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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

\(\begin{aligned}{\rm{P(A) = }}{\rm{.75}}\;\;\;{\rm{P(B}}\mid {\rm{A) = }}{\rm{.9}}\;\;\;{\rm{P}}\left( {{\rm{B}}\mid {{\rm{A}}^{\rm{\cent}}}} \right){\rm{ = }}{\rm{.8}}\\{\rm{P(C}}\mid {\rm{A\c{C}B) = }}{\rm{.8}}\;\;\;{\rm{P}}\left( {{\rm{C}}\mid {\rm{A\c{C}}}{{\rm{B}}^{\rm{\cent}}}} \right){\rm{ = }}{\rm{.6}}\\{\rm{P}}\left( {{\rm{C}}\mid {{\rm{A}}^{\rm{\cent}}}{\rm{\c{C}B}}} \right){\rm{ = }}{\rm{.7}}\;\;\;{\rm{P}}\left( {{\rm{C}}\mid {{\rm{A}}^{\rm{\cent}}}{\rm{\c{C}}}{{\rm{B}}^{\rm{\cent}}}} \right){\rm{ = }}{\rm{.3}}\end{aligned}\)

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 \({\rm{P(A\c{C}B\c{C}C)}}\). c. Compute \({\rm{P(B\c{C}C)}}\). d. Compute \({\rm{P(C)}}\). e. Compute \({\rm{P(A}}\mid {\rm{B\c{C}C)}}\), the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased

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

A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that \({\rm{20\% }}\) of all potential purchasers select a day visit, \({\rm{50\% }}\) choose a one-night visit, and \({\rm{30\% }}\)opt for a two-night visit. In addition, \({\rm{10\% }}\) of day visitors ultimately make a purchase, \({\rm{30\% }}\) of one-night visitors buy a unit, and \({\rm{20\% }}\) 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?

Consider the following information about travellers on vacation (based partly on a recent Travelocity poll): \({\rm{40\% }}\) check work email, \({\rm{30\% }}\) use a cell phone to stay connected to work, \({\rm{25\% }}\) bring a laptop with them, \({\rm{23\% }}\) both check work email and use a cell phone to stay connected, and \({\rm{51\% }}\) neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, \({\rm{88}}\) out of every \({\rm{100}}\) who bring a laptop also check work email, and \({\rm{70}}\) out of every \({\rm{100}}\) who use a cell phone to stay connected also bring a laptop. a. What is the probability that a randomly selected traveller who checks work email 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 the randomly selected traveller checked work email and brought a laptop, what is the probability that he/ she uses a cell phone to stay connected?

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 \({\rm{99\% }}\) chance of correctly identifying a future terrorist and a \({\rm{99}}{\rm{.9\% }}\)chance of correctly identifying someone who is not a future terrorist. If there are \({\rm{1000}}\) future terrorists in a population of \({\rm{300}}\) million, and one of these \({\rm{300}}\) 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.

A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; \({\rm{50\% }}\)of the time she travels on airline\({\rm{\# 1}}\), \({\rm{30\% }}\) of the time on airline \({\rm{\# 2}}\), and the remaining \({\rm{20\% }}\) of the time on airline #3. For airline \({\rm{\# 1}}\), flights are late into D.C. \({\rm{30\% }}\) of the time and late into L.A. \({\rm{10\% }}\) of the time. For airline\({\rm{\# 3}}\), these percentages are \({\rm{25\% }}\) and \({\rm{20\% }}\), whereas for airline #3 the percentages are \({\rm{40\% }}\) and \({\rm{25\% }}\). 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 \({\rm{\# 1}}\), \({\rm{\# 2}}\), and \({\rm{\# 3}}\)? 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, \({\rm{2}}\) late, \({\rm{2}}\) late, and \({\rm{2}}\) late.)

In Exercise\({\rm{59}}\), consider the following additional information on credit card usage: \({\rm{70\% }}\)of all regular fill-up customers use a credit card. \({\rm{50\% }}\) of all regular non-fill-up customers use a credit card. \({\rm{60\% }}\) of all plus fill-up customers use a credit card. \({\rm{50\% }}\) of all plus non-fill-up customers use a credit card. \({\rm{50\% }}\) of all premium fill-up customers use a credit card. \({\rm{40\% }}\) 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. {plus 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 premium was requested?

A college library has five copies of a certain text onreserve. Two copies (1 and 2) are first printings, and the other three (3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 5, and another is 213.

a. List the outcomes in S.

b. Let Adenote the event that exactly one book must be examined. What outcomes are in A?

c. Let Bbe the event that book 5 is the one selected. What outcomes are in B?

d. Let Cbe the event that book 1 is not examined. What outcomes are in C?

Reconsider the credit card scenario of, and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold.

An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A) \({\rm{5 }}{\rm{.4}}\) and P(B) \({\rm{5 }}{\rm{.7}}\).

a. If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning.

b. What is the probability that at least one of the two projects will be successful?

c. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?