3 - 1 © 1998 Prentice-Hall, Inc. Chapter 3 Probability

3 - 2 © 1998 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 © 1998 Prentice-Hall, Inc. Thinking Challenge You ‘know’ that the probability of getting a head on a single toss of a fair coin is 0.5 or 50%. What does this mean? Ten students will each toss a single coin 20 times. The number of heads will be recorded. Compute the probability of a head. AloneGroupClass © T/Maker Co.

3 - 4 © 1998 Prentice-Hall, Inc. Probability Graph P(Head) Cumulative coin tosses

3 - 5 © 1998 Prentice-Hall, Inc. Possible Result* Cumulative coin tosses Total heads Number of tosses

3 - 6 © 1998 Prentice-Hall, Inc. Experiments, Outcomes, & Events

3 - 7 © 1998 Prentice-Hall, Inc. Experiments & Outcomes 1.Random 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

3 - 8 © 1998 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 - 9 © 1998 Prentice-Hall, Inc. Outcome Examples Toss a coin, note face ExperimentSample Space

© 1998 Prentice-Hall, Inc. Outcome Examples Toss a coin, note faceHead, Tail ExperimentSample Space

© 1998 Prentice-Hall, Inc. Outcome Examples Toss a coin, note faceHead, Tail Toss 2 coins, note faces ExperimentSample Space

© 1998 Prentice-Hall, Inc. Outcome Examples Toss a coin, note faceHead, Tail Toss 2 coins, note facesHH, HT, TH, TT ExperimentSample Space

© 1998 Prentice-Hall, Inc. Outcome Examples Toss a coin, note faceHead, Tail Toss 2 coins, note facesHH, HT, TH, TT Select 1 card, note kind ExperimentSample Space

© 1998 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) ExperimentSample Space

© 1998 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 color ExperimentSample Space

© 1998 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 ExperimentSample Space

© 1998 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 game ExperimentSample Space

© 1998 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 ExperimentSample Space

© 1998 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 quality ExperimentSample Space

© 1998 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 ExperimentSample Space

© 1998 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 gender ExperimentSample Space

© 1998 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

© 1998 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.

© 1998 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

© 1998 Prentice-Hall, Inc. Event Examples Sample space Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 Prentice-Hall, Inc. Event Examples Sample spaceHH, HT, TH, TT Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 Prentice-Hall, Inc. Event Examples Sample spaceHH, HT, TH, TT 1 head & 1 tail Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 Prentice-Hall, Inc. Event Examples Sample spaceHH, HT, TH, TT 1 head & 1 tailHT, TH Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 Prentice-Hall, Inc. Event Examples Sample spaceHH, HT, TH, TT 1 head & 1 tailHT, TH Heads on 1st coin Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 Prentice-Hall, Inc. Event Examples Sample spaceHH, HT, TH, TT 1 head & 1 tailHT, TH Heads on 1st coinHH, HT Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 Prentice-Hall, Inc. Event Examples Sample spaceHH, HT, TH, TT 1 head & 1 tailHT, TH Heads on 1st coinHH, HT At least 1 head Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 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 Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 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 both Experiment: Toss 2 coins. Note faces. EventOutcomes in Event

© 1998 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

© 1998 Prentice-Hall, Inc. Sample Space

© 1998 Prentice-Hall, Inc. Visualizing Sample Space 1.Listing S: {Head, Tail} S: {Head, Tail} 2.Venn Diagram 3.Contingency Table 4.Decision Tree Diagram

© 1998 Prentice-Hall, Inc. S HH TT TH HT Sample space S = {HH, HT, TH, TT} Venn Diagram Outcome Experiment: Toss 2 coins. Note faces. Compound event

© 1998 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)

© 1998 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

© 1998 Prentice-Hall, Inc. Compound Events

© 1998 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)

© 1998 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 

© 1998 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 

© 1998 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.

© 1998 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.

© 1998 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

© 1998 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.

© 1998 Prentice-Hall, Inc. S  Mutually Exclusive Events Example Events  & are 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 

© 1998 Prentice-Hall, Inc. Probabilities

© 1998 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

© 1998 Prentice-Hall, Inc. Assigning Event Probabilities 1.a priori classical Method 2.Empirical classical method 3.Subjective method What’s the probability?

© 1998 Prentice-Hall, Inc. a priori Classical Method 1.Requires prior knowledge of process 2.Can assign 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.

© 1998 Prentice-Hall, Inc. Empirical Classical Method 1.Actual data collected 2.Assign 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!

© 1998 Prentice-Hall, Inc. Subjective Method 1.Requires individual knowledge of situation 2.Done before experiment 3.Unique process Not repeatable Not repeatable 4.Different probabilities from different people © T/Maker Co.

© 1998 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 Chris will default on his PLUS loan? 4.That a student will earn an A in this class? 5.That a new computer store will succeed? Which method should be used to find the probability...

© 1998 Prentice-Hall, Inc. Thinking Challenge 1.That a box of 24 bolts will be defective? (E) 2.That a toss of a coin will be a tail?(C) 3.That Chris will default on his PLUS loan?(S) 4.That a student will earn an A in this class?(S) 5.That a new computer store will succeed?(S) Which method should be used to find the probability...

© 1998 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

© 1998 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

© 1998 Prentice-Hall, Inc. Color Type RedBlack Total Ace 2/522/524/52 Not-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)

© 1998 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) = AloneGroupClass

© 1998 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

© 1998 Prentice-Hall, Inc. Additive Rule

© 1998 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)

© 1998 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  

© 1998 Prentice-Hall, Inc. Thinking Challenge Using the additive rule, what’s the probability? P(A  D) = P(B  C) = AloneGroupClass

© 1998 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  

© 1998 Prentice-Hall, Inc. Conditional Probability

© 1998 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)

© 1998 Prentice-Hall, Inc. Conditional Probability Using Venn Diagram

© 1998 Prentice-Hall, Inc. S Black Ace Conditional Probability Using Venn Diagram

© 1998 Prentice-Hall, Inc. S Black Ace Conditional Probability Using Venn Diagram Black ‘happens’: Eliminates all other outcomes

© 1998 Prentice-Hall, Inc. S Black Ace Conditional Probability Using Venn Diagram Black ‘happens’: Eliminates all other outcomes Event (Ace AND Black) (S) Black

© 1998 Prentice-Hall, Inc. S Black Ace Conditional Probability Using Venn Diagram Event (Ace AND Black) (S) Black Black ‘happens’: Eliminates all other outcomes

© 1998 Prentice-Hall, Inc. Conditional Probability Using Venn Diagram Event (Ace AND Black) (S) Black New sample space

© 1998 Prentice-Hall, Inc. Conditional Probability Using Venn Diagram Event (Ace AND Black) (S) Black New sample space P(A | B) = P(A and B) P(B)

© 1998 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

© 1998 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

© 1998 Prentice-Hall, Inc. Tree Diagram Experiment: Select 2 pens from 20 pens: 14 green & 6 red. Don’t replace. Dependent! G R G R G R P(R) = 6/20 P(R|R) = 5/19 P(G|R) = 14/19 P(G) = 14/20 P(R|G) = 6/19 P(G|G) = 13/19

© 1998 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 AloneGroupClass

© 1998 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    / /

© 1998 Prentice-Hall, Inc. Multiplicative Rule

© 1998 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)

© 1998 Prentice-Hall, Inc. Multiplicative Rule Example Experiment: Draw 1 card. Note kind, color & suit. Color Type RedBlack Total Ace 224 Non-Ace Total

© 1998 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 AloneGroupClass

© 1998 Prentice-Hall, Inc. Solution* Using the multiplicative rule, the probabilities are: P(CB)= P(C) P(C)P(B|C) 5 10  P(BD)= P(B) P(B)P(D|B) 4 10  P(AB)= P(A) P(A)P(B|A) 0

© 1998 Prentice-Hall, Inc. Random Sampling

© 1998 Prentice-Hall, Inc. Simple Random Sample 1.Each population element has an equal chance of being selected 2.Selecting 1 element does not affect selecting others 3.Computer © T/Maker Co.

© 1998 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 5.Defined random sample

© 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in class today? 2.What do you still have questions about? 3.How can today’s class be improved? Please take a moment to answer the following questions in writing:

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