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首页沙伦·罗斯《概率初步》第五版详解
《沙伦·罗斯 - 首门概率论入门》第五版是Sheldon Ross所著的一本经典教材,由University of California, Berkeley的教授撰写,专为学习概率论的学生和专业人士设计。本书是Sheldon Ross系列教程中的一部,以其深入浅出的讲解和丰富的实例,在统计学和数学教育领域享有盛誉。
该第五版教材包含了全面且系统的概率论基础理论,覆盖了概率的基本概念、随机变量、概率分布、独立事件、条件概率、期望与方差、极限定理、多元随机变量以及更高级的主题如大数定律、中心极限定理等。书中不仅注重理论解析,还强调实践应用,通过大量的习题和案例分析,帮助读者逐步掌握概率论的核心技巧和实际问题的解决方法。
在内容编排上,Sheldon Ross以清晰的逻辑结构展开讨论,确保了从基础概率概念到复杂模型的渐进式学习过程。每章末尾通常会有复习题和习题,供读者巩固所学,并提供了解决实际问题所需的练习。此外,书中还附有详细的参考文献和索引,便于读者进一步探索相关领域的研究和发展。
对于图书馆馆藏而言,这本书的《Library of Congress Cataloging-in-Publication Data》提供了必要的出版信息,包括作者Sheldon M. Ross、出版社Prentice Hall以及出版年份和ISBN号,方便图书馆进行分类和检索。此外,编辑团队的协作也体现了教材出版的专业性和严谨性,从 acquisitions editor 到 marketing manager 的多个角色共同确保了教材的质量和市场推广。
《Sheldon Ross - A First Course in Probability》第五版是一本全面且实用的概率论入门指南,适合大学生、研究生、研究人员和任何希望系统学习概率论的人士使用,是提高概率理论素养的重要参考资料。无论是作为课堂教学材料还是自我学习的辅助读物,它都能提供扎实的基础知识和深入的理解。
.
.
C.
-
16
Chapter
1
Combinatorial Analysis
.
.
when 0
I
i
I
12, and let it equal 0 otherwise. This quantity represents the number
of different subgroups of size i that can be chosen from a set of size
72. It is often
called a
birzo1ninl coejjficieizt because of its prominence in the binomid theorem,
>>
which states that
.....
.
-
For nonnegative integers 121,
. .
.
,
12,
summing to n,
is the number of ways of dividing up
n
items into
r
distinct nonoverlapping
subgroups of sizes
121, n2,
. . .
,
n,.
-
PROBLEMS
1.
(a) How many different 7-place license plates are possible if the first 2 places
are for letters and the other 5 for numbers?
(b)
Repeat part (a) under the assumption that no letter or number can be
repeated
in
a single license plate.
2.
How many outcome sequences are possible when a die is rolled four times,
where we say, for instance, that the outcome is 3, 4, 3, 1 if the first roll
landed on 3, the second on 4, the third on 3, and the fourth on
l?
3.
Twenty workers are to be assigned to 20 different jobs, one to each job. How
many different assignments are possible?
4.
John, Jim, Jay, and Jack have formed a band consisting of 4 instruments.
If
each of the boys can play
all
4 instruments, how many different arrangements
are possible? What if John and Jim can play all
4
instruments, but Jay and
Jack can each play only piano and drums?
5.
For years, telephone area codes in the United States and Canada consisted
of a sequence of three digits. The first digit was an integer between 2 and 9;
the second digit was either
0 or 1; the third digit was any integer between 1
-
and 9. How many area codes were possible? How many area codes starting
with a 4 were possible?
6.
A
well-known nursery rhyme starts as follows:
As I was going to
St.
Ives
I
met a
man
with 7 wives.
Each wife had-7 sacks.
Each sack had 7 cats.
Each cat had 7 kittens.
How many kittens did the traveler meet?
Problems
17
7.
(a)
In how many ways can 3 boys,and 3 ,girls sit in a row?
(b)
In how many ways can 3 boys and 3 girls sit in a row if the boys and
the girls are each to sit together?
J
(c) In how ,many ways if only the boys must sit together?
(d) In how many ways if no two people of the same sex are allowed to
sit together?
8.
How many different letter arrangements can be made from the letters
O
""E;
(c) MISSISSIPPI;
(d) ARRANGE?
9.
A
child has 12 blocks, of which-6 are black,
4
are red,
1
is white, and 1 is
/
blue.
If
the child puts the blocks in a line, how many arrangements are possible?
10.
In how many ways can
8
people be seated in a row if
(a) there are no restrictions on the seating arrangement;
!
(b)
persons
A
and
B
must sit next to each other;
(c) there are 4 men and 4 women and no 2 men or 2 women can sit next to
each other;
(d) there are 5 men and they
must sit next to each other;
(e)
there are 4 married couples and each couple must sit together?
11.
In how many ways can 3 novels, 2 mathematics books, and 1 chemistry book
be arranged on a bookshelf
if
(a) the books can be arranged in any order;
(b)
the mathematics books must be together and the novels must be together;
(c) the novels must be together but the other books can be arranged in
any order?
I
12.
Five separate awards (best scholarship, best leadership qualities, and so on)
are to be presented to selected students from a class of 30. How many different
,
outcomes are possible
if
(a) a student can receive any number of awards;
.
(b)
each student can receive at most 1 award?
13.
Consider a group of 20 people.
If
everyone shakes hands with everyone else,
how many handshakes take place?
14.
How many 5-card poker hands are there?
15.
A
dance class consists of 22 students, 10 women and 12 men.
If
5 men and
5 women are to be chosen and then paired off, how many results are possible?
16.
A
student has to sell 2 books from a collection of
6
math. 7 science, and
4
economics books. How many choices are possible if
(a) both books are to be on the same subject;
(b)
the books are to be on different subjects?
17.
A
total of 7 different gifts are to be distributed among 10 children. How
many distinct results are possible
if
no child is to receive more than one gift?
18.
A
committee of 7, consisting of 2 Republicans, 2 Democrats, qnd 3 Indepen-
dents, is to be chosen from a group of
5
Republicqs,
6
Democrats, and 4
Independents. How many committees are possible?
18
Chapter
1
Cornbinatonal Analysis
19.
From a group of
8
women and
6
men a committee consisting of 3 men and
3 women is to be formed. How many different committees are possible if
(a)
2 of the men refuse to serve together;
(b)
2
of the women refuse to serve together;
(c)
1 man and 1 woman refusp to serve together?
20.
A
person has
8
friends, of whom 5 will be invited to a party.
(a) How many choices are there if 2 of the friends are feuding and will not
attend together?
(b)
How many choices if 2 of the friends will only attend together?
21.
Consider the ,orid of points shown below. Suppose that starting at the point
labeled
A
you can go one step up or one step to the right at each move. This
is continued until the point labeled
B
is reached. How many different paths
from
A
to
B
are possible?
HINT:
Note that to reach
B
from
A
you must take
4
steps to the right and
3 steps upward.
22.
In
Problem 21, how many different paths are there from
A
to
B
that go through
the point circled below?
Theoretical Exercises
19
24.
Expand (3x'
+
y)5.
25.
The game of bridge is played by
4
players, each of whom is dealt 13 cards.
How many bridge deals are possible?
26.
Expand (xl
"+
2x2
+
3x314.
27.
If
12 people are to be divided into 3 committees of respective sizes 3,
4,
and
5, how many divisions are possible?
28.
If
8
new teachers are to be divided among
4
schools, how many divisions
are possible?
What if each school must receive 2 teachers?
29.
Ten weight lifters are competing in a team weight-lifting contest. Of the
lifters, 3 are from the United States,
4
are from Russia,
2
are from China,
and 1 is from Canada.
If
the scoring takes account of the countries that the
Lifters represent but not their individual identities, how many different out-
comes are possible from the point of view of scores? How many different
outcomes correspond to results in which the United States has 1 competitor
in the
to^
three and 2 in the bottom three?
30.
Delegates from 10 countries, including Russia, France, England, and the
United States, are to be seated in a row. How many different seating arrange-
ments are possible if the French and English delegates are to be seated next
to each other, and the Russian and U.S. delegates are not to be next to
each other?
"31.
If
8
identical blackboards are to be divided among
4
schools, how many
divisions are possible? How many, if each school must receive at least 1 black-
board?
"32.
An
elevator starts at the basement with
8
people (not including the elevator
operator) and discharges them
all
by the time it reaches the top floor,
number
6.
In
how many ways could the operator have perceived the people
leaving the elevator if
all
people look alike to him? What if the
8
people
consisted of 5 men and 3 women and the operator could tell a man from
a woman?
"33.
We have 20 thousand dollars that must be invested among
4
possible opportu-
nities. Each investment must be
integral in units of
1
thousand dollars, and
there are minimal investments that need to be made if one is to invest in
these opportunities. The minimal investments are 2, 2, 3, and
4
thousand
dollars. How many different investment strategies are available
if
(a) an investment must be made in each opportunity;
(b)
investments must be made in at least 3 of the
4
opportunities?
THEORETICAL EXERCISES
A
1.
Prove the generalized version of the basic counting principle.
23.
A
psychology laboratory conducting dream research contains 3 rooms, with
2.
Two experiments are to be performed. The first can result in any one of
m
2 beds in each room.
If
3
sets of identical twins are to be assigned to these
possible outcomes.
If
the first experiment results in outcome number
i,
then
6
beds so that each set of twins sleeps in different beds in the same room,
the second experiment can result in any of
rzi
possible outcomes,
i
=
1,
how many assignments are possible?
Y
2,
.
.
.
,
nz.
What is the number of possible outcomes of the two experiments?
-,.
20
Chapter
1
Combinatorid Analysis
3.
In
how many ways can
r
objects be selected from a set of
rz
if the order of
selection is considered relevant?
4.
There are
(;)
different linear arrangements of
n
balls of which
r
are black
, ,
and
11
-
r
are white. Give a combinatorial explanation of this fact.
5.
Determine the number of vectors (xl,
. . .
,
x,,), such that each xi is either
0
or 1 and
6.
How many vectors xl,
. .
.
,
xk are there for which each
xi
is a positive integer
such that 1
5
xi
5
rz
and
xl
<
x2
<
+ +
<
xk?
7.
Give an analytic proof of Equation (4.1).
18.
Prove that.
:
Consider a group of n men and
m
women. How many groups of size
r
are possible?
9.
Use
heo ore tical
Exercise
8
to prove that
10.
From a group of n people, suppose that we want to choose a committee of
k, k
5
n, one of whom is to be designated as chairperson.
(a) By focusing first on the choice of the committee and then on the choice
of the chair, argue that there are
k possible choices.
(3
(b)
By focusing first on the choice of the nonchair committee members
and then on the choice of the chair, argue that there are
(k
n
(11
-
k
+
1)
possible choices.
(c)
By focusing first on the choice of the chair and then on the choice of
the other committee members, argue that there are
iz
possible
choices.
(d)
Conclude from parts (a), (b), and (c) that
d
(e)
Use the factorial definition of
to verify the identity in part (d).
Theoretical Exercises
2
1
,
11.
The following identity is known as Fermat's combinatorial identity.
Give a combinatorial argument (no computations are needed) to establish this
identity.
HINT:
Consider the set of numbers 1 through n. How many subsets of size
k have
i
as their highest-numbered member?
12.
Consider the following combinatorial identity:
(a) Present a combinatorial argument for the above by considering a set of
-
n people and determining, in two ways, the number of possible selections
of a committee of any size and a chairperson for the committee.
HINT: (i) How many possible selections are there of a committee of
size k and its chairperson?
(ii) How many possible selections are there of a chairperson and
the other committee members?
(b)
Verify the following identity for n
=
1, 2,
3,
4,
5:
For a combinatorial proof of the above, consider a set of
iz
people, and
argue that both sides of the identity above represent the number of different
selections of a committee, its chairperson, and its secretary (possibly the
same as the chairperson).
HINT:
(i) How many different selections result
in
the committee con-
taining exactly k people?
(ii) How many different selections are there in which the chair-
person and the secretary are the same?
(ANSWER:
n2n- I.)
(iii) How many different selections result in the chairperson and
the secretary being different?
(c)
Now argue that
13.
Show that for
rz
>
0,
HINT:
Use the binomial theorem.
\
22
Chapter
1
Cornbinatonal Analysis
14.
From a set of
rz
people a committee of size
j
is to be chosen, and from this
committee a subcommittee of size
i,
i
5
j, is also to be chosen.
(a) Derive a combinatorial identity by computing, in two ways, the number
of possible choices of the committee and subcommittee-first by suppos-
ing that the committee is chosen first and then the subcommittee, and
second by supposing that the subcommittee is chosen first and then the
remaining members of the committee are chosen.
(b) Use part (a) to prove the following combinatorial identity:
(c) Use part (a) and Theoretical Exercise 13 to show that
15.
Let Hk(rz) be the number of vectors xl,
.
.
.
,
xk for which each
xi
is a positive
integer satisfying 1
5
xi
5
rz
and xl
5
x2
S
. . .
5
xk.
(a) Without any computations, argue that
m~:
How many vectors are there in which xk
=
j?
(b)
Use the preceding recursion to compute H3(5).
HINT:
First compute H2(rz) for
IZ
=
1, 2, 3,
4,
5.
16.
Consider a tournament of
n
contestants in which the outcome is an ordering
of these contestants, with ties allowed. That is, the outcome partitions the
players into groups, with the first group consisting of the players that tied
for first place, the next group being those that tied for the next best position,
and so on. Let
N(rz) denote the number of different possible outcomes. For
instance, N(2)
=
3 since in a tournament with 2 contestants, player 1 could
be uniquely first, player 2 could be uniquely first, or they could tie for first.
(a) List all the possible outcomes when
rz
=
3.
(b)
With N(0) defined to equal 1, argue, without any computations, that
HINT:
HOW many outcomes are there in which
i
players tie for last place?
(c) Show that the formula of part
(b)
is equivalent to the following:
(d) Use the recursion to find N(3) and
N(4).
Self-test Problems
and
Exercises
23
'
17.
Present a combinatorial explanation of why
)
=
,
r-
3-
18.
Argue that
.
HINT:
Use an argument similar to the one used to establish Equation (4.1).
19.
Prove the multinomial theorem.
"20.
In
how many ways can
rz
identical balls be distributed into
I-
urns so that the
ith urn contains at least
r?zi
balls, for each
i
=
1,
.
.
.
,
I-?
Assume that
rz
2
"21.
Argue that there are exactly
(I])
(n
!:
1
k)
solutions of
for
whi* exactly
k
of the xi are equal to
0.
"22.
Consider a function f(xl,
.
. .
,
x,:,,) of
rz
variables. How many different partial
derivatives of order
I-
does it possess?
"23.
Determine the number of vectors (xl,
. . .
,
x,,), such that eachxi is a nonnegative
.
integer and
SELF-TEST PROBLEMS
AND
EXERCISES
1.
How many different linear arrangements are there of the letters A,
B,
C, D,
E, F for which
(a)
A and
B
are next to each other;
(b) A is before
B;
(c) A is before
B
and
B
is before C;
(d) A is before
B
and C is before D;
(e)
A and
B
are next to each other and C and D are also next to each other;
(f)
E
is not last in line?
2.
If
4 Americans, 3 Frenchmen, and
3
Englishmen are to be seated in a row,
how many seating arrangements are possible when people of the same national-
ity must sit next to each other?
,
24
Chapter
1
Combinatorial Analysis
3.
A president, treasurer, and secretary, all different, are to be chosen from a
club consisting of 10 people. How many different choices of officers are
possible if
(a)
there are no restrictions;
(b)
A
and
B
will not serve together;
(c)
C
and
D
will serve together or not at all;
(d)
E
must be an officer;
(e)
F
will serve only
if
he is president?
4.
A student is to answer
7
out of 10 questions in an examination. How many
choices has she? How many
if
she must answer at least
3
of the first
5
ques-
-
tions?
5.
In how many ways can a man divide
7
gifts among his
3
children
if
the eldest
1
is to receive
3
gd3s and the others 2 each?
6.
How many different 7-place license plates are possible when
3
of the entries
are letters and 4 are digits? Assume that repetition of letters and numbers is
allowed and that there is no restriction on where the letters or numbers can
--
be placed.
7.
Give a combinatorial explanation of the identity
8.
Consider n-digit numbers where each digit is one of the 10 integers 0,
1,
. . .
,
9.
How many such numbers are there for which
(a) no two consecutive digits are equal;
(b)
0 appears as a digit a total of
i
times,
i
=
0,
. . .
,
lz?
9.
Consider three classes, each consisting of
tz
students. From this group of
312
students, a group of
3
students is to be chosen.
(a) How many choices are possible?
(b)
How many choices are there in which all
3
students are in the same class?
(c) How many choices are there in which
2
of the
3
students are in the same
class and the other student is in a different class?
(d) How many choices are there in which all
3
students are in different classes?
(e) Using the results of parts (a) through (d), write a combinatorial identity.
*lo.
An
art collection on auction consisted of 4 Dalis,
5
van Goghs, and 6 Picassos.
At the auction were
5
art collectors.
If
a reporter noted only the number of
Dalis, van Goghs, and Picassos acquired by each collector, how many different
results could have been recorded if all works were sold?
*11.
Determine the number of vectors
(xl,
.
. .
,
x,)
such that each
xi
is a positive
integer and
where
k
2
12.
s
of
Probability
2.1
INTRODUCTION
In
this chapter we introduce the concept of the probability of an event and
then show how these probabilities can be computed in certain situations. As a
preliminary, however, we need the concept of the sample space and the events
of an experiment.
2.2
SAMPLE SPACE AND EVENTS
Consider an experiment whose outcome is not predictable with certainty in ad-
vance. However, although the outcome of the experiment will not be known in
advance, let us suppose that the set of all possible outcomes is known. This set
of
all.possible outcomes of an experiment is known as the
sarnple space
of the
experiment and is denoted by
S.
Some examples follow.
1.
If
the outcome of an experiment consists in the determination of the sex
of a
new&orn child, then
s
=
Is, b}.
where the outcome g means that the child is a girl and b that it is a boy.
2.
If
the outcome of an experiment is the order of finish in a race among
the
7
horses having post positions 1,
2,
3,
4,
5,
6, 7, then
S
=
{all 7! permutations of (1, 2,
3,
4,
5,
6, 7))
Woutcome (2,
3,
1, 6,
5,
4, 7) means, for instance, that the number 2 horse
comes in first, then the number
3
horse, then the number 1 horse, and so on.
3.
If
the experiment consists of flipping two coins, then the sample space
consists of the following four points:
s
=
{(H,
H),
(H, TI, (T,
H),
(T,
T)}
L
25
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