OPRE504 Chapter Study Guide Chapter 7 Randomness and Probability Terminology of Probability For a Random phenomenon, there are a number of possible Outcomes. For example, tossing a coin could result in either a head or a tail (a total of two possible outcomes). Tossing is called a Trial. A trial generates an outcome. An Event is a collection of possible outcomes. Sample Space is the special Event which contains all possible outcomes. Theoretical (Model-based) Probability of a Random Phenomenon: P (A) = Probability Rules: Rule 1 0 P (A) 1 In a two-outcome situation, 50% means that two outcomes are equally likely Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 If a random phenomenon has N possible outcomes, then P (outcome 1) + P (outcome 2) + P(outcome N) = 1 Complement Rule P (A) = 1 P (A c ) [ P (A c ) is the probability of Event A is not occurring.] Multiplication Rule for Independent Events A and B Probability of all events occurring simultaneously is the product of the probabilities of all individual events P (A and B) = P (A) x P (B) Addition Rule for Disjoint (Mutually Exclusive) Events A and B Probability of either of the two disjoint events occurring P (A or B) = P (A) + P (B) General Addition Rule for Any Two Events A and B Probability of either of the two events occurring is the sum of the probabilities of two individual events subtracted by the potential double counting of both events happening simultaneously. P (A or B) = P (A) + P (B) P (A and B) If A and B are mutually exclusive, P (A and B) = 0, Rule 6 connects Rule 5: so P (A or B) = P (A) + P (B) + 0 = P (A) + P (B) Han OPRE504 Page 1 of 5
Marginal Probability and Joint Probability in A Contingency Table An Example of Contingency Table Note: A customer is randomly selected to receive a prize by an electronic retailer TABLET COMPUTER PREFERENCE ipad Android Other Total Male 120 100 30 250 Customer Female 80 40 30 150 Total 200 140 60 400 The probability of selecting a male customer is called Marginal Probability: P (Male) = 250/400 = 62.5%, because two numbers (250 and 400) in the margins of the contingency table are used. The probability of selecting a customer who is male and prefers ipad is called Joint Probability: P (Male and ipad ) = 120/ 400 = 30% The probability for a male customer to choose ipad is called Conditional Probability: P (ipad Male) = = = 48% because it describes the probability of preferring an ipad given a male customer. Additional Probability Rules Rule 7 Rule 8 General Multiplication Rule for Any Two Events: A and B P (A and B) = P (A) x P (B A) or P (A and B) = P (B) x P (A B) Independence Rule If A and B are independent, we would expect that P (A) = P (A B), probability of A does not change whether B occurs or not. P (B) = P (B A), probability of B does not change whether A occurs or not. To evaluate whether Events A and B are independent, we can also check whether P (A and B) = P (A) x P (B) Question 7.1 [ Sharpe 2011, Exercise 11, p.198] In developing their warranty policy, an automobile company estimates that over a 1-year period 17% of their new cars will need to be repaired once, 7% will need repairs, and 4% will require three or more repairs. If you buy a new car from them, what is the probability that your car will need: a) Some repair? Some repair: P (some repairs) = P (repair once) + P (repair twice) + P (repair three or more) = 0.17 + 0.07 + 0.04 = 0.28 Han OPRE504 Page 2 of 5
b) No repairs? P (No repairs) = 1 P (No repairs) c = 1 P (some repairs) = 1-0.28 = 0.72 c) No more than one repair? P (no more than one repair) = P (no repairs) + P (one repair) = 0.72 + 0.17 = 0.89 or = 1- [P (repair twice) + P (repair three or more)] = 1 (0.07 + 0.04) = 1-0.11 = 0.89 Question 7.2 [Sharpe 2011, Exercise 33, p.200] A GfK Roper Worldwide survey in 2005 asked consumers in five countries whether they agreed with the statement I am worried about the safety of the food I eat. Here are the responses classified by the age of the response: Age Group Agree Neither Agree nor Disagree Disagree Don t Know / No Response Total 13-19 661 368 452 32 1513 20-29 816 365 336 16 1533 30-39 871 355 290 9 1525 40-49 914 335 266 6 1521 50+ 966 339 283 10 1598 Total 4228 1762 1627 73 7690 If we select a person at random from this sample: a) What is the probability that the person agreed with the statement? P (agree) = 4228/7690 = 0.5498 [marginal probability] b) What is the probability that the person is younger than 50 years old? P(< 50) = 1 P (50+) = 1-1598/7690 = 1-0.2078 = 0.7922 Or P(<50) = P(13-19) + P(20-29) + P(30-39) + P(40-49) = =0.7922 c) What is the probability that the person is younger than 50 and agrees with the statement? P (<50 and Agree) = = 0.4242 [joint probability] d) What is the probability that the person is younger than 50 or agrees with the statement? P(<50 or Agree) = P(<50 ) + P(Agree) P(<50 and Agree) = 0.7922+0.5498 0.4242 =0.9178 e) What is the probability that the person agrees is aged between 20 and 29? P (Agree 20-29) = = 0.5323 P(Agree 20-29) = = = 0.5324 [conditional probability] f) Are response and age independent? Han OPRE504 Page 3 of 5
You can choose any cell to test whether response and age are independent. If they are independent, for example, we expect that P(agree) = P (agree 13-19) and P (agree and 13-19) = P (agree) x P (13-19). 1). Check if P(agree) = P (agree 13-19): P (agree) = 4228/7690 = 0.5498 P (agree 13-19) = 661/1513 = 0.4369 (conditional probability) Since P (agree) P (agree 13-19), response is not independent from age. 2) Alternatively, check if P (13-19 and agree) = P (13-19) x P (agree) P(13-19 and agree) = = 0.0860 (joint probability) P (13-19) = = 0.1967 and P (agree) = 0.5498, P (13-19) x P (agree) = 0.1967 x 0.5498 = 0.1082 Since P (13-19 and agree) P (13-19) x P (agree), response is not independent from age. Question 7.3 [Sharpe 2011, Exercise 43, p.202] In a real estate research, 64% of homes for sale have garages, 21% of homes have swimming pools and 17% have both features. a) What is the probability that a home for sale has a garage, but not a pool? P (Garage and No Pool) = P (Garage) - P (Garage and Pool) = 0.64 0.17 = 0.47 b) If a home for sale has a garage, what s the probability that it has a pool, too? P (Pool Garage) = = = 0.2656 c) Are having a garage and a pool independent events? Explain. Check whether P(Pool Garage ) = P (Pool)? P (Pool Garage) = 0.2656; P (Pool) = 0.21. Since P(Pool Garage ) P (Pool), having a garage and a pool are not independent events. Or check whether P (Pool and Garage) = P (Pool) x P (Garage)? P (Pool and Garage) = 0.17 P (Pool) x P (Garage) = 0.21 x 0.64 = 0.1344 Since P (Pool and Garage) P (Pool) x P (Garage), having a garage and a pool are not independent. d) Are having a garage and a pool mutually exclusive? Explain. Check whether P (Garage and Pool) = 0? Since P (Garage and Pool) = 0.17, they are not mutually exclusive events. More exercises: Sharpe 2011, Guided Example, M&M s Modern Market Research, pp.183-185 Sharpe 2011, Chapter 7, Exercises 9, 10, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50. Han OPRE504 Page 4 of 5
Probability Tree Question 7.4 [Sharpe 2011, Exercise 54] Extended warranties. A company that manufactures and sells consumer video cameras sells two versions of their popular hard disk camera, a basic camera for $750 and a deluxe version for $1250. About 75% of customers select the basic camera. Of those, 60% purchase the extended warranty for an additional $200. Of the people who buy the deluxe version, 90% purchase the extended warranty. a) Sketch the probability tree for total purchases. Purchase 0.6 0.45 Basic Camera 0.75 No Purchase 0.4 0.3 Deluxe Camera 0.25 Purchase 0.9 No Purchase 0.1 0.225 0.025 b) What is the percentage of customers who buy an extended warranty? 0.45 + 0.225 = 0.675 c) What is the expected revenue of the company from a camera purchase (includes warranty if applicable)? Basic version: Sales + Warranty = $750x 0.75 + $200 x 0.45 = $652.50 Deluxe version: = $1250 x 0.25 + $200 x 0.225 = $357.50 Total = $1010 d) Given that a customer purchases an extended warranty, what is the probability that he or she bought the deluxe version? P (Deluxe Purchase Warranty) = = 0.333 (Bay s Rule) More Exercises: Sharpe 2011, Chapter 7, Figures 7.2, 7.3 and 7.4, pp.190-192 Sharpe 2011, Chapter 7, Exercises 51, 52, 53, 55, and 56 Han OPRE504 Page 5 of 5