Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal
Chapter Goals After completing this chapter, you should be able to: Explain basic probability concepts and definitions Use a Venn diagram or tree diagram to illustrate simple probabilities Apply common rules of probability Explain three approaches to assessing probabilities Apply common rules of probability, including the Addition Rule and the Multiplication Rule QMIS 120, by Dr. M. Zainal Chap 4-2
Chapter Goals (continued) Compute conditional probabilities Determine whether events are statistically independent Use Bayes Theorem for conditional probabilities QMIS 120, by Dr. M. Zainal Chap 4-3
Important Terms Random Experiment a process leading to an uncertain outcome Basic Outcome (Simple Event ) a possible outcome of a random experiment Sample Space the collection of all possible outcomes of a random experiment Event (Compound Event) any subset of basic outcomes from the sample space Probability the chance that an uncertain event will occur (always between 0 and 1) QMIS 120, by Dr. M. Zainal Chap 4-4
Sample Space The Sample Space is the collection of all possible outcomes e.g., All 6 faces of a die: e.g., All 52 cards of a bridge deck: QMIS 120, by Dr. M. Zainal Chap 4-5
Sample Space e.g., All 2 faces of a coin: e.g., All 4 colors in the jar: QMIS 120, by Dr. M. Zainal Chap 4-6
Examples of Sample Spaces Rolling one die: S = [1, 2, 3, 4, 5, 6] Tossing one coin: S = [Head, Tail] QMIS 120, by Dr. M. Zainal Chap 4-7
Examples of Sample Spaces Choosing one ball: S = [Red, Blue, Green, Yellow] Playing a game: S = [Win, Lose, Tie] QMIS 120, by Dr. M. Zainal Chap 4-8
Events Experimental outcome An outcome from a sample space with one characteristic Example: A red card from a deck of cards Event May involve two or more outcomes simultaneously Example: An ace that is also red from a deck of cards QMIS 120, by Dr. M. Zainal Chap 4-9
Visualizing Events Contingency Tables Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Sample Space Tree Diagrams Full Deck of 52 Cards 2 24 2 24 Sample Space QMIS 120, by Dr. M. Zainal Chap 4-10
Experimental Outcomes A automobile consultant records fuel type and vehicle type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV 6 possible experimental outcomes: e 1 Gasoline, Truck e 2 Gasoline, Car e 3 Gasoline, SUV e 4 Diesel, Truck e 5 Diesel, Car Diesel, SUV e 6 QMIS 120, by Dr. M. Zainal Chap 4-11
Important Terms Intersection of Events If A and B are two events in a sample space S, then the intersection, A B, is the set of all outcomes in S that belong to both A and B S A A B B QMIS 120, by Dr. M. Zainal Chap 4-12
Important Terms A and B are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set A B is empty If A occurs, then B cannot occur A and B have no common elements (continued) S A B A ball cannot be Green and Red at the same time. QMIS 120, by Dr. M. Zainal Chap 4-13
Important Terms (continued) Union of Events If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either A or B S The entire shaded area represents A U B A B QMIS 120, by Dr. M. Zainal Chap 4-14
Important Terms (continued) Events E 1, E 2, E k are Collectively Exhaustive events if E 1 U E 2 U... U E k = S i.e., the events completely cover the sample space S The Complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. The complement is denoted A A A A = + A A QMIS 120, by Dr. M. Zainal Chap 4-15
Examples Rolling a die S = [1, 2, 3, 4, 5, 6] Let A be the event Number rolled is even Let B be the event Number rolled is at least 4 Then: QMIS 120, by Dr. M. Zainal Chap 4-16
Examples (continued) Complements: Intersections: Unions: QMIS 120, by Dr. M. Zainal Chap 4-17
Examples (continued) Mutually exclusive: Collectively exhaustive: QMIS 120, by Dr. M. Zainal Chap 4-18
Probability Probability the chance that an uncertain event will occur (always between 0 and 1) Also, it is a numerical measure of the likelihood that an event will occur 1.5 Certain 0 P(A) 1 For any event A 0 Impossible QMIS 120, by Dr. M. Zainal Chap 4-19
Assessing Probability There are three approaches to assessing the probability of an uncertain event: 1. classical probability N probability of event A A N numberof outcomesthat satisfythe event total numberof outcomesinthe samplespace Assumes all outcomes in the sample space are equally likely to occur QMIS 120, by Dr. M. Zainal Chap 4-20
Classical Probability Examples Example: Find the probability of obtaining an even number in one roll of a die and find the probability of obtaining a number greater than 4. S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] C = [5, 6] QMIS 120, by Dr. M. Zainal Chap 4-21
Counting the Possible Outcomes Use the Combinations formula to determine the number of distinct combinations of n distinct objects that can be formed, taking them r at a time (No ordering) C n n C () n r r r n! r!( n r)! where n! = n(n-1)(n-2) (1) 0! = 1 by definition QMIS 120, by Dr. M. Zainal Chap 4-22
Counting the Possible Outcomes Combinations: If one has 5 different objects (e.g. A, B, C, D, and E), how many ways can they be grouped as 3 objects when position does not matter (e.g. ABC, ABD, ABE, ACD, ACE, ADE are correct but CBA is not ok as is equal to ABC) QMIS 120, by Dr. M. Zainal Chap 4-23
Counting the Possible Outcomes Use the Permutations formula to determine the number of ways we can arrange n distinct objects, taking them r at a time (with ordering) where P n! = n(n-1)(n-2) (1) 0! = 1 by definition n n P () n r r r n! ( n r)! QMIS 120, by Dr. M. Zainal Chap 4-24
Counting the Possible Outcomes Permutations: Given that position (order) is important, if one has 5 different objects (e.g. A, B, C, D, and E), how many unique ways can they be placed in 3 positions (e.g. ADE, AED, DEA, DAE, EAD, EDA, ABC, ACB, BCA, BAC etc.) QMIS 120, by Dr. M. Zainal Chap 4-25
Assessing Probability Three approaches (continued) 2. relative frequency probability n probability of event A A n number of events inthe populationthat satisfyevent total number of events inthe population A the limit of the proportion of times that an event A occurs in a large number of trials, n 3. subjective probability an individual opinion or belief about the probability of occurrence QMIS 120, by Dr. M. Zainal Chap 4-26
Relative Frequency Probability Examples Example: In a group of 500 women, 80 have played football at least once in their life. Suppose one of these 500 woman is selected. What is the probability that she played football at least once? Solution: n = 500 and f = 80 Women Did not play football Played football f 420 80 n = 500 Relative frequency 420 / 500 = 0.84 80 / 500 = 0.16 Sum = 1.0 QMIS 120, by Dr. M. Zainal Chap 4-27
Probability Postulates 1. If A is any event in the sample space S, then 0 P(A) 2. Let A be an event in S, and let E i denote the basic outcomes. Then A (the notation means that the summation is over all the basic outcomes in A) 1 P(A) P(E i) 3. P(S) = 1 QMIS 120, by Dr. M. Zainal Chap 4-28
Probability Rules The Complement rule: P( A) 1 P(A) i.e., P(A) P(A) 1 A A QMIS 120, by Dr. M. Zainal Chap 4-29
Joint and Marginal Probabilities The probability of a joint event, A B: P(A B) numberof outcomessatisfyinga andb total number of elementaryoutcomes Computing a marginal probability: P(A) P(A B 1) P(A B2) P(A B k ) Where B 1, B 2,, B k are k mutually exclusive and collectively exhaustive events QMIS 120, by Dr. M. Zainal Chap 4-30
Marginal Probability Marginal probability is the probability of a single event without consideration of any other event Suppose one employee is selected out of a sample of 100, he/she maybe classified either on the bases of gender alone or on the bases of a marital status. Married Single Total Male 15 45 60 Female 4 36 40 Total 19 81 100 QMIS 120, by Dr. M. Zainal Chap 4-31
Marginal Probability Example The probability of each of the following event is called marginal probability P(male) = 60/100 P(female) = 40/100 P(married) = 19/100 P(single) = 81/100 Male Female Married 15 4 Single 45 36 Total 60 40 Total 19 81 100 QMIS 120, by Dr. M. Zainal Chap 4-32
Conditional Probability This probability is called conditional probability and is written as the probability that the employee selected is married given that he is a male. P(married male ) This event has already occurred The event whose probability is to be determined Read as given QMIS 120, by Dr. M. Zainal Chap 4-33
Conditional Probability A conditional probability is the probability of one event, given that another event has occurred: P(A B) P(A B) P(B) The conditional probability of A given that B has occurred P(B A) P(A B) P(A) The conditional probability of B given that A has occurred QMIS 120, by Dr. M. Zainal Chap 4-34
Conditional Probability Example Example: Find the conditional probability P(married male) for the data on 100 employees Solution Married Single Total Male 15 45 60 Female 4 36 40 Total 19 81 100 QMIS 120, by Dr. M. Zainal Chap 4-35
Conditional Probability Example (continued) QMIS 120, by Dr. M. Zainal Chap 4-36
Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. What is the probability that a car has a CD player, given that it has AC? i.e., we want to find P(CD AC) QMIS 120, by Dr. M. Zainal Chap 4-37
Conditional Probability Example (continued) Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. AC No AC CD No CD Total Total 1.0 QMIS 120, by Dr. M. Zainal Chap 4-38
Using a Tree Diagram Given AC or no AC: All Cars QMIS 120, by Dr. M. Zainal Chap 4-39
Conditional Probability Example Example: Consider the experiment of tossing a fair die. Denote by A and B the following events: A={Observing an even number}, B={Observing a number of dots less than or equal to 3}. Find the probability of the event A, given the event B. Solution QMIS 120, by Dr. M. Zainal Chap 4-40
Multiplication Rule Multiplication rule for two events A and B: P(A B) P(A B)P(B) also P(A B) P(B A) P(A) QMIS 120, by Dr. M. Zainal Chap 4-41
Probability Concepts Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other E 1 = heads on one flip of fair coin E 2 = heads on second flip of same coin Result of second flip does not depend on the result of the first flip. QMIS 120, by Dr. M. Zainal Chap 4-42
Probability Concepts Independent and Dependent Events Dependent: Occurrence of one affects the probability of the other E 1 = rain forecasted on the news E 2 = take umbrella to work Probability of the second event is affected by the occurrence of the first event QMIS 120, by Dr. M. Zainal Chap 4-43
Statistical Independence Two events are statistically independent if and only if: P(A B) P(A)P(B) Events A and B are independent when the probability of one event is not affected by the other event If A and B are independent, then P(A B) P(A) if P(B)>0 P(B A) P(B) if P(A)>0 QMIS 120, by Dr. M. Zainal Chap 4-44
Statistical Independence Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. CD No CD Total AC.2.5.7 No AC.2.1.3 Total.4.6 1.0 Are the events AC and CD statistically independent? QMIS 120, by Dr. M. Zainal Chap 4-45
Statistical Independence Example CD No CD Total AC.2.5.7 No AC.2.1.3 Total.4.6 1.0 (continued) QMIS 120, by Dr. M. Zainal Chap 4-46
For Independent Events: Conditional probability for independent events E 1, E 2 : P(E E ) P(E ) 2 1 1 where P(E ) 2 0 where P(E 1) 0 2 P(E E ) P(E ) 1 2 QMIS 120, by Dr. M. Zainal Chap 4-47
Probability Rules The Addition rule: The probability of the union of two events is P(A B) P(A) P(B) P(A B) A + = B A B P(A or B) = P(A) + P(B) - P(A and B) Don t count common elements twice! QMIS 120, by Dr. M. Zainal Chap 4-48
Addition Rule for Mutually Exclusive Events If A and B are mutually exclusive, then P(A B) = 0 A B P(A or B) = P(A) + P(B) - P(A and B) So P(A U B) = P(A) + P(B) QMIS 120, by Dr. M. Zainal Chap 4-49
A Probability Table Probabilities and joint probabilities for two events A and B are summarized in this table: B B A P(A B) P(A B) P(A) A P( A B) P( A B) P(A) P(B) P(B) P(S) 1.0 QMIS 120, by Dr. M. Zainal Chap 4-50
Law of Total Probability Consider the following Venn diagram B B 1 B 2 B n-1 B n A A 1 A 2 A n-1 A n Can we find the area (probability) of A assuming that we know the probability of each Bi & P(A Bi)? QMIS 120, by Dr. M. Zainal Chap 4-51
Bayes Theorem P(B A) i P(A B ) P(A) i P(A B )P(B ) i P(A B )P(B ) P(A B )P(B ) 1 1 2 2 i where: B i = i th event of k mutually exclusive and collectively exhaustive events A = new event that might impact P(B i ) QMIS 120, by Dr. M. Zainal Chap 4-52
Bayes Theorem Example A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful? QMIS 120, by Dr. M. Zainal Chap 4-53
Bayes Theorem Example (continued) Let S = successful well U = unsuccessful well P(S) =, P(U) = (prior probabilities) Define the detailed test event as D Conditional probabilities: Goal is to find QMIS 120, by Dr. M. Zainal Chap 4-54
Bayes Theorem Example Apply Bayes Theorem: (continued) QMIS 120, by Dr. M. Zainal Chap 4-55
Bayes Theorem Example (continued) Given the detailed test, the revised probability of a successful well has from the original estimate of QMIS 120, by Dr. M. Zainal Chap 4-56
Complete Example Consider a standard deck of card that we see normally being played on computer or table. It consists of 52 cards with 4 suits and so 13 cards in a suit. Consider a standard deck of 52 cards, with four suits: Spades and Clubs are traditionally in black color, while Hearts and Diamonds are in red. Spade (S): Heart (H): Club (C): Diamond (D): Let event A = card is an Ace Let event B = card is from a Red suit QMIS 120, by Dr. M. Zainal Chap 4-57
Complete Example What is the probability of Red given it was Ace (Cond.) Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 QMIS 120, by Dr. M. Zainal Chap 4-58
Complete Example What is the probability of Red and Ace (Intersection) Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 QMIS 120, by Dr. M. Zainal Chap 4-59
Complete Example What is the probability of Red or Ace (Union) Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 QMIS 120, by Dr. M. Zainal Chap 4-60
Complete Example What is the probability of Ace (Marginal) Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 QMIS 120, by Dr. M. Zainal Chap 4-61
Complete Example Are the events Red and Ace statistically independent? Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 QMIS 120, by Dr. M. Zainal Chap 4-62
Complete Example What is the complement of the event Ace? Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 QMIS 120, by Dr. M. Zainal Chap 4-63
Bayes Theorem Example Example: Manufacturing firm that receives shipment of parts from two different suppliers. Currently, 65 percent of the parts purchased by the company are from supplier 1 and the remaining 35 percent are from supplier 2. Historical Data suggest the quality rating of the two supplier are shown in the table: Good Parts Bad Parts Supplier 1 98 2 Supplier 2 95 5 a) Draw a tree diagram for this experiment with the probability of all outcomes b) Given the information the part is bad, What is the probability the part came from supplier 1? QMIS 120, by Dr. M. Zainal Chap 4-64
Bayes Theorem Example QMIS 120, by Dr. M. Zainal Chap 4-65
Bayes Theorem Example (continued) Apply Bayes Theorem: QMIS 120, by Dr. M. Zainal Chap 4-66
Bayes Theorem Example Example: An insurance company rents 35% of the cars for its customers from Avis and the rest from Hertz. From past records they know that 8% of Avis cars break down and 5% of Hertz cars break down. A customer calls and complains that his rental car broke down. What is the probability that his car was rented from Avis? QMIS 120, by Dr. M. Zainal Chap 4-67
Bayes Theorem Example QMIS 120, by Dr. M. Zainal Chap 4-68
Chapter Summary Defined basic probability concepts Sample spaces and events, intersection and union of events, mutually exclusive and collectively exhaustive events, complements Examined basic probability rules Complement rule, addition rule, multiplication rule Defined conditional, joint, and marginal probabilities Reviewed odds and the overinvolvement ratio Defined statistical independence Discussed Bayes theorem QMIS 120, by Dr. M. Zainal Chap 4-69
Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e 2008 Prentice-Hall, Inc. QMIS 120, by Dr. M. Zainal Chap 4-70