2011 Pearson Education Inc Statistics for Business and
© 2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability © 2011 Pearson Education, Inc
Contents 1. 2. 3. 4. 5. 6. 7. 8. Events, Sample Spaces, and Probability Unions and Intersections Complementary Events The Additive Rule and Mutually Exclusive Events Conditional Probability The Multiplicative Rule and Independent Events Random Sampling Baye’s Rule © 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 © 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? © 2011 Pearson Education, Inc
Many Repetitions!* Total Heads Number of Tosses 1. 00 0. 75 0. 50 0. 25 0. 00 0 25 50 75 Number of Tosses © 2011 Pearson Education, Inc 100 125
3. 1 Events, Sample Spaces, and Probability © 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 Sample Space Depends on Experimenter! 3. Sample space (S) • Collection of all possible outcomes © 2011 Pearson Education, Inc
Sample Space Properties 1. Mutually Exclusive • Experiment: Observe Gender 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 © 2011 Pearson Education, Inc © 1984 -1994 T/Maker Co.
Visualizing Sample Space 1. Listing S = {Head, Tail} 2. Venn Diagram H T S © 2011 Pearson Education, Inc
Sample Space Examples Experiment • • Sample Space 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} © 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 © 2011 Pearson Education, Inc
Venn Diagram Experiment: Toss 2 Coins. Note Faces. Sample Space S = {HH, HT, TH, TT} TH Outcome HH Compound Event: At least one Tail HT TT © 2011 Pearson Education, Inc S
Event Examples Experiment: Toss 2 Coins. Note Faces. Sample Space: HH, HT, TH, TT • • Event 1 Head & 1 Tail Head on 1 st Coin At Least 1 Heads on Both Outcomes in Event HT, TH HH, HT, TH HH © 2011 Pearson Education, Inc
Probabilities © 2011 Pearson Education, Inc
What is Probability? 1. Numerical measure of the likelihood that event will cccur • P(Event) • P(A) • Prob(A) 1 Certain . 5 2. Lies between 0 & 1 3. Sum of sample points is 1 0 © 2011 Pearson Education, Inc Impossible
Probability Rules for Sample Points Let pi represent the probability of sample point i. 1. All sample point probabilities must lie between 0 and 1 (i. e. , 0 ≤ pi ≤ 1). 2. The probabilities of all sample points within a sample space must sum to 1 (i. e. , pi = 1). © 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 © 2011 Pearson Education, Inc © 1984 -1994 T/Maker Co.
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 © 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 by and is equal to where the factorial symbol (!) means that For example, 0! is defined to be 1. © 2011 Pearson Education, Inc
3. 2 Unions and Intersections © 2011 Pearson Education, Inc
Compound Events Compound events: Composition of two or more other events. Can be formed in two different ways. © 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) © 2011 Pearson Education, Inc
Event Union: Venn Diagram Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space: 2 , 2 , . . . , A Ace Black S Event Ace: A , A , A Event Ace Black: A , . . . , A , 2 , . . . , K © 2011 Pearson Education, Inc Event Black: 2 , . . . , A
Event Union: Two–Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Simple Sample Space Type (S): Ace 2 , 2 , . . . , A Non-Ace Total Event Ace Black: A , . . . , A , � 2 , . . . , K Total Ace & Ace Red Black Non & Non. Red Black Ace Red Black Simple Event Black: 2 , . . . , A © 2011 Pearson Education, Inc Event Ace: A , A , A
Event Intersection: Venn Diagram Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space: 2 , 2 , . . . , A Ace Black S Event Ace: A , A , A Event Ace Black: A , A © 2011 Pearson Education, Inc Event Black: 2 , . . . , A
Event Intersection: Two–Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Sample Space Type (S): Ace 2 , 2 , . . . , A Non-Ace Event Ace Black: A , A Total Ace & Ace Red Black Non & Non. Red Black Ace Red Black Simple Event Ace: A , A , A Simple Event Black: 2 , . . . , A © 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 © 2011 Pearson Education, Inc
Event Probability Using Two–Way Table 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 Joint Probability P(B 1) P(B 2) 1 Marginal (Simple) Probability © 2011 Pearson Education, Inc
Two–Way Table Example Experiment: Draw 1 Card. Note Kind & Color Type Red Black Ace 2/52 Total 4/52 Non-Ace 24/52 48/52 Total 26/52 52/52 P(Red) P(Ace Red) © 2011 Pearson Education, Inc P(Ace)
Thinking Challenge What’s the Probability? 1. P(A) = 2. P(D) = Event C D 4 2 3. P(C B) = Event A 4. P(A D) = B 1 3 4 5. P(B D) = Total 5 5 10 © 2011 Pearson Education, Inc Total 6
Solution* The Probabilities Are: 1. P(A) = 6/10 2. P(D) = 5/10 Event C D 4 2 3. P(C B) = 1/10 Event A 4. P(A D) = 9/10 B 1 3 4 5. P(B D) = 3/10 Total 5 5 10 © 2011 Pearson Education, Inc Total 6
3. 3 Complementary Events © 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 AC AC A © 2011 Pearson Education, Inc S
Rule of Complements The sum of the probabilities of complementary events equals 1: P(A) + P(AC) = 1 AC A © 2011 Pearson Education, Inc S
Complement of Event Example Experiment: Draw 1 Card. Note Color. Black Sample Space: 2 , 2 , . . . , A Event Black: 2 , . . . , A S Complement of Event Black, Black. C: 2 , . . . , A © 2011 Pearson Education, Inc
3. 4 The Additive Rule and Mutually Exclusive Events © 2011 Pearson Education, Inc
Mutually Exclusive Events • Events do not occur simultaneously • A B does not contain any sample points © 2011 Pearson Education, Inc
Mutually Exclusive Events Example Experiment: Draw 1 Card. Note Kind & Suit. Sample Space: 2 , 2 , . . . , A Event Spade: 2 , 3 , 4 , . . . , A S Outcomes in Event Heart: 2 , 3 , 4 , . . . , A Events and are Mutually Exclusive © 2011 Pearson Education, Inc
Additive Rule 1. Used to get compound probabilities for union of events 2. P(A OR B) = P(A) + P(B) – P(A B) 3. For mutually exclusive events: P(A OR B) = P(A) + P(B) © 2011 Pearson Education, Inc
Additive Rule Example Experiment: Draw 1 Card. Note Kind & Color Type Ace Red Black 2 2 Total 4 Non-Ace 24 24 48 Total 26 26 52 P(Ace Black) = P(Ace) + P(Black) – P(Ace Black) 4 26 2 28 = + – = 52 52 © 2011 Pearson Education, Inc
Thinking Challenge Using the additive rule, what is the probability? 1. P(A D) = 2. P(B C) = Event A Event C D 4 2 Total 6 B 1 3 4 Total 5 5 10 © 2011 Pearson Education, Inc
Solution* Using the additive rule, the probabilities are: 1. P(A D) = P(A) + P(D) – P(A D) 6 5 2 9 = + – = 10 10 2. P(B C) = P(B) + P(C) – P(B C) 4 5 1 8 = + – = 10 10 © 2011 Pearson Education, Inc
3. 5 Conditional Probability © 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) © 2011 Pearson Education, Inc
Conditional Probability Using Venn Diagram Ace Black S Event (Ace Black) © 2011 Pearson Education, Inc Black ‘Happens’: Eliminates All Other Outcomes Black (S)
Conditional Probability Using Two–Way Table Experiment: Draw 1 Card. Note Kind & Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 © 2011 Pearson Education, Inc Revised Sample Space
Thinking Challenge Using the table then the formula, what’s the probability? 1. P(A|D) = 2. P(C|B) = Event A Event C D 4 2 Total 6 B 1 3 4 Total 5 5 10 © 2011 Pearson Education, Inc
Solution* Using the formula, the probabilities are: © 2011 Pearson Education, Inc
3. 6 The Multiplicative Rule and Independent Events © 2011 Pearson Education, Inc
Multiplicative Rule 1. Used to get compound probabilities for intersection of events 2. P(A and B) = P(A) P(B|A) = P(B) P(A|B) 3. For Independent Events: P(A and B) = P(A) P(B) © 2011 Pearson Education, Inc
Multiplicative Rule Example Experiment: Draw 1 Card. Note Kind & Color Type Ace Red Black 2 2 Total 4 Non-Ace 24 24 48 Total 26 26 52 P(Ace Black) = P(Ace)∙P(Black | Ace) © 2011 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) © 2011 Pearson Education, Inc
Thinking Challenge Using the multiplicative rule, what’s the probability? 1. P(C B) = Event C D 4 2 2. P(B D) = Event A 3. P(A B) = B 1 3 4 Total 5 5 10 © 2011 Pearson Education, Inc Total 6
Solution* Using the multiplicative rule, the probabilities are: © 2011 Pearson Education, Inc
Tree Diagram Experiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don’t replace. Dependent! 6/20 R 5/19 14/20 B 6/19 13/19 R P(R R)=(6/20)(5/19) =3/38 B R P(R B)=(6/20)(14/19) =21/95 B P(B B)=(14/20)(13/19) =91/190 P(B R)=(14/20)(6/19) =21/95 © 2011 Pearson Education, Inc
3. 7 Random Sampling © 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. © 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. © 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. © 2011 Pearson Education, Inc
3. 8 Bayes’s Rule © 2011 Pearson Education, Inc
Bayes’s Rule Given k mutually exclusive and exhaustive events B 1, . . . Bk , such that P(B 1) + P(B 2) + … + P(Bk) = 1, and an observed event A, then © 2011 Pearson Education, Inc
Bayes’s Rule Example A company manufactures MP 3 players at two factories. Factory I produces 60% of the MP 3 players and Factory II produces 40%. Two percent of the MP 3 players produced at Factory I are defective, while 1% of Factory II’s are defective. An MP 3 player is selected at random and found to be defective. What is the probability it came from Factory I? © 2011 Pearson Education, Inc
Bayes’s Rule Example Factory I 0. 6 0. 4 Factory II 0. 02 Defective 0. 98 Good 0. 01 Defective 0. 99 Good © 2011 Pearson Education, Inc
Key Ideas Probability Rules for k Sample Points, S 1, S 2, S 3, . . . , Sk 1. 0 ≤ P(Si) ≤ 1 2. © 2011 Pearson Education, Inc
Key Ideas Random Sample All possible such samples have equal probability of being selected. © 2011 Pearson Education, Inc
Key Ideas Combinations Rule Counting number of samples of n elements selected from N elements © 2011 Pearson Education, Inc
Key Ideas Bayes’s Rule © 2011 Pearson Education, Inc
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