Deck 8: Randomness and Probability

Question 1

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 in the last year a passenger redeemed frequent flyer miles to purchase a ticket for domestic or international travel?
b. What is the probability that in the last year a passenger did not redeem frequent flyer miles to purchase a ticket for domestic or international travel?
c. Is redeeming frequent flyer miles to purchase a ticket for domestic and international travel disjoint or mutually exclusive? Explain.

Answer 1

a. 0.86 (0.58 + 0.44 - 0.16) general addition rule
b. 0.14 (1 - 0.86) complement rule
c. No, the intersection of these two events is not zero; a passenger could purchase tickets for both domestic and international travel.

Question 2

Does it appear that choice of instructional method (traditional or online) and division (sales, quality or operations) are independent? Explain.

Answer 2

Since the marginal probability of choosing online training (0.75) does not equal the conditional probability of choosing online training given the employee is in the sales division (0.69), the choice of instructional method is not independent of division.

Question 3

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?

Answer 3

a. 0.42 (175/420)
b. 0.07 (30/420)
c. 0.42 (30/72)

Question 4

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 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 5

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

Question 6

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 7

Consider the following to answer the question(s) below:
A recent survey of local cell phone retailers showed that of all cell phones sold last month, 64% had a camera, 28% had a music player, and 22% had both.
Which of the following statements about cell phones sold last month is true?

A) Having a camera and having a music player are mutually exclusive events.
B) The intersection of having a camera and having a music player is zero.
C) Having a camera and having a music player are independent events.
D) Having a camera and having a music player are disjoint events.
E) Having a camera and having a music player are not mutually exclusive events.

Question 8

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 9

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 10

Consider the following to answer the question(s) below:
A recent survey of local cell phone retailers showed that of all cell phones sold last month, 64% had a camera, 28% had a music player, and 22% had both.
The probability that a cell phone sold last month did not have either a camera or a music player is

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

Question 11

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 12

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 13

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.

Question 14

A recent survey of local cell phone retailers showed that of all cell phones sold last month, 64% had a camera, 28% had a music player, and 22% had both.
a. What is the probability that a cell phone sold last month had a camera or a music player?
b. What is the probability that a cell phone sold last month did not have either a camera or a music player?
c. Are the events of a cell phone having a camera and a music player disjoint or mutually exclusive? Explain.

Question 15

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.

Question 16

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?

Question 17

Consider the following to answer the question(s) below:
A recent survey of local cell phone retailers showed that of all cell phones sold last month, 64% had a camera, 28% had a music player, and 22% had both.
The probability that a cell phone sold last month had a camera or a music player is

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

Question 18

One explanation put forth for the scarcity of women CEOs in the high tech industry is that there are a lack of mentoring opportunities for women. A recent survey of CEOs in that industry found that 80% were men. Moreover, 75% in the survey had been mentored, but only 15% were mentored women.
a. Construct the contingency table.
b. Are gender and mentoring independent? Explain.

Question 19

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 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:

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 22

Consider the following to answer the question(s) below:

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:
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 24

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.

Question 25

Consider the following to answer the question(s) below:

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 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.
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 27

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 28

Consider the following to answer the question(s) below:

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 29

Consider the following to answer the question(s) below:

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 30

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 31

Consider the following to answer the question(s) below:

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 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