Deck 8: Randomness and Probability

Question 1

As an incentive to get new customers, the local branch of a bank launched "bouncing for bucks." During this week-long event, any customer opening a new chequing account with the bank would have the opportunity to throw a bouncy rubber ball into a large box divided into squares. Each square was labeled with a dollar amount that would be deposited into his/her new chequing account. The way the box was labeled is shown below.
a. What is the probability that a customer gets $20 or more?
b. What is the probability that a customer gets less than $20?
c. What is the probability that a customer gets $20 or $30?
d. What is the probability that two customers in a row get $50?

Answer 1

Solution: 1/10 × 1/10 = 0.01 by the Multiplication Rule)

Question 2

Does it appear that the decision to purchase an extended warranty and the type of electronics purchased (digital camera or laptop computer) are independent? Explain.

Answer 2

Since the marginal probability of purchasing an extended warranty (0.42) does equal the conditional probability of purchasing an extended warranty given the consumer purchased a digital camera (0.42), the decision to purchase an extended warranty is independent of type of electronics (digital camera or laptop computer) purchased.

Question 3

Consider the following to answer the question(s) below:
The option to buy extended warranties is commonplace with most electronics purchases. Does the type of purchase affect a consumer's willingness to pay extra for an extended warranty? Data for 420 consumers who purchased digital cameras and laptop computers from a leading electronics retailer are summarized in the table.
The probability that a consumer purchases a digital camera and an extended warranty is

A) 0.07.
B) 0.42.
C) 0.58.
D) 0.17.
E) 0.83.

Answer 3

0.07.

Question 4

Consider the following to answer the question(s) below:
During its grand opening week, Stickler's bicycle shop offers a "wheel of discount savings." After customers select the items they wish to purchase, they spin the wheel to determine the discount they will receive. The wheel is divided into 12 slices. Six slices are red and award a 10% discount, three slices are white and award a 20% discount, and two slices are blue and award a 40% discount. The remaining slice is gold and awards a 100% discount!
The probability that a customer gets at least a 40% discount is

A) 3/12.
B) 2/12.
C) 0.0625.
D) 9/12.
E) 10/12.

Question 5

Does it appear that choice of instructional method (traditional or online) and division (Sales, Quality or Operations) are independent? Explain.

Question 6

During its grand opening week, Stickler's bicycle shop offers a "wheel of discount savings". After customers select the items they wish to purchase, they spin the wheel to determine the discount they will receive. The wheel is divided into 12 slices. Six slices are red and award a 10% discount, three slices are white and award a 20% discount, and two slices are blue and award a 40% discount. The remaining slice is gold and awards a 100% discount! a. What is the probability that a customer gets at least a 40% discount?
b. What is the probability that a customer does not get at least a 40% discount?
c. What is the probability that a customer gets a 10% or 20% discount?
d. What is the probability that two customers in a row get a 20% discount?

Question 7

Consider the following to answer the question(s) below:
During its grand opening week, Stickler's bicycle shop offers a "wheel of discount savings." After customers select the items they wish to purchase, they spin the wheel to determine the discount they will receive. The wheel is divided into 12 slices. Six slices are red and award a 10% discount, three slices are white and award a 20% discount, and two slices are blue and award a 40% discount. The remaining slice is gold and awards a 100% discount!
The probability that a customer gets a 10% or 20% discount is

A) 3/12.
B) 2/12.
C) 0.0625.
D) 9/12.
E) 10/12.

Question 8

It has been reported that men are more likely than women to participate in online auctions. A recent study found that 52% of Internet shoppers are women and that 35% of Internet shoppers have participated in online auctions. Moreover, 25% of online shoppers were men and had participated in online auctions.
a. Construct the contingency table.
b. Are gender and participation in online auctions independent? Explain.
Hint: Compare two events M = {a randomly selected shopper is a man} and A = {a randomly selected shopper has participated in online auctions}

Question 9

One explanation put forth for the scarcity of women CEOs in the high tech industry is that there is a lack of mentoring opportunities for women. A recent survey of CEOs in this industry found that 80% were men. Moreover, 75% of those who were surveyed had been mentored, but only 15% were mentored women.
a. Construct the contingency table.
b. Are two events W = {a randomly selected CEO is a woman} and N = {a randomly selected CEO has not been mentored} independent? Explain.

Question 10

A small manufacturing company recently instituted Six Sigma training for its employees. Two methods of training were offered: online and traditional classroom. Management was interested in whether the division (Sales, Quality, Operations) in which employees worked affected their choice of method. Below is a table summarizing the data.
a. What is the probability that an employee chose online training?
b. What is the probability that an employee is in the Quality division and chose online training?
c. What is the probability that an employee chose online training given that he/she is in the Sales division?

Question 11

As you enter the bank, you watch four persons in front of you all win $50. The local branch manager tells you how lucky you are to be throwing the ball while it is on a hot streak but the friend with you says that you're unlucky because the streak can't continue. Comment on their statements.

Question 12

The option to buy extended warranties is commonplace with most electronics purchases. Does the type of purchase affect a consumer's willingness to pay extra for an extended warranty? Data for 420 consumers who purchased digital cameras and laptop computers from a leading electronics retailer are summarized in the table. a. What is the probability that a consumer purchases an extended warranty?
b. What is the probability that a consumer purchases a digital camera and an extended warranty?
c. What is the probability that a consumer purchases an extended warranty given that he/she has purchased a digital camera?

Question 13

Consider the following to answer the question(s) below:
SamaraCom is a communications and media company. It operates in the field of wireless communications, cable TV, telephone, and Internet connectivity. A recent survey showed that 64% of the SamaraCom clients are subscribed to Internet service, 28% are TV customers, and 22% are subscribed to a bundle (both, Internet and TV services).
What is the probability that a client is subscribed to TV service or Internet?

A) 0.22
B) 0.70
C) 0.92
D) 0.30
E) 0.08

Question 14

Consider the following to answer the question(s) below:
SamaraCom is a communications and media company. It operates in the field of wireless communications, cable TV, telephone, and Internet connectivity. A recent survey showed that 64% of the SamaraCom clients are subscribed to Internet service, 28% are TV customers, and 22% are subscribed to a bundle (both, Internet and TV services).
The probability that a SamaraCom client is subscribed to neither TV service nor Internet is

A) 0.22.
B) 0.70.
C) 0.92.
D) 0.30.
E) 0.08.

Question 15

Consider the following to answer the question(s) below:
During its grand opening week, Stickler's bicycle shop offers a "wheel of discount savings." After customers select the items they wish to purchase, they spin the wheel to determine the discount they will receive. The wheel is divided into 12 slices. Six slices are red and award a 10% discount, three slices are white and award a 20% discount, and two slices are blue and award a 40% discount. The remaining slice is gold and awards a 100% discount!
The probability that two customers in a row get a 20% discount is

A) 3/12.
B) 2/12.
C) 0.0625.
D) 9/12.
E) 10/12.

Question 16

Suppose you visit Stickler's bicycle shop in the hopes of getting a 100% discount on your purchases. As you wait your turn in line, there are three gold winners in a row. The two customers in line behind you begin to discuss what's happened. One believes that the streak of three gold winners has killed anyone else's chances of getting a 100% discount, while the other says just the opposite… that the wheel's "hot streak" increases their chances of getting a 100% discount. Comment on these opinions.

Question 17

SamaraCom is a communications and media company. It operates in the field of wireless communications, cable TV, telephone, and Internet connectivity. A recent survey showed that 64% of the SamaraCom clients are subscribed to Internet service, 28% are TV customers, and 22% are subscribed to a bundle (both, Internet and TV services).
a. What is the probability that a client is subscribed to TV service or Internet?
b. What is the probability that a SamaraCom client is subscribed to neither TV service nor Internet?
c. Are the events of T = {TV service subscription} and I = {Internet service subscription} disjoint (or mutually exclusive)? Explain.

Question 18

Consider the following to answer the question(s) below:
SamaraCom is a communications and media company. It operates in the field of wireless communications, cable TV, telephone, and Internet connectivity. A recent survey showed that 64% of the SamaraCom clients are subscribed to Internet service, 28% are TV customers, and 22% are subscribed to a bundle (both, Internet and TV services).
Which of the following statements about TV service subscription is true?

A) TV service subscription and having Internet from SamaraCom are disjoint events.
B) The probability of intersection of two events T = {TV service subscription} and I = {Internet service subscription} is equal to zero.
C) T = {TV service subscription} and I = {Internet service subscription} are independent events.
D) TV service subscription and having Internet from SamaraCom are mutually exclusive events.
E) TV service subscription and having Internet from SamaraCom are not mutually exclusive events.

Question 19

A major airline keeps track of data on how their passengers redeem frequent flyer miles. They found that in the last year 58% of passengers redeemed them to purchase tickets for domestic travel, 44% redeemed them to purchase tickets for international travel, and that 16% redeemed them to purchase tickets for both domestic and international travel.
a. What is the probability that a passenger redeemed frequent flyer miles to purchase a ticket for domestic or international travel?
b. What is the probability that a passenger did not redeem frequent flyer miles to purchase a ticket for domestic or international travel?
c. Are two events D ={redeeming miles to purchase a ticket for domestic travel} and I ={redeeming miles to purchase a ticket for international travel} disjoint (or mutually exclusive)? Explain.

Question 20

Consider the following to answer the question(s) below:
The option to buy extended warranties is commonplace with most electronics purchases. Does the type of purchase affect a consumer's willingness to pay extra for an extended warranty? Data for 420 consumers who purchased digital cameras and laptop computers from a leading electronics retailer are summarized in the table.
The probability that a consumer does not purchase an extended warranty is

A) 0.07.
B) 0.42.
C) 0.58.
D) 0.17.
E) 0.83.

Question 21

Consider the following to answer the question(s) below:
Harrison Water Sports has three retail outlets: Vancouver, Victoria, and Nanaimo. The Vancouver store does 50 percent of the total sales in a year, while the Victoria store does 35 percent of the total sales, and Nanaimo the remainder. Further analysis indicates that of the sales in Vancouver, 20 percent are in boat accessories. The percentage of boat accessories at the Victoria store is 30 and the percentage at the Nanaimo store is 25.
If a sale is recorded as a boat accessory, the probability that the sale was made at the Victoria store is

A) 0.433.
B) 0.35.
C) 0.30.
D) 0.2425.
E) 0.105.

Question 22

Which of the following statements is true?

A) The decision to purchase stocks and the age of the client are not independent.
B) The decision to purchase stocks and the age of the client are independent.
C) The decision to purchase stocks and the age of the client are mutually exclusive.
D) The decision to purchase stocks and the age of the client are unrelated.
E) The decision to purchase stocks and the age of the client are disjoint events.

Question 23

Consider the following to answer the question(s) below:
The following table represents information about investments of 200 randomly selected residents of Etobicoke.

The probability that a client is over age 50 and is holding bonds is

A) 0.125.
B) 0.15.
C) 0.225.
D) 0.010.
E) 0.175.

Question 24

Consider the following to answer the question(s) below:
The following table represents information about investments of 200 randomly selected residents of Etobicoke.

The probability that a client is over age 50 and is not holding bonds is

A) 0.40.
B) 0.225.
C) 0.525.
D) 0.175.
E) 0.325.

Question 25

Consider the following to answer the question(s) below:
The following table represents information about investments of 200 randomly selected residents of Etobicoke.

The probability that a client is holding mutual funds given that he/she is age 50 or younger is

A) 0.316.
B) 0.461.
C) 0.857.
D) 0.15.
E) 0.475.

Question 26

Consider the following to answer the question(s) below:
Harrison Water Sports has three retail outlets: Vancouver, Victoria, and Nanaimo. The Vancouver store does 50 percent of the total sales in a year, while the Victoria store does 35 percent of the total sales, and Nanaimo the remainder. Further analysis indicates that of the sales in Vancouver, 20 percent are in boat accessories. The percentage of boat accessories at the Victoria store is 30 and the percentage at the Nanaimo store is 25.
If a sale is recorded as a boat accessory, the probability that the sale was made at the Nanaimo store is

A) 0.155.
B) 0.375.
C) 0.2245.
D) 0.25.
E) 0.15.

Question 27

Consider the following to answer the question(s) below:
The option to buy extended warranties is commonplace with most electronics purchases. Does the type of purchase affect a consumer's willingness to pay extra for an extended warranty? Data for 420 consumers who purchased digital cameras and laptop computers from a leading electronics retailer are summarized in the table.
The probability that a consumer purchases an extended warranty given that he/she has purchased a digital camera is

A) 0.07.
B) 0.42.
C) 0.58.
D) 0.17.
E) 0.83.

Question 28

Of the last 100 customers entering a computer shop, 25 have purchased a computer. If the purchases of customers are independent of one another, the probability that the next customer will purchase a computer is

A) 0.50.
B) 1.00.
C) 0.25.
D) 0.0625.
E) 0.75.

Question 29

Consider the following to answer the question(s) below:
The following table represents information about investments of 200 randomly selected residents of Etobicoke.

The probability that a client only holds bonds is

A) 0.225.
B) 0.45.
C) 0.475.
D) 0.125.
E) 0.10.

Question 30

Consider the following to answer the question(s) below:
Harrison Water Sports has three retail outlets: Vancouver, Victoria, and Nanaimo. The Vancouver store does 50 percent of the total sales in a year, while the Victoria store does 35 percent of the total sales, and Nanaimo the remainder. Further analysis indicates that of the sales in Vancouver, 20 percent are in boat accessories. The percentage of boat accessories at the Victoria store is 30 and the percentage at the Nanaimo store is 25.
Overall, the probability that a sale by Harrison Water Sports will be for a boat accessory is

A) 0.2425.
B) 0.105.
C) 0.10.
D) 0.375.
E) 0.30.

Question 31

When a customer enters a store there are three outcomes that can occur: buy nothing, buy a small amount, or buy a large amount. It is the case that if a customer buys a large amount, he or she cannot also buy a small amount or buy nothing. We say that these events are

A) disjoint.
B) independent.
C) all inclusive.
D) related.
E) unrelated.

Question 32

Consider the following to answer the question(s) below:
The option to buy extended warranties is commonplace with most electronics purchases. Does the type of purchase affect a consumer's willingness to pay extra for an extended warranty? Data for 420 consumers who purchased digital cameras and laptop computers from a leading electronics retailer are summarized in the table.
Which of the following statements is true?

A) The decision to purchase an extended warranty and the type of electronics (digital camera or laptop computer) purchased are independent.
B) The decision to purchase an extended warranty and the type of electronics (digital camera or laptop computer) purchased are mutually exclusive.
C) The decision to purchase an extended warranty and the type of electronics (digital camera or laptop computer) purchased are disjoint events.
D) The decision to purchase an extended warranty and the type of electronics (digital camera or laptop computer) purchased are not independent.
E) The decision to purchase an extended warranty and the type of electronics (digital camera or laptop computer) purchased are related.