc. S Black Ace Event Intersection: Venn Diagram Joint Event (Ace  Black): A B , A B  Event Black: 2 B ,..., A B  Sample Space: 2 R, 2 R , 2 B ,..., A B  Experiment: Draw 1 Card. Note Kind, Color & Suit. Event Ace: A R, A R , A B , A B 

© 2000 Prentice-Hall, Inc. Color Type RedBlack Total Ace Ace & Red Black Ace Non-Ace Non & Red Black Non- Ace TotalRedBlackS Event Intersection: Contingency Table Sample Space (S): 2 R, 2 R , 2 B ,..., A B  Experiment: Draw 1 Card. Note Kind, Color & Suit. Joint Event Ace AND Black: A B , A B  Simple Event Ace: A R, A R , A B , A B  Simple Event Black: 2 B ,..., A B 

© 2000 Prentice-Hall, Inc. S Black Ace Event Union : Venn Diagram Event (Ace  Black): A R,..., A B , 2 B ,..., K B  Event Black: 2 B , 2 B , ..., A B  Sample Space: 2 R, 2 R , 2 B ,..., A B  Event Ace: A R, A R , A B , A B  Experiment: Draw 1 Card. Note Kind, Color & Suit.

© 2000 Prentice-Hall, Inc. Color Type RedBlack Total Ace Ace & Red Black Ace Non-Ace Non & Red Black Non- Ace TotalRedBlackS Event Union : Contingency Table Sample Space (S): 2 R, 2 R , 2 B ,..., A B  Joint Event Ace OR Black: A R,..., A B ,  2 B ,..., K B  Simple Event Ace: A R, A R , A B , A B  A R, A R , A B , A B  Simple Event Black: 2 B ,..., A B  Experiment: Draw 1 Card. Note Kind, Color & Suit.

© 2000 Prentice-Hall, Inc.   Special Events 1.Null Event Club & Diamond on 1 Card Draw Club & Diamond on 1 Card Draw 2.Complement of Event For Event A, All Events Not In A: A’ For Event A, All Events Not In A: A’ 3.Mutually Exclusive Event Events Do Not Occur Simultaneously Events Do Not Occur Simultaneously Null Event

© 2000 Prentice-Hall, Inc. S Black Complement of Event Example Event Black: 2 B , 2 B ,..., A B  Complement of Event Black, Black ’: 2 R, 2 R ,..., A R, A R  Sample Space: 2 R, 2 R , 2 B ,..., A B  Experiment: Draw 1 Card. Note Kind, Color & Suit.

© 2000 Prentice-Hall, Inc. S  Mutually Exclusive Events Example Events  & Mutually Exclusive Experiment: Draw 1 Card. Note Kind & Suit. Outcomes in Event Heart: 2, 3, 4,..., A Outcomes in Event Heart: 2, 3, 4,..., A Sample Space: 2, 2 , 2 ,..., A  Event Spade: 2 , 3 , 4 ,..., A 

© 2000 Prentice-Hall, Inc. Probabilities

© 2000 Prentice-Hall, Inc. What is Probability? 1.Numerical Measure of Likelihood that Event Will Occur P(Event) P(Event) P(A) P(A) Prob(A) Prob(A) 2.Lies Between 0 & 1 3.Sum of Events is Certain Impossible

© 2000 Prentice-Hall, Inc. Assigning Event Probabilities 1.a priori Classical Method 2.Empirical Classical Method 3.Subjective Method What’s the probability?

© 2000 Prentice-Hall, Inc. a priori Classical Method 1.Prior Knowledge of Process 2.Before Experiment 3.P(Event) = X / T X = No. of Event Outcomes X = No. of Event Outcomes T = Total Outcomes in Sample Space T = Total Outcomes in Sample Space Each of T Outcomes Is Equally Likely Each of T Outcomes Is Equally Likely P(Outcome) = 1/T P(Outcome) = 1/T © T/Maker Co.

© 2000 Prentice-Hall, Inc. Empirical Classical Method 1.Actual Data Collected 2.After Experiment 3. P(Event) = X / T Repeat Experiment T Times Repeat Experiment T Times Event Observed X Times Event Observed X Times 4.Also Called Relative Frequency Method Of 100 Parts Inspected, Only 2 Defects!

© 2000 Prentice-Hall, Inc. Subjective Method 1.Individual Knowledge of Situation 2.Before Experiment 3.Unique Process Not Repeatable Not Repeatable 4.Different Probabilities from Different People © T/Maker Co.

© 2000 Prentice-Hall, Inc. Thinking Challenge 1.That a Box of 24 Bolts Will Be Defective? 2.That a Toss of a Coin Will Be a Tail? 3.That Tom Will Default on His PLUS Loan? 4.That a Student Will Earn an A in This Class? 5.That a New Store on Rte. 1 Will Succeed? Which Method Should Be Used to Find the Probability...

© 2000 Prentice-Hall, Inc. Compound Event Probability 1.Numerical Measure of Likelihood that Compound Event Will Occur 2.Can Often Use Contingency Table 2 Variables Only 2 Variables Only 3.Formula Methods Additive Rule Additive Rule Conditional Probability Formula Conditional Probability Formula Multiplicative Rule Multiplicative Rule

© 2000 Prentice-Hall, Inc. Event Event B 1 B 2 Total A 1 P(A 1  B 1 )P(A 1  B 2 ) P(A 1 ) A 2 P(A 2  B 1 )P(A 2  B 2 ) P(A 2 ) Total P(B 1 )P(B 2 )1 Event Probability Using Contingency Table Joint Probability Marginal (Simple) Probability

© 2000 Prentice-Hall, Inc. Color Type RedBlack Total Ace 2/522/524/52 Non-Ace 24/5224/5248/52 Total 26/5226/5252/52 Contingency Table Example Experiment: Draw 1 Card. Note Kind, Color & Suit. P(Ace) P(Ace AND Red) P(Red)

© 2000 Prentice-Hall, Inc. Event EventCDTotal A 426 B 134 Total 5510 Thinking Challenge What’s the Probability? P(A) = P(D) = P(C  B) = P(A  D) = P(B  D) =

© 2000 Prentice-Hall, Inc. Solution* The Probabilities Are: P(A) = 6/10 P(D) = 5/10 P(C  B) = 1/10 P(A  D) = 9/10 P(B  D) = 3/10 Event EventCDTotal A 426 B 134 Total 5510

© 2000 Prentice-Hall, Inc. Additive Rule

© 2000 Prentice-Hall, 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)

© 2000 Prentice-Hall, Inc. Additive Rule Example Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Type RedBlack Total Ace 224 Non-Ace Total P(Ace OR B lack)= P(Ace) P(Ace)+ P(Black) P(Black)- P(Ace P(Ace Black) Black) 4 52  

© 2000 Prentice-Hall, Inc. Thinking Challenge Using the Additive Rule, What’s the Probability? P(A  D) = P(B  C) = Event EventCDTotal A 426 B 134 Total 5510

© 2000 Prentice-Hall, Inc. Solution* Using the Additive Rule, the Probabilities Are: P(AD)= P(A) P(A)+ P(D) P(D)- P(A P(A D) D) 6 10   P(BC)= P(B) P(B)+ P(C) P(C)- P(B P(B C) C) 4 10  

© 2000 Prentice-Hall, Inc. Conditional Probability

© 2000 Prentice-Hall, Inc. Conditional Probability 1.Event Probability Given that Another Event Occurred 2.Revise Original Sample Space to Account for New Information Eliminates Certain Outcomes Eliminates Certain Outcomes 3.P(A | B) = P(A and B) P(B)

© 2000 Prentice-Hall, Inc. S Black Ace Conditional Probability Using Venn Diagram Black ‘Happens’: Eliminates All Other Outcomes Event (Ace AND Black) (S) Black

© 2000 Prentice-Hall, Inc. Color Type RedBlack Total Ace 224 Non-Ace Total Conditional Probability Using Contingency Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Revised Sample Space

© 2000 Prentice-Hall, Inc. 1.Event Occurrence Does Not Affect Probability of Another Event Toss 1 Coin Twice Toss 1 Coin Twice 2.Causality Not Implied 3.Tests For P(A | B) = P(A) P(A | B) = P(A) P(A and B) = P(A)*P(B) P(A and B) = P(A)*P(B) Statistical Independence

© 2000 Prentice-Hall, Inc. Tree Diagram Experiment: Select 2 Pens from 20 Pens: 14 Blue & 6 Red. Don’t Replace. Dependent! B R B R B R P(R) = 6/20 P(R|R) = 5/19 P(B|R) = 14/19 P(B) = 14/20 P(R|B) = 6/19 P(B|B) = 13/19

© 2000 Prentice-Hall, Inc. Thinking Challenge Using the Table Then the Formula, What’s the Probability? P(A|D) = P(C|B) = Are C & B Independent? Event EventCDTotal A 426 B 134 Total 5510

© 2000 Prentice-Hall, Inc. Solution* Using the Formula, the Probabilities Are: Dependent P(A | D) D)= P(A P(D)   / / P(C | B) B)= P(C P(B) P(C)= 5 10    / /

© 2000 Prentice-Hall, Inc. Multiplicative Rule

© 2000 Prentice-Hall, Inc. Multiplicative Rule 1.Used to Get Compound Probabilities for Intersection of Events Called Joint Events Called Joint 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)

© 2000 Prentice-Hall, Inc. Multiplicative Rule Example Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Type RedBlack Total Ace 224 Non-Ace Total

© 2000 Prentice-Hall, Inc. Thinking Challenge Using the Multiplicative Rule, What’s the Probability? P(C  B) = P(B  D) = P(A  B) = Event EventCDTotal A 426 B 134 Total 5510

© 2000 Prentice-Hall, Inc. Solution* Using the Multiplicative Rule, the Probabilities Are: P(CB)= P(C) P(C)P(B| C) = 5/10 * 1/5 = 1/10  P(BD)= P(B) P(B)P(D| B) = 4/10 * 3/4 = 3/10  P(AB)= P(A) P(A)P(B|A) 0

© 2000 Prentice-Hall, Inc. Conclusion 1.Defined Experiment, Outcome, Event, Sample Space, & Probability 2.Explained How to Assign Probabilities 3.Used a Contingency Table, Venn Diagram, or Tree to Find Probabilities 4.Described & Used Probability Rules

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