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Chapter 2: Q8E (page 57)

An engineering construction firm is currently working on power plants at three different sites. Let Aidenote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of \({A_1}\), \({A_2}\), and \({A_3}\), draw a Venn diagram, and shade the region corresponding to each one.

a. At least one plant is completed by the contract date.

b. All plants are completed by the contract date.

c. Only the plant at site 1 is completed by the contract date.

d. Exactly one plant is completed by the contract date.

e. Either the plant at site 1 or both of the other two plants are completed by the contract date.

Short Answer

Expert verified

a. The Venn diagram is represented as,

b. The Venn diagram is represented as,

c. The Venn diagram is represented as,

d. The Venn diagram is represented as,

e. The Venn diagram is represented as,

Step by step solution

01

Given information

Let \({A_i}\) represents the event that the plant at site i is completed by the contract date.

02

Describe the events by using the operations

a.

Let U represent the event that at least one plant is completed by the contract date.

The operation that will be used to describe the event U is union.

Mathematically,

\(U = {A_1} \cup {A_2} \cup {A_3}\)

03

Construction a Venn diagram

The Venn diagram representing the event U is given as,

04

Describe the events by using the operations for part (b).

b.

Let M be the event that all plants are completed by the contract date.

The operation that will be used to describe the event M is intersection.

Mathematically,

\(M = {A_1} \cap {A_2} \cap {A_3}\)

05

Construction of a Venn diagram

The Venn diagram representing the event M is given as,

06

Describe the events by using the operations for part (c)

c.

Let N be the event that only the plant at site 1 is completed by the contract date.

In the provided event only the area of\({A_1}\)will be shaded.

The operation that will be used to describe the event N is intersection and complementation.

Mathematically,

\(N = {A_1} \cap A_2^C \cap A_3^C\)

07

Construction of a Venn diagram

The Venn diagram representing the event N is given as,

08

Describe the events by using the operations for part (d)

d.

Let R be the event that exactly one plant is completed by the contract date.

The operation that will be used to describe the event R is intersection, complementation and union.

Mathematically,

\(R = \left( {{A_1} \cap A_2^C \cap A_3^C} \right) \cup \left( {A_1^C \cap {A_2} \cap A_3^C} \right) \cup \left( {A_1^C \cap A_2^C \cap {A_3}} \right)\)

09

Construction of a Venn diagram

The Venn diagram representing the event R is given as,

10

Describe the events by using the operations for part (e)

e.

Let T represent the event that either the plant at site 1 or both of the other two plants are completed by the contract date.

The operation that will be used to describe the event T is intersection, complementation and union.

Mathematically,

\(T = {A_1} \cup \left( {{A_2} \cap {A_3}} \right)\)

11

Construct a Venn diagram

The Venn diagram representing the event T is given as,

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Most popular questions from this chapter

Consider randomly selecting a single individual and having that person test drive \({\rm{3}}\) different vehicles. Define events \({{\rm{A}}_{\rm{1}}}\), \({{\rm{A}}_{\rm{2}}}\), and \({{\rm{A}}_{\rm{3}}}\) by

\({{\rm{A}}_{\rm{1}}}\)=likes vehicle #\({\rm{1}}\)\({{\rm{A}}_{\rm{2}}}\)= likes vehicle #\({\rm{2}}\)\({{\rm{A}}_{\rm{3}}}\)=likes vehicle #\({\rm{3}}\)Suppose that\({\rm{ = }}{\rm{.65,}}\)\({\rm{P(}}{{\rm{A}}_3}{\rm{)}}\)\({\rm{ = }}{\rm{.70,}}\)\({\rm{P(}}{{\rm{A}}_{\rm{1}}} \cup {{\rm{A}}_{\rm{2}}}{\rm{) = }}{\rm{.80,P(}}{{\rm{A}}_{\rm{2}}} \cap {{\rm{A}}_{\rm{3}}}{\rm{) = 40,}}\)and\({\rm{P(}}{{\rm{A}}_{\rm{1}}} \cup {{\rm{A}}_{\rm{2}}} \cup {{\rm{A}}_{\rm{3}}}{\rm{) = }}{\rm{.88}}{\rm{.}}\)

a. What is the probability that the individual likes both vehicle #\({\rm{1}}\)and vehicle #\({\rm{2}}\)?

b. Determine and interpret\({\rm{p}}\)(\({{\rm{A}}_{\rm{2}}}\)|\({{\rm{A}}_{\rm{3}}}\)).

c. Are \({{\rm{A}}_{\rm{2}}}\)and \({{\rm{A}}_{\rm{3}}}\)independent events? Answer in two different ways.

d. If you learn that the individual did not like vehicle #\({\rm{1}}\), what now is the probability that he/she liked at least one of the other two vehicles?

In October, \({\rm{1994}}\), a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division. The manufacturer initially claimed that the chance of any particular division being incorrect was only \({\rm{1}}\) in \({\rm{9}}\) billion, so that it would take thousands of years before a typical user encountered a mistake. However, statisticians are not typical users; some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility. Assuming that the \({\rm{1}}\) in \({\rm{9}}\) billion figure is correct and that results of different divisions are independent of one another, what is the probability that at least one error occurs in one billion divisions with this chip?

The proportions of blood phenotypes in the U.S. population are as follows:

Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O? What is the probability that the phenotypes of two randomly selected individual鈥檚 match?

a. A lumber company has just taken delivery on a shipment of \({\rm{10,000 2 \times 4}}\)boards. Suppose that 20% of these boards (\({\rm{2000}}\)) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let A 5 {the first board is green} and B 5 {the second board is green}. Compute \({\rm{P(A),P(B)}}\), and \({\rm{P(A}} \cap {\rm{B)}}\) (a tree diagram might help). Are \({\rm{A}}\) and \({\rm{B}}\) independent?

b. With \({\rm{A}}\) and \({\rm{B}}\) independent and \({\rm{P(A) = P(B) = }}{\rm{.2,}}\) what is \({\rm{P(A}} \cap {\rm{B)}}\)? How much difference is there between this answer and \({\rm{P(A}} \cap {\rm{B)}}\)part (a)? For purposes of calculating \({\rm{P(A}} \cap {\rm{B)}}\), can we assume that \({\rm{A}}\)and \({\rm{B}}\) of part (a) are independent to obtain essentially the correct probability?

c. Suppose the shipment consists of ten boards, of which two are green. Does the assumption of independence now yield approximately the correct answer for \({\rm{P(A}} \cap {\rm{B)}}\)? What is the critical difference between the situation here and that of part (a)? When do you think an independence assumption would be valid in obtaining an approximately correct answer to \({\rm{P(A}} \cap {\rm{B)}}\)?

In any Ai independent of any other \({\rm{Aj}}\)? Answer using the multiplication property for independent events.
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