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Chapter 2: Q3E (page 57)

Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2鈥�3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1 must function and so must the 2鈥�3 subsystem.

The experiment consists of determining the condition of each component (S(success) for a functioning componentand F (failure) for a non-functioning component).

a. Which outcomes are contained in the event Athat exactly two out of the three components function?

b. Which outcomes are contained in the event Bthat at least two of the components function?

c. Which outcomes are contained in the event Cthat the system functions?

d. List outcomes in C鈥�, A \( \cup \)C, A \( \cap \)C, B \( \cup \)C, and B \( \cap \)C.

Short Answer

Expert verified

a. The possible outcomes are:

\(A = \left\{ {SSF,SFS,FSS} \right\}\).

b. The possible outcomes are:

\(B = \left\{ {SSS,SSF,SFS,FSS} \right\}\).

c. The possible outcomes are:

\(C = \left\{ {SSF,SFS,SSS} \right\}\).

d. The outcomes are:

\(C' = \left\{ {FFF,FFS,FSS,FSF,SFF} \right\}\)

\(A \cup C = \left\{ {SSF,SFS,FSS,SSS} \right\}\)

\(B \cup C = B\)

\(A \cap C = \left\{ {SSF,SFS} \right\}\)

\(B \cap C = C\)

Step by step solution

01

Given information

The number of components that are connected to form a system are 3.

Components 2 and 3 are connected in parallel. The subsystem will work if at least one of the two individual components functions.

S represents success and F represents Failure.

02

List of the possible outcomes

a.

Event A represents that exactly two out of the three components function.

The possible outcomes of A is given as,

\(A = \left\{ {SSF,SFS,FSS} \right\}\)

03

List of the possible outcomes

b.

Event B represents that at least two of the components function.

The possible outcomes of B is given as,

\(B = \left\{ {SSS,SSF,SFS,FSS} \right\}\)

04

List of the possible outcomes

c.

Event C represents the system functions.

The possible outcomes of C is given as,

\(C = \left\{ {SSF,SFS,SSS} \right\}\)

05

List of the possible outcomes in different events

d.

Referring to parts a, b, and c,

\(A = \left\{ {SSF,SFS,FSS} \right\}\)

\(B = \left\{ {SSS,SSF,SFS,FSS} \right\}\)

\(C = \left\{ {SSF,SFS,SSS} \right\}\)

The complementary of an event C consists of all the outcomes that are not contained in C.

The possible outcomes are,

\(C' = \left\{ {FFF,FFS,FSS,FSF,SFF} \right\}\)

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

\(A \cup C = \left\{ {SSF,SFS,FSS,SSS} \right\}\)

Similarly, for events B and C, we have,

\(\begin{aligned}B \cup C &= \left\{ {SSS,SSF,SFS,FSS} \right\}\\ &= B\end{aligned}\)

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

\(A \cap C = \left\{ {SSF,SFS} \right\}\)

Similarly for events B and C, we have,

\(\begin{aligned}B \cap C &= \left\{ {SSF,SFS,SSS} \right\}\\ &= C\end{aligned}\)

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Most popular questions from this chapter

An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component:

A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions:

a. In how many ways can one component of each type be selected?

b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?

c. In how many ways can components be selected if none is to be Sony?

d. In how many ways can a selection be made if at least one Sony component is to be included?

e. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component?

Four universities鈥�1, 2, 3, and 4鈥攁re participating in a holiday basketball tournament. In the first round, 1 will play 2 and 3 will play 4. Then the two winners will play for the championship, and the two losers will also play. One possible outcome can be denoted by 1324 (1 beats 2 and 3 beats 4 in first-round games, and then 1 beats 3 and 2 beats 4).

a. List all outcomes in S.

b. Let A denote the event that 1 wins the tournament. List outcomes in A.

c. Let Bdenote the event that 2 gets into the championship game. List outcomes in B.

d. What are the outcomes in A\( \cup \)B and in A\( \cap \)B? What are the outcomes in A鈥�?

A box in a supply room contains \({\rm{15}}\) compact fluorescent lightbulbs, of which \({\rm{5}}\) are rated \({\rm{13}}\)-watt, \({\rm{6}}\)are rated \({\rm{18}}\)-watt, and \({\rm{4}}\) are rated \({\rm{23}}\)-watt. Suppose that three of these bulbs are randomly selected.

a. What is the probability that exactly two of the selected bulbs are rated \({\rm{23}}\)-watt?

b. What is the probability that all three of the bulbs have the same rating?

c. What is the probability that one bulb of each type is selected?

d. If bulbs are selected one by one until a \({\rm{23}}\)-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?

In five-card poker, a straight consists of five cards with adjacent denominations (e.g., \({\rm{9}}\)of clubs, \({\rm{10}}\)of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card \({\rm{10}}\)? What is the probability that it will be a straight? What is the probability that it will be a straight flush (all cards in the same suit)?

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?
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