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Chapter 1

Probability

Educators

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Problem 50

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.
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 $\}$
(a) Calculate $P(A), P(C),$ and $P(A \cap C)$ .
(b) Calculate both $P(A | C)$ and $P(C | A)$ and explain in context what each of these probabilities
represents.
(c) If the selected individual does not have type $\mathrm{B}$ blood, what is the probability that he or she is from ethnic group 1$?$

Lucas F.
Numerade Educator

Problem 51

Suppose an individual is randomly selected from the population of all adult males living in the USA. 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, $P(A B)$ or $P(B | A) ?$ Why?

Andrew K.
Numerade Educator

Problem 52

Return to the credit card scenario of Exercise $14,$ where $A=\{$ MasterCard $\}$ $P(A)=.5, P(B)=.4,$ and $P(A \cap B)=.25 .$ Calculate and interpret each of the following probabilities (a Venn diagram might help).
(a) $P(B | A)$
(b) $P\left(B^{\prime} | A\right)$
(c) $P(A | B)$
(d) $P\left(A^{\prime} B\right)$
(e) Given that the selected individual has at least one card, what is the probability that he or she has a Visa card?

Lucas F.
Numerade Educator

Problem 53

Reconsider the system defect situation described in Exercise $28 .$
(a) Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?
(b) Given that the system has a type 1 defect, what is the probability that it has all three types of defects?
(c) Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?
(d) 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?

Andrew K.
Numerade Educator

Problem 54

The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk.
Consider randomly selecting such a coffee purchaser.
(a) What is the probability that the individual purchased a small cup? A cup of decaf coffee?
(b) If we learn that the selected individual purchased a small cup, what now is the probability that s/he chose decaf coffee, and how would you interpret this probability?
(c) 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 from (a)?

Lucas F.
Numerade Educator

Problem 55

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.
(a) What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?
(b) What is the probability that the next shirt sold is a medium print shirt?
(c) What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?
(d) 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?
(e) Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?
(f) Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleved?

Andrew K.
Numerade Educator

Problem 56

One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly chosen from the first box and placed in the second box. Then a ball is randomly selected from the second box and placed in the first box.
(a) What is the probability that a red ball is selected from the first box and a red ball is selected from the second box?
(b) At the conclusion of the selection process, what is the probability that the numbers of red and green balls in the first box are identical to the numbers at the beginning?

Lucas F.
Numerade Educator

Problem 57

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 extra remaining pump is now more likely to fail than was originally the case. That is, $r=P(\# 2$ fails $| \# 1$ fails $)>P(\# 2$ fails $)=q .$ 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?

Andrew K.
Numerade Educator

Problem 58

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 $P(A)=.6$ and $P(B)=.05 .$ What is $P(B | A) ?$

GN
Gennady N.
Numerade Educator

Problem 59

In Exercise $15, A_{i}=\{$ 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.
(a) $P\left(A_{2} A_{1}\right)$
(b) $P\left(A_{2} \cap A_{3} | A_{1}\right)$
(c) $P\left(A_{2} \cup A_{3} | A_{1}\right)$
(d) $P\left(A_{1} \cap A_{3} \cap A_{3} | A_{1} \cup A_{2} \cup A_{3}\right)$

Andrew K.
Numerade Educator

Problem 60

Three plants manufacture hard drives and ship them to a warehouse for distribution. Plant I produces 54$\%$ of the warehouse's inventory with a 4$\%$ defect rate. Plant II produces 35$\%$ of the warehouse's inventory with an 8$\%$ defect rate. Plant III produces the remainder of the warehouse's inventory with a 12$\%$ defect rate.
(a) Draw a tree diagram to represent this information.
(b) A warehouse inspector selects one hard drive at random. What is the probability that it is a defective hard drive and from Plant II?
(c) What is the probability that a randomly selected hard drive is defective?
(d) Suppose a hard drive is defective. What is the probability that it came from Plant II?

Lucas F.
Numerade Educator

Problem 61

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

Andrew K.
Numerade Educator

Problem 62

If $P(B | A)>P(B)$ show that $P\left(B^{\prime} | A\right)<P\left(B^{\prime}\right) .$ [Hint: Add $P\left(B^{\prime} | A\right)$ to both sides of the given inequality and then use the result of the previous exercise.

GN
Gennady N.
Numerade Educator

Problem 63

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

Andrew K.
Numerade Educator

Problem 64

At a certain gas station, 40$\%$ of the customers use regular gas $\left(A_{1}\right), 35 \%$ use mid-grade gas $\left(A_{2}\right)$
and 25$\%$ use premium gas $\left(A_{3}\right) .$ Of those customers using regular gas, only 30$\%$ fill their tanks
(event $B$ ). Of those customers using mid-grade gas, 60$\%$ fill their tanks, whereas of those using
premium, 50$\%$ fill their tanks.
(a) What is the probability that the next customer will request mid-grade gas and fill the tank $\left(A_{2} \cap B\right) ?$
(b) What is the probability that the next customer fills the tank?
(c) If the next customer fills the tank, what is the probability that regular gas is requested? mid-grade gas? Premium gas?

Lucas F.
Numerade Educator

Problem 65

Suppose a single gene controls the color of hamsters: black $(B)$ is dominant and brown $(b)$ is recessive. Hence, a hamster will be black unless its genotype is $b b .$ Two hamsters, each with genotype $B b,$ mate and produce a single offspring. The laws of genetic recombination state that each parent is equally likely to donate either of its two alleles $(B$ or $b),$ so the offspring is equally likely to be any of $B B, B b, b B,$ or $b b$ (the middle two are genetically equivalent).
(a) What is the probability their offspring has black fur?
(b) Given that their offspring has black fur, what is the probability its genotype is $B b ?$

Andrew K.
Numerade Educator

Problem 66

Refer back to the scenario of the previous exercise. In the figure below, the genotypes of both members of Generation I are known, as is the genotype of the male member of Generation II. We know that hamster II2 must be black-colored thanks to her father, but suppose that we don't know her genotype exactly (as indicated by $B$ - in the figure).
(a) What are the possible genotypes of hamster $\mathrm{II} 2,$ and what are the corresponding probabilities?
(b) If we observe that hamster III has a black coat (and hence at least one $B$ gene), what is the probability her genotype is $B b ?$
(c) If we later discover (through DNA testing on poor little hamster III) that her genotype in $B B,$ what is the posterior probability that her mom is also $B B ?$

Lucas F.
Numerade Educator

Problem 67

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.
(a) If it has an emergency locator, what is the probability that it will not be discovered?
(b) If it does not have an emergency locator, what is the probability that it will be discovered?

Andrew K.
Numerade Educator

Problem 68

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$\%$ of 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?
(a) Neither tested component is defective.
(b) 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.]

PS
Peter S.
Numerade Educator

Problem 69

Show that $P(A \cap B | C)=P(A | B \cap C) \cdot P(B | C)$

Andrew K.
Numerade Educator

Problem 70

For customers purchasing a full set of tires at a particular tire store, consider the events
$A=\{$ tires purchased were made in the USA $\}$
$B=\{$ purchaser has tires balanced immediately $\}$
$C=\{$ purchaser requests front-end alignment $\}$
along with $A^{\prime}, B^{\prime},$ and $C^{\prime} .$ Assume the following unconditional and conditional probabilities:
(a) Construct a tree diagram consisting of first-, second-, and third-generation branches and place an event label and appropriate probability next to each branch.
(b) Compute $P(A \cap B \cap C)$ .
(c) Compute $P(B \cap C) .$
(d) Compute $P(C)$ .
(e) Compute $P(A B \cap C),$ the probability of a purchase of US tires given that both balancing and an alignment were requested.

Lucas F.
Numerade Educator

Problem 71

A professional organization (for statisticians, of course) sells term life insurance and major medical insurance. Of those who have just life insurance, 70$\%$ will renew next year, and 80$\%$ of of those with only a major medical policy will renew next. However, 90$\%$ of policyholders
who have both types of policy will renew at least one of them next year. Of the policy holders, 75$\%$ have term life insurance, 45$\%$ have major medical, and 20$\%$ have both.
(a) Calculate the percentage of policyholders that will renew at least one policy next year.
(b) If a randomly selected policy holder does in fact renew next year, what is the probability that he or she has both life and major medical insurance?

Andrew K.
Numerade Educator

Problem 72

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 $(5-66$ pages. Data on recent reviews indicate that 60$\%$ of them are short, 30$\%$ are medium, and the other 10$\%$ are long. Reviews are submitted in either $\mathrm{Word}$ or LaTeX. For short reviews, 80$\%$ are in $\mathrm{Word}$ , whereas 50$\%$ of medium reviews and 30$\%$ of long reviews are in $\mathrm{Word}$ . Suppose a recent review is randomly selected.
(a) What is the probability that the selected review was submitted in Word?
(b) If the selected review was submitted in Word, what are the posterior probabilities of it being short, medium, and long?

Lucas F.
Numerade Educator

Problem 73

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 night visitors buy a unit, and 20$\%$ of those visiting for two nights decide to buy. Suppose a visitor is randomly selected and found to have bought a timeshare. How likely is it that this person made a day visit? A one-night visit? A two-night visit?

Andrew K.
Numerade Educator

Problem 74

Consider the following information about travelers (based partly on a recent Travelocity poll): 40$\%$ check work e-mail, 30$\%$ use a cell phone to stay connected to work, 25$\%$ bring a laptop with them, 23$\%$ both check work e-mail and use a cell phone to stay connected, and 51$\%$ neither check work e-mail nor use a cell phone to stay connected nor bring a laptop. Finally, 88 out of every 100 who bring a laptop check work e-mail, and 70 out of every 100 who use a cell phone to stay connected also bring a laptop.
(a) What is the probability that a randomly selected traveler who checks work e-mail also uses a cell phone to stay connected?
(b) What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected?
(c) If a randomly selected traveler checked work e-mail and brought a laptop, what is the probability that s/he uses a cell phone to stay connected?

Lucas F.
Numerade Educator

Problem 75

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$\%$ chance of correctly identifying someone who is not a future terrorist. Imagine there are 1000 future terrorists in a population of 300 million (roughly the US population). If one of these 300 million people is randomly selected and the system determines him/her to be a future terrorist, what is the probability the system is correct? Does your answer make you uneasy about using the surveillance system? Explain.

Andrew K.
Numerade Educator

Problem 76

At a large university, in the never-ending quest for a satisfactory textbook, the Statistics Department has tried a different text during each of the last three quarters. During the fall quarter, 500 students used the text by Professor Mean; during the winter 300 students used the text by Professor Median; and during the spring tharter, 200 students used the text by Professor Mode. A survey at the end of each quarter showed that 200 students were satisfied with Mean's book, 150 were satisfied with Median's book, and 160 were satisfied with Mode's book. If a student who took statistics during one of these quarters is selected at random and admits to having been satisfied with the text, is the student most likely to have used the book by Mean, Median, or Mode? Who is the least likely author? [Hint: Dree-diagram or use Bayes' theorem. $.$ .

VW
Victoria W.
Numerade Educator

Problem 77

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. 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. $]$

Andrew K.
Numerade Educator

Problem 78

In Exercise $64,$ 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 mid-grade fill-up customers use a credit card.
50$\%$ of all mid-grade 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).
(a) $\{$ mid-grade and fill-up and credit card $\}$
(b) $\{$ premium and non-fill-up and credit card $\}$
(c) $\{$ premium and credit card $\}$
(d) $\{$ fill-up and credit card $\}$
(e) $\{$ credit card $\}$
(f) If the next customer uses a credit card, what is the probability that s/he purchased premium gasoline?

YQ
Yuan Q.
Numerade Educator