2011 Pearson Education, Inc

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2 Statistics for Business and Economics Chapter 3 Probability

3 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

4 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

5 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?

6 Total Heads Number of Tosses Many Repetitions!* Number of Tosses

7 3.1 Events, Sample Spaces, and Probability

8 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!

9 Sample Space Properties 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 Experiment: Observe Gender T/Maker Co.

10 1. Listing S = {Head, Tail} Visualizing Sample Space 2. Venn Diagram H T S

11 Sample Space Examples Experiment Toss a Coin, Note Face Sample Space {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}

12 Events 1. Specific collection of sample points 2. Simple Event Contains only one sample point 3. Compound Event Contains two or more sample points

13 Venn Diagram Experiment: Toss 2 Coins. Note Faces. Outcome Sample Space S = {HH, HT, TH, TT} TH HH HT TT S Compound Event: At least one Tail

14 Event Examples Experiment: Toss 2 Coins. Note Faces. Sample Space: HH, HT, TH, TT Event 1 Head & 1 Tail Head on 1st Coin At Least 1 Head Heads on Both Outcomes in Event HT, TH HH, HT HH, HT, TH HH

15 Probabilities

16 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

17 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).

18 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.

19 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

20 Combinations Rule A sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible N is denoted by and is equal to n N n N! n! N n! where the factorial symbol (!) means that n! n n 1 n 2 L ! For example, 0! is defined to be 1.

21 3.2 Unions and Intersections

22 Compound Events Compound events: Composition of two or more other events. Can be formed in two different ways.

23 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)

24 Event Union: Venn Diagram Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space: 2, 2, 2,..., A Ace Black S Event Black: 2, 2,..., A Event Ace: A, A, A, A Event Ace Black: A,..., A, 2,..., K

25 Event Union: Two Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space (S): 2, 2, 2,..., A Event Ace Black: A,..., A, 2,..., K Color Type Red Black Total Ace Ace & Red Ace & Black Ace Non-Ace Non & Non & Non- Red Black Ace Total Red Black S Simple Event Black: 2,..., A Simple Event Ace: A, A, A, A

26 Event Intersection: Venn Diagram Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space: 2, 2, 2,..., A Ace Black S Event Black: 2,...,A Event Ace: A, A, A, A Event Ace A, A Black:

27 Event Intersection: Two Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space (S): 2, 2, 2,..., A Event Ace A, A Black: Color Type Red Black Total Ace Ace & Red Ace & Black Ace Non-Ace Non & Non & Non- Red Black Ace Total Red Black S Simple Event Black: 2,..., A Simple Event Ace: A, A, A, A

28 Compound Event Probability 1. Numerical measure of likelihood that compound event will occur 2. Can often use two way table Two variables only

29 Event Probability Using Two Way Table 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 Joint Probability P(B 1 ) P(B 2 ) 1 Marginal (Simple) Probability

30 Two Way Table Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace 2/52 2/52 4/52 Non-Ace 24/52 24/52 48/52 Total 26/52 26/52 52/52 P(Ace) P(Red) P(Ace Red)

31 Thinking Challenge What s the Probability? 1. P(A) = 2. P(D) = 3. P(C B) = 4. P(A D) = 5. P(B D) = Event Event C D Total A B Total

32 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 Event C D Total A B Total

33 3.3 Complementary Events

34 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 A A C S

35 Rule of Complements The sum of the probabilities of complementary events equals 1: P(A) + P(A C ) = 1 A A C S

36 Complement of Event Example Experiment: Draw 1 Card. Note Color. Sample Space: 2, 2, 2,..., A Black S Event Black: 2, 2,..., A Complement of Event Black, Black C : 2, 2,..., A, A

37 3.4 The Additive Rule and Mutually Exclusive Events

38 Mutually Exclusive Events Mutually Exclusive Events Events do not occur simultaneously A does not contain any sample points

39 Mutually Exclusive Events Example Experiment: Draw 1 Card. Note Kind & Suit. Sample Space: 2, 2, 2,..., A S Outcomes in Event Heart: 2, 3, 4,..., A Event Spade: 2, 3, 4,..., A Events and are Mutually Exclusive

40 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)

41 Additive Rule Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace Non-Ace Total P(Ace Black) = P(Ace) + P(Black) P(Ace Black) = + =

42 Thinking Challenge Using the additive rule, what is the probability? 1. P(A D) = 2. P(B C) = Event Event C D Total A B Total

43 Solution* Using the additive rule, the probabilities are: 1. P(A D) = P(A) + P(D) P(A D) = + = P(B C) = P(B) + P(C) P(B C) = + =

44 3.5 Conditional Probability

45 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)

46 Conditional Probability Using Venn Diagram Ace Black Black Happens : Eliminates All Other Outcomes S Black (S) Event (Ace Black)

47 Conditional Probability Using Two Way Table Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace Non-Ace Revised Sample Space Total P(Ace Black) = P(Ace Black) 2/ 52 2 P(Black) 26/ 52 26

48 Thinking Challenge Using the table then the formula, what s the probability? 1. P(A D) = 2. P(C B) = Event Event C D Total A B Total

49 Solution* Using the formula, the probabilities are: P A D P A P D B P C B P C P B B

50 3.6 The Multiplicative Rule and Independent Events

51 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)

52 Multiplicative Rule Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace Non-Ace Total P(Ace Black) = P(Ace) P(Black Ace)

53 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)

54 Thinking Challenge Using the multiplicative rule, what s the probability? 1. P(C B) = 2. P(B D) = 3. P(A B) = Event Event C D Total A B Total

55 Solution* Using the multiplicative rule, the probabilities are: P C B P C P B C P B D P B P D B P A B P A P B A 0

56 Tree Diagram Experiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don t replace. 6/20 14/20 R B Dependent! 5/19 14/19 6/19 13/19 R B R B 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

57 3.7 Random Sampling

58 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.

59 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.

60 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.

61 3.8 Bayes s Rule

62 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 P(B i A) P(B i A) P( A) P(B i )P( A B i ) P(B 1 )P( A B 1 ) P(B 2 )P( A B 2 )... P(B k )P( A B k )

63 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?

64 Bayes s Rule Example 0.6 Factory I Defective Good P(I D) 0.4 P(I)P(D I) Factory II P(I)P(D I) P(II)P(D II) Defective Good

65 Key Ideas Probability Rules for k Sample Points, S 1, S 2, S 3,..., S k 1. 0 P(S i ) 1 2. P S i 1

66 Key Ideas Random Sample All possible such samples have equal probability of being selected.

67 Key Ideas Combinations Rule Counting number of samples of n elements selected from N elements N n N! n! N n! N N 1 N 2 L N n 1 n n 1 n 2 L 2 1

68 Key Ideas Bayes s Rule P(S i A) P(S i )P( A S i ) P(S 1 )P( A S 1 ) P(S 2 )P( A S 2 )... P(S k )P( A S k )

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