chapter 2

Question 1

Probablity

Answer 1

The term probability refers to the study of randomness and uncertainty.

Question 2

In any situation in which one of a number of _______ outcomes may occur, the discipline of ________ provides methods for quantifying the chances, or likelihoods, associated with the various outcomes.

Answer 2

In any situation in which one of a number of possible outcomes may occur, the discipline of probability provides methods for quantifying the chances, or likelihoods, associated with the various outcomes.

Question 3

Experiment

Answer 3

An experiment is any activity or process whose outcome is subject to uncertainty

Question 4

Although the word _______ generally suggests a planned or carefully controlled laboratory testing situation.

Answer 4

Although the word experiment generally suggests a planned or carefully controlled laboratory testing situation.

Question 5

sample space

Answer 5

the sample space of an experiment, denoted S, is the set of all possible outcomes of that experiment

Question 6

The simplest experiment to which probability applies is one with two possible out-comes. One such experiment consists of examining a single weld to see whether it is defective. The _____ _____ for this experiment can be abbreviated as S 5 { N , D }, where N represents not defective, D represents defective, and the braces are used to enclose the elements of a set. Another such experiment would involve tossing a thumbtack and noting whether it landed point up or point down, with sample space 5 { U , D }, and yet another would consist of observing the gender of the next child born at the local hospital, with 5 { M , F }.

Answer 6

The simplest experiment to which probability applies is one with two possible out-comes. One such experiment consists of examining a single weld to see whether it is defective. The sample space for this experiment can be abbreviated as S 5 { N , D }, where N represents not defective, D represents defective, and the braces are used to enclose the elements of a set. Another such experiment would involve tossing a thumbtack and noting whether it landed point up or point down, with sample space 5 { U , D }, and yet another would consist of observing the gender of the next child born at the local hospital, with 5 { M , F }.

Question 7

If we examine three welds in sequence and note the result of each examination, then an outcome for the entire experiment is any sequence of N ’s and D ’s of length 3, so

S = 5 { NNN , NND , NDN , NDD , DNN , DND , DDN , DDD }

If we had tossed a thumbtack three times, the sample space would be obtained by replacing N by U in S above, with a similar notational change yielding the sample space for the experiment in which the genders of three newborn children are observed.

Answer 7

If we examine three welds in sequence and note the result of each examination, then an outcome for the entire experiment is any sequence of N ’s and D ’s of length 3, so

S = 5 { NNN , NND , NDN , NDD , DNN , DND , DDN , DDD }

If we had tossed a thumbtack three times, the sample space would be obtained by replacing N by U in S above, with a similar notational change yielding the sample space for the experiment in which the genders of three newborn children are observed.

Question 8

Two gas stations are located at a certain intersection. Each one has six gas pumps. Consider the experiment in which the number of pumps in use at a particular time of day is deter-mined for each of the stations. An experimental outcome specifies how many pumps are in use at the first station and how many are in use at the second one. One possible outcome is (2, 2), another is (4, 1), and yet another is (1, 4). The 49 outcomes in are displayed in the accompanying table. The sample space for the experiment in which a six-sided die is thrown twice results from deleting the 0 row and 0 column from the table, giving 36 outcomes

Answer 8

Two gas stations are located at a certain intersection. Each one has six gas pumps. Consider the experiment in which the number of pumps in use at a particular time of day is deter-mined for each of the stations. An experimental outcome specifies how many pumps are in use at the first station and how many are in use at the second one. One possible outcome is (2, 2), another is (4, 1), and yet another is (1, 4). The 49 outcomes in are displayed in the accompanying table. The sample space for the experiment in which a six-sided die is thrown twice results from deleting the 0 row and 0 column from the table, giving 36 outcomes

Question 9

A reasonably large percentage of C++ programs written at a particular company compile on the first run, but some do not (a compiler is a program that translates source code, in this case C++ programs, into machine language so programs can be executed). Suppose an experiment consists of selecting and compiling C++ programs at this location one by one until encountering a program that compiles on the first run. Denote a program that compiles on the first run by S (for success) and one that doesn’t do so by F (for failure). Although it may not be very likely, a possible outcome of this experiment is that the first 5 (or 10 or 20 or …) are F ’s and the next one is an S. That is, for any positive integer n, we may have to examine n programs before seeing the first S. The sample space is S = { S , FS , FFS , FFFS ,…}, which contains an infinite number of possible outcomes. The same abbreviated form of the sample space is appropriate for an experiment in which, starting at a specified time, the gender of each newborn each newborn infant is recorded until the birth of a male is observed

Answer 9

A reasonably large percentage of C++ programs written at a particular company compile on the first run, but some do not (a compiler is a program that translates source code, in this case C++ programs, into machine language so programs can be executed). Suppose an experiment consists of selecting and compiling C++ programs at this location one by one until encountering a program that compiles on the first run. Denote a program that compiles on the first run by S (for success) and one that doesn’t do so by F (for failure). Although it may not be very likely, a possible outcome of this experiment is that the first 5 (or 10 or 20 or …) are F ’s and the next one is an S. That is, for any positive integer n, we may have to examine n programs before seeing the first S. The sample space is S = { S , FS , FFS , FFFS ,…}, which contains an infinite number of possible outcomes. The same abbreviated form of the sample space is appropriate for an experiment in which, starting at a specified time, the gender of each newborn each newborn infant is recorded until the birth of a male is observed

Question 10

Event

Answer 10

An event is any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome

Question 11

When an experiment is performed, a particular event A is said to occur if the result-ing experimental outcome is contained in A. In general, exactly one simple event will occur, but many compound events will occur simultaneously.

Answer 11

When an experiment is performed, a particular event A is said to occur if the result-ing experimental outcome is contained in A. In general, exactly one simple event will occur, but many compound events will occur simultaneously.

Question 12

Consider an experiment in which each of three vehicles taking a particular freeway exit turns left ( L ) or right ( R ) at the end of the exit ramp. The eight possible outcomes that comprise the sample space are LLL, RLL, LRL, LLR, LRR, RLR, RRL, and RRR. Thus there are eight simple events, among which are E₁= { LLL } and E₅ { LRR }. Some compound events include

A = { RLL , LRL , LLR } = the event that exactly one of the three vehicles turns right

B = { LLL , RLL , LRL , LLR } = the event that at most one of the vehicles turns right

C = { LLL , RRR } = the event that all three vehicles turn in the same direction

Suppose that when the experiment is performed, the outcome is LLL. Then the sim-ple event E 1 has occurred and so also have the events B and C (but not A ).

Answer 12

Consider an experiment in which each of three vehicles taking a particular freeway exit turns left ( L ) or right ( R ) at the end of the exit ramp. The eight possible outcomes that comprise the sample space are LLL, RLL, LRL, LLR, LRR, RLR, RRL, and RRR. Thus there are eight simple events, among which are E₁= { LLL } and E₅ { LRR }. Some compound events include

A = { RLL , LRL , LLR } = the event that exactly one of the three vehicles turns right

B = { LLL , RLL , LRL , LLR } = the event that at most one of the vehicles turns right

C = { LLL , RRR } = the event that all three vehicles turn in the same direction

Suppose that when the experiment is performed, the outcome is LLL. Then the sim-ple event E 1 has occurred and so also have the events B and C (but not A ).

Question 13

(example 2.3 continued)

When the number of pumps in use at each of two six-pump gas stations is observed, there are 49 possible outcomes, so there are 49 simple events: E₁ = {(0, 0)}, E₂ = {(0, 1)},…, E₄₉ = {(6, 6)}. Examples of compound events are

A = {(0, 0), (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} = the event that the number of pumps in use is the same for both stations

B = {(0, 4), (1, 3), (2, 2), (3, 1), (4, 0)} = the event that the total number of pumps in use is four

C = {(0, 0), (0, 1), (1, 0), (1, 1)} = the event that at most one pump is in use at each station

Question 14

(Example 2.4 continued)

The sample space for the program compilation experiment contains an infinite number of outcomes, so there are an infinite number of simple events. Compound events include

A = { S , FS , FFS } = the event that at most three programs are examined

E = { FS , FFFS , FFFFFS ,…} = the event that an even number of programs are examined

Answer 14

The sample space for the program compilation experiment contains an infinite number of outcomes, so there are an infinite number of simple events. Compound events include

A = { S , FS , FFS } = the event that at most three programs are examined

E = { FS , FFFS , FFFFFS ,…} = the event that an even number of programs are examined

Question 15

Some Relations from Set Theory: An event is just a set, so relationships and results from elementary set theory can be used to study events. The following operations will be used to create new events from given events.

Answer 15

An event is just a set, so relationships and results from elementary set theory can be used to study events. The following operations will be used to create new events from given events.

Question 16

Complement

Answer 16

The complement of an event A, denoted by A’ , is the set of all outcomes in S that are not contained in A.

Question 17

Union

Answer 17

The union of two events A and B, denoted by A U B and read “ A or B, ” is the event consisting of all outcomes that are either in A or in B or in both events (so that the union includes outcomes for which both A and B occur as well as outcomes for which exactly one occurs)—that is, all outcomes in at least one of the events.

Question 18

Intersection

Answer 18

The intersection of two events A and B, denoted by A ∩ B and read “ A and B, ” is the event consisting of all outcomes that are in both A and B.

Question 19

(example 2.3 continued)
For the experiment in which the number of pumps in use at a single six-pump gas station is observed, let

A = {0, 1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {1, 3, 5}. Then

A’ = {5, 6}
A ∪ B = {0, 1, 2, 3, 4, 5, 6} = S
A ∪ C = {0, 1, 2, 3, 4, 5}
A ∩ B 5 {3, 4}
A ∩ C = {1, 3}
( A ∩ C ) ‘ = {0, 2, 4, 5, 6}

Answer 19

(example 2.3 continued)
For the experiment in which the number of pumps in use at a single six-pump gas station is observed, let

A = {0, 1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {1, 3, 5}. Then

A’ = {5, 6}
A ∪ B = {0, 1, 2, 3, 4, 5, 6} = S
A ∪ C = {0, 1, 2, 3, 4, 5}
A ∩ B 5 {3, 4}
A ∩ C = {1, 3}
( A ∩ C ) ‘ = {0, 2, 4, 5, 6}

Question 20

(Example 2.4 continued )
In the program compilation experiment, define A, B, and C by
A = { S , FS , FFS }, B = { S , FFS , FFFFS }, C = { FS , FFFS , FFFFFS ,…}
Then
A’ = { FFFS , FFFFS , FFFFFS ,…}, C’ = { S , FFS , FFFFS ,…}
A ∪ B { S , FS , FFS , FFFFS }, A ∩ B { S , FFS }

Answer 20

(Example 2.4 continued )
In the program compilation experiment, define A, B, and C by
A = { S , FS , FFS }, B = { S , FFS , FFFFS }, C = { FS , FFFS , FFFFFS ,…}
Then
A’ = { FFFS , FFFFS , FFFFFS ,…}, C’ = { S , FFS , FFFFS ,…}
A ∪ B { S , FS , FFS , FFFFS }, A ∩ B { S , FFS }

_{Sometimes A and B have no outcomes in common, so that the intersection of A and B contains no outcomes.}

Question 21

Let Ø denote the null event (the event consisting of no outcomes whatsoever). When A ∩ B= Ø, A and B are said to be mutually exclusive or disjoint events.

Answer 21

Let Ø denote the null event (the event consisting of no outcomes whatsoever). When A ∩ B= Ø, A and B are said to be mutually exclusive or disjoint events.

Question 22

Three components are connected to form a system as shown in the accompanying diagram. Because the compo-nents in the 2–3 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 2–3 subsystem. 213 The experiment consists of determining the condition of each component [S (success) for a functioning compo-nent and F (failure) for a nonfunctioning component]. a. Which outcomes are contained in the event A that exactly two out of the three components function? b. Which outcomes are contained in the event B that at least two of the components function? c. Which outcomes are contained in the event C that the system functions? d. List outcomes in C 9 , A ø C , A ù C , B ø C, and B C. Copyright | CENGAGE Learning | Probability and Statistics for Engineering and the Sciences | Edition 9 | l2aindrop@hotmail.com | Printed from www.chegg.com

Question 23

Axioms, Interpretations and Properties of Probability:
Given an experiment and a sample space S, the objective of probability is to assign to each event A a number P ( A ) , called the probability of the event A, which will give a precise measure of the chance that A will occur.

Answer 23

Given an experiment and a sample space S, the objective of probability is to assign to each event A a number P ( A ) , called the probability of the event A, which will give a precise measure of the chance that A will occur.

Question 24

Axioms, Interpretations and Properties of Probability:

Axiom 1

Answer 24

Axiom 1: For any event A , P ( A ) >= 0

Question 25

Axioms, Interpretations and Properties of Probability:

Axiom 2

Answer 25

P ( S ) = 1.

Question 26

Axioms, Interpretations and Properties of Probability:

Axiom 3

Answer 26

If A₁ , A₂ , A₃ ,… is an infinite collection of disjoint events, then