EXERCISES Section 2.4 (45-69)

45

The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations.

	O	A	B	AB
1	.082	.106	.008	.004
2	.135	.141	.018	.006
3	.215	.200	.065	.020

Suppose that an individual is randomly selected from the population, and define events by $A=\{$ type A selected $\}$, $B=\{$ type B selected $\}$, and $C=\{$ ethnic group 3 selected $\}$ .

1. Calculate $P\left(A\right)$, $P\left(C\right)$, and $P\left(A\cap C\right)$.
2. Calculate both $P\left(A\mid C\right)$ and $P\left(C\mid A\right)$ , and explain in context what each of these probabilities represents.
3. If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1 ?

46

Suppose an individual is randomly selected from the population of all adult males living in the United States. Let $A$ be the event that the selected individual is over $6\mathrm{ft}$ in height, and let $B$ be the event that the selected individual is a professional basketball player. Which do you think is larger, $P\left(A\mid B\right)$ or $P\left(B\mid A\right)$ ? Why?

47

Return to the credit card scenario of Exercise 12 (Section 2.2), and let $C$ be the event that the selected student has an American Express card. In addition to $P\left(A\right)=.6$ , $P\left(B\right)=.4$ , and $P\left(A\cap \mathrm{B}\right)=.3$ , suppose that $P\left(C\right)=.2$ , $P\left(A\cap C\right)=.15,P\left(B\cap C\right)=.1$ , and $P\left(A\cap B\cap C\right)=.08$ .

1. What is the probability that the selected student has at least one of the three types of cards?

2. What is the probability that the selected student has both a Visa card and a MasterCard but not an American Express card?

3. Calculate and interpret $P\left(B\mid A\right)$ and also $P\left(A\mid B\right)$ .

4. If we learn that the selected student has an American Express card, what is the probability that she or he also has both a Visa card and a MasterCard?

5. Given that the selected student has an American Express card, what is the probability that she or he has at least one of the other two types of cards?

48

11. Reconsider the system defect situation described in Exercise 26 (Section 2.2).

12. Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?

13. Given that the system has a type 1 defect, what is the probability that it has all three types of defects?

14. Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?

15. Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?

49

21. The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.

	Small	Medium	Large
Regular	14%	20%	26%
Decaf	20%	10%	10%

Consider randomly selecting such a coffee purchaser.

1. What is the probability that the individual purchased a small cup? A cup of decaf coffee?

2. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability?

3. If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

50

A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.

Short-sleeved

Size	Pl	Pr	St
S	.04	.02	.05
M	.08	.07	.12
L	.03	.07	.08

Long-sleeved

Size	Pl	Pr	St
S	.03	.02	.03
M	.10	.05	.07
L	.04	.02	.08

1. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?

2. What is the probability that the next shirt sold is a medium print shirt?

3. What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?

4. What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print?

5. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?

6. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?

51

According to a July 31, 2013, posting on cnn.com subsequent to the death of a child who bit into a peanut, a 2010 study in the journal Pediatrics found that $8\%$ of children younger than 18 in the United States have at least one food allergy. Among those with food allergies, about $39\%$ had a history of severe reaction.

If a child younger than 18 is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction?

It was also reported that $30\%$ of those with an allergy in fact are allergic to multiple foods. If a child younger than 18 is randomly selected, what is the probability that he or she is allergic to multiple foods?

52

A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, $r=P\left(\#2\text{fails}\mid \#1\text{fails}\right)>P\left(\#2\text{fails}\right)=q$ . If at least one pump fails by the end of the pump design life in $7\%$ of all systems and both pumps fail during that period in only $1\%$ , what is the probability that pump #1 will fail during the pump design life?

53

A certain shop repairs both audio and video components. Let $A$ denote the event that the next component brought in for repair is an audio component, and let $B$ be the event that the next component is a compact disc player (so the event $B$ is contained in $A$ ). Suppose that $P\left(A\right)=.6$ and $P\left(B\right)=.05$ . What is $P\left(B\mid A\right)$ ?

54

In Exercise $13,A_i=\{$ awarded project $i\}$ , for $i=1,2,3$ . Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one.

• $P\left(A_2\mid A_1\right)$
• $P\left(A_2\mid A_1\right)$
• $P\left(A_2\cap A_3\mid A_1\right)$
• $P\left(A_2\cap A_3\mid A_1\right)$
• $P\left(A_2\cup A_3\mid A_1\right)$
• $P\left(A_2\cup A_3\mid A_1\right)$
• $P\left(A_1\cap A_2\cap A_3\mid A_1\cup A_2\cup A_3\right)$

55

Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that $16\%$ of all ticks in a certain location carry Lyme disease, $10\%$ carry HGE, and $10\%$ of the ticks that carry at least one of these diseases in fact carry both of them. If a randomly selected tick is found to have carried HGE, what is the probability that the selected tick is also a carrier of Lyme disease?

56

For any events $A$ and $B$ with $P\left(B\right)>0$ , show that $P\left(A\mid B\right)+P\left(A^{\prime }\mid B\right)=1.$

57

If $P\left(B\mid A\right)>P\left(B\right)$ , show that $P\left(B^{\prime }\mid A\right)<P\left(B^{\prime }\right)$ .

• Hint: Add $P\left(B^{\prime }\mid A\right)$ to both sides of the given inequality and then use the result of Exercise 56.

58

Show that for any three events $A,B$ , and $C$ with $P\left(C\right)>0$ , $P\left(A\cup B\mid C\right)=P\left(A\mid C\right)+P\left(B\mid C\right)-P\left(A\cap B\mid C\right).$

59

At a certain gas station, $40\%$ of the customers use regular gas $\left(A_1\right),35\%$ use plus gas $\left(A_2\right)$ , and $25\%$ use premium $\left(A_3\right)$ . Of those customers using regular gas, only $30\%$ fill their tanks (event $B$ ). Of those customers using plus, $60\%$ fill their tanks, whereas of those using premium, $50\%$ fill their tanks.

What is the probability that the next customer will request plus gas and fill the tank $\left(A_2\cap B\right)$ ?

What is the probability that the next customer fills the tank?

If the next customer fills the tank, what is the probability that regular gas is requested? Plus? Premium?

60

Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, $60\%$ have an emergency locator, whereas $90\%$ of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared.

If it has an emergency locator, what is the probability that it will not be discovered?

If it does not have an emergency locator, what is the probability that it will be discovered?

61

Components of a certain type are shipped to a supplier in batches of ten. Suppose that $50\%$ of all such batches contain no defective components, $30\%$ contain one defective component, and $20\%$ contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0,1 , and 2 defective components being in the batch under each of the following conditions?

Neither tested component is defective.

One of the two tested components is defective.

• Hint: Draw a tree diagram with three first-generation branches for the three different types of batches.

62

Blue Cab operates $15\%$ of the taxis in a certain city, and Green Cab operates the other $85\%$ . After a nighttime hit-and-run accident involving a taxi, an eyewitness said the vehicle was blue. Suppose, though, that under night vision conditions, only $80\%$ of individuals can correctly distinguish between a blue and a green vehicle. What is the (posterior) probability that the taxi at fault was blue? In answering, be sure to indicate which probability rules you are using.

• Hint: A tree diagram might help.
• Note: This is based on an actual incident.

63

For customers purchasing a refrigerator at a certain appliance store, let $A$ be the event that the refrigerator was manufactured in the U.S., $B$ be the event that the refrigerator had an icemaker, and $C$ be the event that the customer purchased an extended warranty. Relevant probabilities are

$$P\left(A\right)=.75P\left(B\mid A\right)=.9P\left(B\mid A^{\prime }\right)=.8$$ $$P\left(C\mid A\cap B\right)=.8P\left(C\mid A\cap B^{\prime }\right)=.6$$ $$P\left(C\mid A^{\prime }\cap B\right)=.7P\left(C\mid A^{\prime }\cap B^{\prime }\right)=.3$$

1. Construct a tree diagram consisting of first-, second-, and third-generation branches, and place an event label and appropriate probability next to each branch.

2. Compute $P\left(A\cap B\cap C\right)$ .

3. Compute $P\left(B\cap C\right)$ .

4. Compute $P\left(C\right)$ .

5. Compute $P\left(A\mid B\cap C\right)$ , the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased.

64

The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1-2 pages), medium (3-4 pages), or long (5-6 pages). Data on recent reviews indicates that $60\%$ of them are short, $30\%$ are medium, and the other $10\%$ are long. Reviews are submitted in either Word or LaTeX. For short reviews, $80\%$ are in Word, whereas $50\%$ of medium reviews are in Word and $30\%$ of long reviews are in Word. Suppose a recent review is randomly selected.

13. What is the probability that the selected review was submitted in Word format?

14. If the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long?

65

A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that $20\%$ of all potential purchasers select a day visit, $50\%$ choose a one-night visit, and $30\%$ opt for a two-night visit. In addition, $10\%$ of day visitors ultimately make a purchase, $30\%$ of one-night visitors buy a unit, and $20\%$ of those visiting for two nights decide to buy. Suppose a visitor is randomly selected and is found to have made a purchase. How likely is it that this person made a day visit? A one-night visit? A two-night visit?

66

Consider the following information about travelers on vacation (based partly on a recent Travelocity poll): $40\%$ check work email, $30\%$ use a cell phone to stay connected to work, $25\%$ bring a laptop with them, $23\%$ both check work email and use a cell phone to stay connected, and $51\%$ neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, 88 out of every 100 who bring a laptop also check work email, and 70 out of every 100 who use a cell phone to stay connected also bring a laptop.

21. What is the probability that a randomly selected traveler who checks work email also uses a cell phone to stay connected?

22. What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected?

23. If the randomly selected traveler checked work email and brought a laptop, what is the probability that he/ she uses a cell phone to stay connected?

67

There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Suppose a particular surveillance system has a $99\%$ chance of correctly identifying a future terrorist and a $99.9\%$ chance of correctly identifying someone who is not a future terrorist. If there are 1000 future terrorists in a population of 300 million, and one of these 300 million is randomly selected, scrutinized by the system, and identified as a future terrorist, what is the probability that he/she actually is a future terrorist? Does the value of this probability make you uneasy about using the surveillance system? Explain.

68

A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; $50\%$ of the time she travels on airline #1, $30\%$ of the time on airline #2, and the remaining $20\%$ of the time on airline #3 . For airline #1, flights are late into D.C. $30\%$ of the time and late into L.A. $10\%$ of the time. For airline #2, these percentages are $25\%$ and $20\%$ , whereas for airline #3 the percentages are $40\%$ and $25\%$ . If we learn that on a particular trip she arrived late at exactly one of the two destinations, what are the posterior probabilities of having flown on airlines $\#1,\#2$ , and $\#3$ ? Assume that the chance of a late arrival in L.A. is unaffected by what happens on the flight to D.C.

• Hint: From the tip of each first-generation branch on a tree diagram, draw three second-generation branches labeled, respectively, 0 late, 1 late, and 2 late.

69

In Exercise 59, consider the following additional information on credit card usage:

$70\%$ of all regular fill-up customers use a credit card.

$50\%$ of all regular non-fill-up customers use a credit card.

$60\%$ of all plus fill-up customers use a credit card.

$50\%$ of all plus non-fill-up customers use a credit card.

$50\%$ of all premium fill-up customers use a credit card.

$40\%$ of all premium non-fill-up customers use a credit card.

Compute the probability of each of the following events for the next customer to arrive (a tree diagram might help).

1. {plus and fill-up and credit card }

2. {premium and non-fill-up and credit card }

3. {premium and credit card }

4. {fill-up and credit card }

5. {credit card }

6. If the next customer uses a credit card, what is the probability that premium was requested?