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Chapter 2: Q4E (page 57)

Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V).

a. What are the 16 outcomes in S?

b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate?

c. Which outcomes are in the event that all four mortgages are of the same type?

d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage?

e. What is the union of the events in parts (c) and (d), and what is the intersection of these two events?

f. What are the union and intersection of the two events in parts (b) and (c)?

Short Answer

Expert verified

a. The possible outcomes are:

\(S = \left\{ {FVVV,VFVV,VVFV,VVVF,FFFF,FFFV,FFVF,FVFF,VFFF,VVFF,FFVV,FVFV,VFVF,VVVV,VFFV,FVVF} \right\}\)

b. The possible outcomes are:

\(A = \left\{ {FFFV,FFVF,FVFF,VFFF} \right\}\)

c. The possible outcomes are:

\(B = \left\{ {FFFF,VVVV} \right\}\)

d. The possible outcomes are,

\(C = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

e. The possible outcomes are,

\(B \cup C = \left\{ {FFFF,VVVV,FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

\(B \cap C = \left\{ {FFFF} \right\}\)

f. The possible outcomes are,

\(A \cup B = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF,VVVV} \right\}\)

\(A \cap B = \phi \)

Step by step solution

01

Given information

Each of a sample of four home mortgages is classified as fixed-rate (F) or variable rate (V).

02

List of the possible outcomes in sample space.

a.

The possible 16 outcomes in S is given as,

\(S = \left\{ {FVVV,VFVV,VVFV,VVVF,FFFF,FFFV,FFVF,FVFF,VFFF,VVFF,FFVV,FVFV,VFVF,VVVV,VFFV,FVVF} \right\}\)

03

List of the possible outcomes

b.

Let A represents the event that that exactly three of the selected mortgages are fixed.

The event A includes exactly 3 F鈥檚 included in the sample of four mortgages.

The outcomes of the event that exactly three of the selected mortgages are fixed is given as,

\(A = \left\{ {FFFV,FFVF,FVFF,VFFF} \right\}\)

04

List of the possible outcomes

c.

Let B represents the event that all four mortgages are of the same types.

The event B includes all 4 F鈥檚 and all four V鈥檚.

The outcomes of the event that all four mortgages are of the same types is given as,

\(B = \left\{ {FFFF,VVVV} \right\}\)

05

List of the possible outcomes

d.

Let C represents the event that at most one of the four is a variable-rate mortgage.

The outcomes of the event that at most one of the four is a variable-rate mortgage is given as,

\(C = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

06

List of the possible outcomes in union and intersection

e.

A union of two events B and C consists of all the outcomes that are either in B or C or in both events.

Referring to part c and d, the two events are,

\(B = \left\{ {FFFF,VVVV} \right\}\)

\(C = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

The union of events is represented as,

\(B \cup C = \left\{ {FFFF,VVVV,FFFV,FFVF,FVFF,VFFF,FFFF} \right\}\)

The intersection of two events B and C consists of all the outcomes that are present in both events.

The intersection of the events is represented as,

\(B \cap C = \left\{ {FFFF} \right\}\)

07

List of the possible outcomes in union and intersection

f.

A union of two events A and B consists of all the outcomes that are either in A or C or in both events.

Referring to part b and c, the two events are,

\(A = \left\{ {FFFV,FFVF,FVFF,VFFF} \right\}\)

\(B = \left\{ {FFFF,VVVV} \right\}\)

The union of events is represented as,

\(A \cup B = \left\{ {FFFV,FFVF,FVFF,VFFF,FFFF,VVVV} \right\}\)

The intersection of two events A and B consists of all the outcomes that are present in both events.

The intersection of the events is represented as,

\(A \cap B = \phi \)

Therefore, there are no outcomes for the intersection of the events A and B.

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Most popular questions from this chapter

An academic department with five faculty members鈥� Anderson, Box, Cox, Cramer, and Fisher鈥攎ust select two of its members to serve on a personnel review committee. Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representatives will be selected by putting the names on identical pieces of paper and then randomly selecting two.

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a. \(P\left( {{A_2}|{A_1}} \right)\)

b. \(P\left( {{A_2} \cap {A_3}|{A_1}} \right)\)

c. \(P\left( {{A_2} \cup {A_3}|{A_1}} \right)\)

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

Small

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\(14\% \)

\(20\% \)

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Consider randomly selecting such a coffee purchaser.

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

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

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

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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?

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