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Chapter 2: Q68E (page 85)

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.)

Short Answer

Expert verified

The probabilities are

\(\begin{array}{l}P\left( {{A_1}\mid 1} \right) = 0.466\\P\left( {{A_2}\mid 1} \right) = 0.288\\P\left( {{A_3}\mid 1} \right) = 0.247\end{array}\)

Step by step solution

01

Definition

The updated chance of an event occurring after additional information is taken into account is known as posterior probability.

02

Finding probabilities

The solution is

The probability of being late at exactly one city is obtained by the following calculation

\({\rm{0}}{\rm{.5*0}}{\rm{.3*0}}{\rm{.9 + 0}}{\rm{.5*0}}{\rm{.7*0}}{\rm{.1 + 0}}{\rm{.3*0}}{\rm{.25*0}}{\rm{.8 + 0}}{\rm{.3*0}}{\rm{.75*0}}{\rm{.2 + 0}}{\rm{.2*0}}{\rm{.4*0}}{\rm{.75 + 0}}{\rm{.2*0}}{\rm{.6*0}}{\rm{.25}}\)

\({\rm{ = 0}}{\rm{.365}}\)

The probability she uses \({\rm{1}}\) is

\(\frac{{{\rm{0}}{\rm{.5*0}}{\rm{.3*0}}{\rm{.9 + 0}}{\rm{.5*0}}{\rm{.7*0}}{\rm{.1}}}}{{{\rm{0}}{\rm{.365}}}}\)

The probability she uses \({\rm{2}}\) is

\(\begin{array}{l}\frac{{{\rm{0}}{\rm{.3*0}}{\rm{.25*0}}{\rm{.8 + 0}}{\rm{.3*0}}{\rm{.75*0}}{\rm{.2}}}}{{{\rm{0}}{\rm{.365}}}}\\{\rm{ = 0}}{\rm{.288}}\end{array}\)

The probability she uses \({\rm{3}}\) is

\(\begin{array}{l}\frac{{{\rm{0}}{\rm{.2*0}}{\rm{.4*0}}{\rm{.75 + 0}}{\rm{.2*0}}{\rm{.6*0}}{\rm{.25}}}}{{{\rm{0}}{\rm{.365}}}}\\{\rm{ = 0}}{\rm{.247}}\end{array}\)

Therefore, the probabilities are

\(\begin{array}{l}P\left( {{A_1}\mid 1} \right) = 0.466\\P\left( {{A_2}\mid 1} \right) = 0.288\\P\left( {{A_3}\mid 1} \right) = 0.247\end{array}\)

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Most popular questions from this chapter

The computers of six faculty members in a certain department are to be replaced. Two of the faculty members have selected laptop machines and the other four have chosen desktop machines. Suppose that only two of the setups can be done on a particular day, and the two computers to be set up are randomly selected from the six (implying 15 equally likely outcomes; if the computers are numbered1, 2,鈥�, 6, then one outcome consists of computers 1 and2, another consists of computers 1 and 3, and so on).

a. What is the probability that both selected setups are for laptop computers?

b. What is the probability that both selected setups are desktop machines?

c. What is the probability that at least one selected setup is for a desktop computer?

d. What is the probability that at least one computer of each type is chosen for setup?

Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has \({\rm{25}}\) failed keyboards, \({\rm{6}}\) of which have electrical defects and 19 of which have mechanical defects.

a. How many ways are there to randomly select \({\rm{5}}\) of these keyboards for a thorough inspection (without regard to order)?

b. In how many ways can a sample of\({\rm{5}}\) keyboards be selected so that exactly two have an electrical defect?

c. If a sample of \({\rm{5}}\) keyboards is randomly selected, what is the probability that at least \({\rm{4}}\) of these will have a mechanical defect

At a certain gas station, \({\rm{40\% }}\)of the customers use regular gas \(\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{,35\% }}\) use plus gas\(\left( {{{\rm{A}}_{\rm{2}}}} \right)\), and \({\rm{25\% }}\) use premium\(\left( {{{\rm{A}}_{\rm{3}}}} \right)\). Of those customers using regular gas, only \({\rm{30\% }}\) fill their tanks (event \({\rm{B}}\) ). Of those customers using plus, \({\rm{60\% }}\)fill their tanks, whereas of those using premium, \({\rm{50\% }}\)fill their tanks.

a. What is the probability that the next customer will request plus gas and fill the tank\(\left( {{{\rm{A}}_{\rm{2}}} \cap {\rm{B}}} \right)\)?

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? Plus? Premium?

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a \({\rm{CD}}\) player. Let \({{\rm{A}}_{\rm{1}}}\) be the event that the receiver functions properly throughout the warranty period, \({{\rm{A}}_{\rm{2}}}\) be the event that the speakers function properly throughout the warranty period, and \({{\rm{A}}_{\rm{3}}}\) be the event that the \({\rm{CD}}\) player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with \({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = }}{\rm{.95}}\), \({\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.98}}\), and \({\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = }}{\rm{.80}}\).

a. What is the probability that all three components function properly throughout the warranty period?

b. What is the probability that at least one component needs service during the warranty period?

c. What is the probability that all three components need service during the warranty period?

d. What is the probability that only the receiver needs service during the warranty period?

e. What is the probability that exactly one of the three components needs service during the warranty period?

f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

If \({\rm{A}}\)and \({\rm{B}}\) are independent events, show that \({{\rm{A}}^\prime }\) and \({\rm{B}}\)are also independent. (Hint: First establish a relationship between \({\rm{P}}\left( {{{\rm{A}}^{\rm{垄}}}{\rm{脟B}}} \right){\rm{,P(B)}}\), and \(\left. {{\rm{P(A脟B)}}{\rm{.}}} \right)\)
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