schaums-probability.pdf

CONTENTS
Chapter I SET THEORY ......................................................
Page
1
Introduction. Sets, elements. Set operations. Finite and countable sets. Product
sets. Classes of sets.
Chapter 2 TECHNIQUES OF COUNTING .................................... 16
Introduction. Fundamental principle of counting. Factorial notation. Permutations.
Permutations with repetitions. Ordered samples. Binomial coefficients and theorem.
Combinations. Ordered partitions. Tree diagrams.
Chapter 3 INTRODUCTION TO PROBABILITY .............................. 38
Introduction. Sample space and events. Axioms of probability. Finite probability
spaces. Finite equiprobable spaces. Infinite sample spaces.
Chapter 4 CONDITIONAL PROBABILITY AND INDEPENDENCE .......... 54
Conditional probability. Multiplication theorem for conditional probability. Finite
stochastic processes and tree diagrams. Partitions and Bayed theorem. Inde-
pendence. Independent or repeated trials.
Chapter 5 RANDOM VARIABLES ............................................ 74
Introduction. Distribution and expectation of a finite random variable. Variance
and standard deviation. Joint distribution. Independent random variables. Func-
tions of a random variable. Discrete random variables in general. Continuous
random variables. Cumulative distribution function. Tchebycheff’s inequality.
Law of large numbers.
. Chapter 6 BINOMIAL, NORMAL AND POISSON DISTRIBUTIONS .......... 105
Binomial distribution. Normal distribution. Normal approximation to the binomial
distribution. Central limit theorem. Poisson distribution. Multinomial distribution.
Chapter 7 MARKOV CHAINS ................................................ 126
Introduction. Probability vectors, stochastic matrices. Regular stochastic matrices.
Fixed points and regular stochastic matrices. Markov chains. Higher transition
probabilities. Stationary distribution of regular Markov chains. Absorbing states.
~
INDEX ........................................................................... 152

Chapter I
Set Theory
INTRODUCTION
This chapter treats some of the elementary ideas and concepts of set theory which are
necessary for a modern introduction to probability theory.
SETS, ELEMENTS
Any well defined list or collection of objects is called a set; the objects comprising the
set are called its elements or members. We write
p E A if p is an element in the set A
If every element of A also belongs to a set B, i.e. if p E A implies p E B, then A is called a
subset of B or is said to be contained in B; this is denoted by
A c B or B 3 A
Two sets are equaZ if each is contained in the other; that is,
A =B if and only if A c B and B c A
The negations of p EA, A cB and A =B are written p A, A $ZB and A +B respectively.
We specify a particular set by either listing its elements or by stating properties which
characterize the elements of the set. For example,
A = {I,3, 5,7, 9}
means A is the set consisting of the numbers 1,3,5,7and 9; and
B = {x : x is a prime number, x < 15)
means that B is the set of prime numbers less than 15.
Unless otherwise stated, all sets under investigation are assumed to be subsets of some
fixed set called the universal set and denoted (in this chapter) by U. We also use 9 to denote
the emptg or n
u
Z
Z set, i.e. the set which contains no elements; this set is regarded as a subset
of every other set. Thus for any set A, we have 9C AC U.
Example 1.1: The sets A and B above can also be written as
A = {x : x is an odd number, z < 10) and B = {2,3,6, 7,11,13}
Observe that 9EA but 9 4 B, and 11E B but 114 A ; whereas 3 EA and 3 EB, and
6 B A and 6 B B .
1

2 SET THEORY [CHAP. 1
Example 1.2: We use the following special symbols:
N = the set of positive integers: 1,2, 3, ...
Z = the set of integers: ...,-2, -1, 0, 1, 2, ...
R = the set of real numbers.
Thus we have N C Z C R .
Example 1
.
3
: Intervals on the real line, defined below, appear very often in mathematics. Here
a and b are real numbers with a < b.
Open interval from a to b = (a,b) = {x : a < x < b}
Closed interval from a to b = [a,b] = {x : a f x 4 b}
Open-closed interval from a to b = (a,b] = {x : a < x f b}
Closed-open interval from a to b = [a,b) = {x : a f x < b}
The open-closed and closed-open intervals are also called half-open intervals.
Example 1.4: In human population studies, the universal set consists of all the people in the
world.
Example 15: Let C = {x : x2 = 4, x is odd}. Then C = (B; that is, C is the empty set.
The following theorem applies.
Theorem 1
.
1
: Let A, B and C be any sets. Then: (i)ACA; (ii)if A c B and B C A then
A = B; and (iii) if A c B and B c C then A c e .
We emphasize that A c B does not exclude the possibility that A = B. However, if
A C B but A # B, then we say that A is a proper subset of B. (Some authors use the symbol
c for a subset and the symbol c only for a proper subset.)
SET OPERATIONS
Let A and B be arbitrary sets. The union of A and B, denoted by A UB, is the set of
elements which belong to A or to B:
A U B = { x : x E A or x E B }
Here “or” is used in the sense of and/or.
The intersection of A and B, denoted by A n B , is the set of elements which belong to
both A and B:
A n B = { x : x E A and x E B }
If A nB = @, that is, if A and B do not have any elements in common, then A and B
are said to be disjoint.
The differenceof A and B or the relative complement of B with respect to A, denoted by
A B, is the set of elements which belong to A but not to B:
A

B = { x : x E A , x 4 B )
Observe that A

B and B are disjoint, i.e. (AB) nB = @.
The absolute complement or, simply, complement of A, denoted by A“, is the set of
elements which do not belong to A:
Ac = ( x : x E U , x B A )
That is, A” is the difference of the universal set U and A.

CHAP. 1
1 SET THEORY 3
Example 1.6: The following diagrams, called Venn diagrams, illustrate the above set operations.
Here sets are represented by simple plane areas and U,the universal set, by
the area in the entire rectangle.
Example 1.7: Let A = {1,2,3,4} and B = {3,4,5, 6} where U = {1,2,3, ...}. Then
A U B = (1,2, 3, 4, 5, 6) A n B = {3,4}
AB = {1,2} Ac = (5, 6, 7, ...}
Sets under the above operations satisfy various laws or identities which are listed
in the table below (Table 1). In fact, we state
Theorem 1.2: Sets satisfy the laws in Table 1.
~ ~~
LAWS OF THE ALGEBRA OF SETS
Idempotent Laws
la. A u A = A lb. A n A = A
Associative Laws
2a. (AuB)uC = A u ( B u C ) 2b. ( A n B ) n C = A n ( B n C )
Commutative Laws
3a. A u B = B u A 3b. A n B = B n A
Distributive Laws
4a. A u ( B n C ) = ( A u B ) n ( A u C ) 4b. A n ( B u C ) = ( A n B ) u ( A n C )
Identity Laws
6a. A U @ = A 5b. A n U = A
6a. A U U = U 6b. A n @ = @
Complement Laws
7a. A uAC = U 7b. A n A c = @
8a. (Ac)c = A 8b. Uc = @, @C = U
De Morgan’s Laws
9a. ( A u B ) ~
= Acn Bc 9b. (AnB)c = Acu BE
Table 1

Remark: Each of the above laws follows from an analogous logical law. For example,
A n B = { x : x E A and x E B } = { x : x E B and x E A } = B n A
~
Here we use the fact that the composite statement “ p and q”, written p ~ isqlogically
equivalent to the composite statement “ q and p”, i.e. q A p.
The relationship between set inclusion and the above set operations follows:
Theorem 1
.
3
: Each of the following conditions is equivalent to ACB:
(i) A n B = A (iii) BccAc (v) BUAc = U
(ii) AUB = B (iv) AnBC= 9
FINITE AND COUNTABLE SETS
Sets can be finite or infinite. A set is finite if it is empty or if it consists of exactly n
elements where n is a positive integer; otherwise it is infinite.
Example 1.8: Let M be the set of the days of the week; that is,
M = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
Then M is finite.
Example 1.9: Let P = {z: x is a river on the earth}. Although it may be difficult to count the
number of rivers on the earth, P is a finite set.
Example 1.10: Let Y be the set of (positive) even integers, i.e. Y = {2,4,6, ...}. Then Y is an
infinite set.
Example 1.U: Let Z be the unit interval of real numbers, i.e. Z = {x : 0 f z 1
)
. Then Z is
f
also an infinite set.
A set is countable if it is finite or if its elements can be arranged in the form of a
sequence, in which case it is said to be countablg infinite;otherwise the set is uncountable.
The set in Example 1.10 is countably infinite, whereas it can be shown that the set in
Example 1
.
1
1is uncountable.
PRODUCT SETS
Let A and B be two sets. The product set of A and Bydenoted by A X B, consists of all
ordered pairs (a,b) where a E A and b E B:
A x B = {(a,b):a E A , b E B }
The product of a set with itself, say A x A, is denoted by A2.
Example 1.12: The reader is familiar with the cartesian plane R
2 = R X R as shown below.
Here each point P represents an ordered pair (a,b) of real numbers, and vice versa.
2
b ? P
1
, o 0 0 1
- 3 - 2 - 1 i i a i
Example 1.13: Let A = (1,2,3} and B = {a,
b}. Then

5
CHAP. 11 SET THEORY
The concept of product set is extended to any finite number of sets in a natural way.
The product set of the sets AI,A2, ...,A,, written A1 x A2 x - x A,, is the set of all
ordered m-tuples (a1,a ~ ,
...,am) where ai E At for each i.
CLASSES OF SETS
Frequently the members of a set are sets themselves. For example, each line in a set
of lines is a set of points. To help clarify these situations, we usually use the word class
or family for such a set. The words subclass and subfamily have meanings analogous to
subset.
Example 1.14: The members of the class {{2,3}, {2}, {5,6}} are the sets {2,3}, (2) and (5, 6}.
Example 1.15: Consider any set A. The power set of A , denoted by "(A), is the class of all sub-
sets of A. In particular, if A = {a,
b, c}, then
W A ) = {A, {a,
w,{a,c), {b,cl, {a>,
{a>, {c>, P}
In general, if A is finite and has n elements, then T(A) will have 2n elements.
A partition of a set X is a subdivision of X into nonempty subsets which are disjoint and
whose union is X , i.e. is a class of nonempty subsets of X such that each a E X belongs to
a unique subset. The subsets in a partition are called ct?ZZs.
Then (i) is not a partition of X since 7 € X but 7 does not belong to any of the cells.
Furthermore, (ii) is not a partition of X since 5 E X and 5 belongs to both {1,3,5}
and {5,7,9}. On the other hand, (iii) is a partition of X since each element of X
belongs to exactly one cell.
When we speak of an indexed class of sets {A*:i E I ) or simply {Ai},we mean that there
is a set Ai assigned to each element i E I . The set I is called the indexing set and the sets
At are said to be indexed by I. When the indexing set is the set N of positive integers, the
indexed class {A1,A2, ...} is called a sequence of sets. By the union of these A*, denoted
by UiEr At (or simply UiAi), we mean the set of elements each belonging to at least one of
the Ai; and by the intersection of the Ai, denoted by niE1 At (or simply n Ai), we mean the
set of elements each belonging to every Ai. We also write
U,"=, Ai = ArUAzU and n
;
=
, Ai = A1nA2n
for the union and intersection, respectively, of a sequence of sets.
Definition: A nonempty class CA of subsets of U is called an algebra (U-algebra)of sets if:
(i) the complement of any set in eA belongs to CA; and
(ii) the union of any finite (countable) number of sets in CA belongs to CA;
that is, if CA is closed under complements and finite (countable) unions.
It is simple to show (Problem 1.30)that an algebra (a-algebra) of sets contains U and
Iz, and is also closed under finite (countable) intersections.

Solved Problems
SETS, ELEMENTS, SUBSETS
1.1. Let A = {x:3x=6}. Does A = 2 ?
A is the set which consists of the single element 2, that is, A = (2). The number 2 belongs
to A; it does not equal A. There is a basic difference between an element p and the singleton
set (PI.
1.2. Which of these sets are equal: {r,s,t}, {t,s, r } ,{s, r, t},{t,r,s} ?
They are all equal. Order does not change a set.
1
.
3
. Determine whether or not each set is the null set:
(i)X = {x:x2= 9, 2x = 4}, (ii) Y = {x:x # x}, (iii) 2 = {x:x +8= 8
}
.
(i) There is no number which satisfies both 2 2 = 9 and 22 = 4; hence X is empty, i.e. X = 9.
(ii) We interpret “=” to mean “is identical with” and so Y is also empty. In fact, some texts define
the empty set as follows: @ ( 2 : z#z).
(iii) The number zero satisfies x +8 = 8; hence 2 = (0). Accordingly, 2 is not the empty set
since it contains 0. That is, 2# (B.
1.4. Prove that A = {2,3,4,5} is not a subset of B = {x:x is even).
It is necessary to show that a t least one element in A does not belong to B. Now 3 € A and,
since B consists of even numbers, 3B B; hence A is not a subset of B.
1.5. Let V = {d}, W = {c, d}, X = {a,
b, c } , Y = {a,
b} and Z = {a,
b,d}. Determine
whether each statement is true or false:
(i)Y c X, (ii) W f:2, (iii) Z V, (iv) V c X, (v) X = W, (vi) W c Y.
(i) Since each element in Y is a member of X,Y cX is true.
(ii) Now a E 2but a B W ;
hence W # 2 is true.
(iii) The only element in V is d and it also belongs to 2;hence 23 V is true.
(iv) V is not a subset of X since d E V but d B X;hence V C X is false.
(v) Now a E X but a B W; hence X = W is false.
(vi) W is not a subset of Y since c E W but c 6Z Y; hence W c Y is false.
1.6. Prove: If A is a subset of the empty set Q), then A = 9.
The null set Q, is a subset of every set; in particular, @ C A . But, by hypothesis, A c g ;
hence A = 9.
1
.
7
. Prove Theorem l.l(iii): If A c B and B c C, then A C C.
We must show that each element in A also belongs to C. Let z E A. Now A C B implies
x E B. But B c C; hence zE C. We have shown that 2: E A implies zE C, that is, that A C C.
1.8. Which of the following sets are finite?
(i) The months of the year. (iv) The set Q of rational numbers.
(ii) {I,2,3, ...,99,100). (v) The set R of real numbers.
(iii) The number of people living on the earth.
The first three sets are finite; the last two are infinite. (It can be shown that Q is countable
but R is uncountable.)

7
CHAP.1
3 SET THEORY
1.9. Consider the following sets of figures in the Euclidean plane:
A = {x : x is a quadrilateral) C = {x : x is a rhombus}
B = {z: x is a rectangle} D = {x : x is a square}
Determine which sets are proper subsets of any of the others.
Since a square has 4 right angles it is a rectangle, since it has 4 equal sides it is a rhombus,
and since it has 4 sides it is a quadrilateral. Thus
D c A , D c B and D C C C
that is, D is a subset of the other three. Also, since there are examples of rectangles, rhombuses and
quadrilaterals which are not squares, D is a proper subset of the other three.
In a similar manner we see that B is a proper subset of A and C is a proper subset of A. There
are no other relations among the sets.
1.10. Determine which of the following sets are equal: @ {0}, {Iz)).
Each is different from the other. The set (0) contains one element, the number zero. The
set (b contains no elements; it is the empty set. The set {(b} also contains one element, the null set.
SET OPERATIONS
1
.
1
1
. Let U = {l,Z, ...,8,9}, A = {1,2,3,4}, B = {2,4,6,8} and C = {3,4,5,6}. Find:
(i) Ac, (ii) AnC, (iii) (AnC)",(iv) A uB, (v) B

C
.
(i) Ac cansists of the elements in U that are not in A; hence Ac = {6,6,7,8,9}.
(ii) A nC consists of the elements in both A and C; hence A n C = {3,4).
(iii) (A nC)" consists of the elements in U that are not in A nC. Now by (ii), A nC = {3,4} and
so (AnC)C= {1,2,5,6,7,8,9).
(iv) A uB consists of the elements in A or B (or both): hence A U B = {1,2,3,4,6,8).
(v) B C consists of the elements in B which are not in C; hence B C = {2,8).
1.12. Let U = {a,b,c,d,e}, A = {a,b,d} and B = {b,d,e}. Find:
(i) AUB (iii) Bc (v) AcnB (vii) AcnBc (ix) (AnB)c
(ii) B n A (iv) B

A (vi) AUBc (viii) BCAc (x) (AM?)"
(i) The union of A and B consists of the elements in A or in B (or both); hence A UB = {a,b,d, 6).
(ii) The intersection of A and B consists of those elements which belong to both A and B; hence
AnB = {b,d}.
(iii) The complement of B consists of the letters in U but not in B; hence Bc = (a,c).
(iv) The difference B

A consists of the elements of B which do not belong to A; hence
B

A = (e).
(v) Ac = {a, e) and B = {b, d, e); then AcnB = {e).
(vi) A = {a,
b, d} and Bc = (a,
c}; then A uBc = (a,b, c, d).
(vii) and (viii). Ac = {c, e) and Bc = {U,c); then
AcnBc = {c} and BcAc = {U)
(ix) From (ii), A nB = {b,d); hence (A nB)c = (a,c, e).
(x) From (i), A u B = {a,b, d, e); hence (AuB)c = {c).

1.13. In the Venn diagram below, shade: (i) Bc, (ii) (AuB)c,(iii) (BA)c, (iv) AcnBc.
Bc consists of the elements which do not belong to B; hence shade the area outside B as follows:
Bc is shaded.
First shade A U B ; then (A uB)c is the area outside A UB:
A U Bis shaded. (A uB)cis shaded.
First shade BA, the area in B which does not lie in A; then (BA)c is the area outside
BA:
B A is shaded. (B A)c is shaded.
First shade Ac, the area outside of A, with strokes slanting upward to the right (////), and
then shade Bc with strokes slanting downward to the right (); then AcnBc is the
cross-hatched area:
Ac and Bc are shaded. AcnBc is shaded.
Observe that (A u B ) ~
= ACnBc, as expected by De Morgan's law.

9
CHAP. 1
1 SET THEORY
1
.
1
4
. Prove: B

A =BnAc. Thus the set operation of difference can be written in
terms of the operations of intersection and complementation.
BA = {z:x E B , z 4 A } = { z : x E R , z E A c } = B n A c
1
.
1
5
. Prove: For any sets A and B, A n B c A c A UB.
Let 2: E A nB; then zE A and 1~ E B. In particular, zE A. Since x E A nB implies 5 € A,
A nB c A. Furthermore if x E A, then zE A or z E Byi.e. x E A UB. Hence A c A UB. In other
words, A n B c A c AuB.
1
.
1
6
. Prove Theorem 1.3(i): A c B if and only if A nB = A.
Suppose A cB. Let z E A ; then by hypothesis, zE B. Hence 1~ EA and x E B, i.e. 1~ E A nB.
Accordingly, A c A nB. On the other hand, it is always true (Problem 1.15) that A n B c A.
Thus A n B = A.
Now suppose that A n B = A. Then in particular, A c A nB. But it is always true that
A n B c B. Thus A c A n B c B and so, by Theorem 1.1, A c B.
PRODUCT SETS
1
.
1
7
. Let M = {Tom,Marc, Erik} and W = {Audrey, Betty}. Find M X W.
M X W consists of all ordered pairs (a,
b) where a E M and b E W. Hence
M X W = ((Tom, Audrey), (Tom,Betty), (Marc, Audrey),
(Marc, Betty), (Erik, Audrey), (Erik,Betty)}
1
.
1
8
. Let A = {1,2,3}, B = {2,4} and C = {3,4,5}. Find A x B X C .
A convenient method of finding A X B X C is through the so-called “tree diagram” shown b
e
l
o
w
:
The “tree” is constructed from the left to the right. A X B X C consists of the ordered triples
listed to the right of the “tree”.
1
.
1
9
. Let A = {a,b}, B = {2,3} and C = {3,4}. Find:
(i) A x (BuC), (ii) (A x B)U ( Ax C), (iii)A x (BnC), (iv) ( Ax B) n(A x C).
(i) First compute BUC = {2,3,4}. Then
A x (BUC) = {(a,
21, (a,
3), (a,
41, (b,21, (b,3)s (b,4))

10 SET THEORY [CHAP.
1
Then compute the union of the two sets:
(A x B) U ( A x C) = {(a,2), (a,3), (6,2), (b,3), (a,4), (b,4))
Observe from (i) and (ii) that .
Ax(BuC) = ( A x B ) u ( A X C )
(iv) Now A X B and A x C were computed above. The intersection of A X B and A X C consists of
those ordered pairs which belong to both sets:
(A x B) n (A x C) = {(a,3), (6,311
Observe from (iii) and (iv) that
A x (BnC) = (A x B ) n (A x C)
1
.
2
0
. Prove: A x (BnC) = ( Ax 23) n (A x C).
-4 x (BnC) = {(x,y) : x € A , y E BnC}
= {(x,y) : x E A, y E B, y E C}
= {(x,2/): (%?h E A x c>
E A x B, (
%
!
I
)
= (A x B) n (A x C)
1.21. Let S = {a,b}, W = {1,2,3,4,5,6}and V = {3,5,7,9}. Find (SX W) n (SX V).
The product set (SX W) n (SX V) can be found by first computing S X W and S X V, and
then computing the intersection of these sets. On the other hand, by the preceding problem,
(Sx W) n (S X V )= S x (WnV). Now W nV = {3,5}, and so
(S x W )n (S x V ) = S x (Wnv) = {(a,3), (a,5), (b,3), (b,5))
1
.
2
2
. Prove: Let A cB and CCD; then ( Ax C) c (Bx D).
Let (x,y) be any arbitrary element in A X C; then x: E A and y E C. By hypothesis, A C B and
C c D ; hence x E B and y E D. Accordingly (x,y) belongs to B XD. We have shown that
(2,y) E A X C implies (2,y) E B X D; hence (A X C) c (B X D).
CLASSES OF SETS
1
.
2
3
. Consider the class A = {{2,3},(431, (6)). Which statements are incorrect and
why? (i) {4,5}C A , (ii) {4,5} E A, (iii) {{4,5}}cA.
The members of A are the sets {2,3}, (4,6) and (
6
)
. Therefore (ii) is correct but (i) is an
incorrect statement. Moreover, (iii) is also a correct statement since the set consisting of the single
element {4,5} is a subclass of A .
1.24. Find the power set T(S) of the set S = {1,2,3}.
The power set T(S) of S is the class of all subsets of S; these are {1,2,3}, {1,2}, {1,3), {2,3},
{l}, {2}, (3) and the empty set @. Hence
T(S) = {S, {1,3), {2,3}, {1,2), {I}, (21, (311 @I
Note that there are 23 = 8 subsets of S.

11
CHAP. 1
3 SET THEORY
1.25. Let X = (a,b, c,d, e,f ,g } , and let:
(i) AI = {a,c,e}, A2 = { b } ,A3 = { d , g } ;
(ii) B1 = {a,e,g}, BZ= {c,d}, B3 = {b,e , f } ;
(iii) CI = {a,b,e,g}, CZ={c}, C
3 = {d,f};
(iv) D
1 = {a,b, c,d, e,f,s > .
Which of (AI,A2,A3}, {BI,
Bz,B3}, {CI,CZ,C3}, (01)
are partitions of X?
(i) {Al,A2,A3} is not a partition of X since f E X but f does not belong to either A,, AS, or A,.
(ii) {Bl,B,, B3) is not a partition of X since e E X belongs to both Bl and B3.
(iii) {C1,C2,C3} is a partition of X since each element in X belongs to exactly one cell, i.e.
X = ClUC2UC3 and the sets are pairwise disjoint.
(iv) {Ill}is a partition of X.
1.27. Let N be the set of positive integers and, for each n E N, let
A, = {x : x is a multiple of n} == {n,2n,3n, ...}
Find (i) A3nA5, (ii) A4nAs, (iii) U ~ E P
Ai, where! P is the set of prime numbers,
2,3,5,7,11,... .
(i) Those numbers which are multiples of both 3 and 6 are the multiples of 15; hence A,nA5 = A15.
(ii) The multiples of 12 and no other numbers belong to both A4 and A,; hence A4nA, = A12.
(iii) Every positive integer except 1is a multiple of at least one prime number; hence
1.28. Prove: Let {Ai :i E I } be an indexed class of sets and let io E I . Then
niEr
Ai c Ai, c UiEi Ai
Let x E niEI Ai; then x E Ai for every i E I. In particular, x E Aio. Hence niEI Ai c Aio.
Now let y E Ai,. Since ioE I, y E U j E I Ai. Hence Aio c u ~ E I
Ai.
1.29. Prove (DeMorgan's law): For any indexed class (Ai :i E I } , (UiAi)' = niA:.
(uiA,)c =
: {x : z B uiAi) = {x : x @ A i for every i} = (x : x € A t for every i} = niAt
1.30. Let CA be an algebra (a-algebra) of subsets of U. Show that: (i) U and 9 belong to
CA; and (ii) d is closed under finite (countable) intersections.
Recall that CA is closed under complements and finite (countable) unions.
(i) Since CA is nonempty, there is a set A E CA. Hence the complement Ac E CA, and the union
E CA. Also the complement (3 = Uc E CA..
U = A u A ~
(ii) Let {Ai} be a finite (countable) class of sets belonging to CA. By De Morgan's law (Problem 1.29),
(uiA;)c = niAT = n,A,. Hence niAi belongs to CA, as required.

12 SET THEORY [CHAP.
1
Supplementary Problems
SETS, ELEMENTS, SUBSETS
1.31. Write in set notation:
(a)R is a subset of T. (d) M is not a subset of S.
(b) x is a member of Y. (e) z does not belong to A.
(c) The empty set. (f) R belongs to CA.
1.32. Rewrite explicitly giving the elements in each set:
(i) A = {x: x 2 - x - 2 = 0 }
(ii) B = {
x : x is a letter in the word “follow”}
(iii) C = {
z : x2 = 9, x -3 = 5 )
(iv) D = {x : x is a vowel}
(v) E = {x : z is a digit in the number 2324)
1.33. Let A = (1,2,...,8,9}, B = {2,4,6,8}, C = (1,3,6,7,9), D = {3,4,5} and E = {3,6}.
Which sets can equal X if we are given the following information?
(i) X and B are disjoint. (ii) X C D but X $ B . (iii) X C A but X e C . (iv) X C C but X $ A .
1.35. Let A = {l,O}. State whether or not each statement is correct:
(i) (0) E A, (ii) @ E A , (iii) (0) C A , (iv) 0 E A, (v) 0 c A.
1.36. State whether each set is finite or infinite:
(i) The set of lines parallel to the x axis.
(ii) The set of letters in the English alphabet.
(iii) The set of numbers which are multiples of 6.
(iv) The set of animals living on the earth.
(v) The set of numbers which are solutions of the equation $27 +26x18-17x11-t7~ -10 = 0.
(vi) The set of circles through the origin (0,O).
SET OPERATIONS
1.37. Let U = {a,b, c, d, e, f,g}, A = {a,
b, c, d, e>, B = {a,c, e, g } and C = {b, e, f,g}. Find:
(i) A U C (iii) C

B (v) CcnA (vii) (A Bc)c
(ii) B nA (iv) Bc U C (vi) ( AClc (viii) (AnAc)c
1.38. In the Venn diagrams below, shade (i) W V, (ii) VcU W , (iii) V n Wc, (iv) Vc Wc.
1.39. Prove: (a)A UB = (AcnBC)c; (b) A B = A nBc. (Thus the union and difference operations can
be defined in terms of the operations of intersection and complement.)
1.40. Prove Theorem 1.3(ii): A c B if and only if A UB = B.
1.41. Prove: If A nB = @, then A c Bc.
1.42. Prove: AcBc = BA.

13
CHAP. 1
1 SET THEORY
1.43. Prove: A c B implies A U (BA) = B.
1.44. (i) Prove: A n (B C) = (AnB) (AnC).
(ii) Give an example to show that A U (B C ) # (AU B ) (AU C).
PRODUCT SETS
1.45. Let W = {Mark, Eric, Paul} and let V = {Eric, David}. Find:
(i) W x V, (ii) V x W , (iii) V2 = V X V.
1.46. Let A = {2,3}, B = {1,3, 5 } and C = {3,4}. Construct the “tree diagram” of A X B X C and then
find A X B X C. (See Problem 1.18.)
1.47. Let S = {a, b, c}, T = {b, c, d} and W = {a,d}. Construct the tree diagram of S X T X W and
then find S X T X W .
1.48. Suppose that the sets V, W and 2 have 3, 4 and 6 elements respectively. Determine the number
of elements in (i) V X W X 2, (ii) 2 X V X W, (iii) W X 2 X: V.
1.49. Let A = B nC. Determine if either statement is true:
(i) A X A = (B X B) n(CX C
)
, (ii) A X A = (B X C)n (CX B).
1.50. Prove: A X (BuC) = (AX B) U (A X C).
CLASSES OF SETS
1.51. Let A,, = {z: z is a multiple of n} = {n,2n, 3n, ...}, where n E N, the positive integers. Find:
(i) A, f~
A,; (ii) A , nA,; (iii) A3U A12; (iv) A, nAI,; (v) A, U Ast, where 8, t E N; (vi) A, nASt,
where s,t E N. (vii) Prove: If J cN is infinite, then n
i
,
: J A, = @.
1.52. Find the power set %?(A) of A = (1,2,3,4} and the power set T(B) of = (1, {2,3}, 4).
1.53.
1.54. Find all partitions of V = {1,2,3}.
1.55. Let [Al,A2,...,Am]and [B,,B,, ...,E?,,]be partitions of a set X. Show that the collection of sets
[A,nBj: i = 1 , . ..,m, j = I., ...,n]
is also a partition (called the cross partition) of X.
1.56. Prove: For any indexed class {Ai:iEI} and any set B,
(a) B u ( n i A i ) = ni(BuAi), (b) B n ( u , A , ) = ui(BnAi).
1.57. Prove (De Morgan’s law): (niAi)c= U,A;.
1.58. Show that each of the following is an algebra of subsets of U:
(i)CA = {@, U}; (ii) 93 = (@,A,AC,U};(iii) T ( U ) ,the power set of U.
1.59. Let CA and 5
3 be algebras (a-algebras) of subsets of U. Prove that the intersection CAng is also
an algebra (a-algebra) of subsets of U.

Answers to Supplementary Problems
1.31. (a)R c T, (b) x E Y,(c) e), (d) M Q!S, (e) z 4A , ( f ) R E CA.
1.32. (i) A = {--1,2}, (ii) B = {f,0, LW}, (iii) C = e), (iv) D = {a,
e, i,o,u}, (v) E = {2,3,4}.
1.33. (i) C and E, (ii) D and E, (iii) A, B and D, (iv) none.
1.34. All the statements are true except (v).
1.35. (i) incorrect, (ii) incorrect, (iii) correct, (iv) correct, (v) incorrect.
1.36. (i)infinite, (ii)finite, (iii)infinite, (iv) finite, (v) finite, (vi) infinite.
1.37. (i) A UC = U
(ii) B n A = {a,
c, e}
(iii) C B = {b,f }
(iv) BcuC= { b , d , e , f , g }
1.38. (a)
vcu w V nWc
wv vcu w V n Wc vc wc
Observe that VcU W = U and V n W c= e) in case (6)where VC W.
1.45. (i) W X V = {(Mark, Eric), (Mark,David), (Eric, Eric), (Eric, David), (Paul, Eric), (Paul,David)}
(ii) V X W = ((Eric, Mark), (David, Mark), (Eric, Eric), (David, Eric), (Eric, Paul), (David, Paul)}
(iii) V X V = {(Eric,Eric), (Eric, David), (David, Eric), (David, David)}
1.46.
t
3
4
<
2
The elements of A X B X C are the ordered triplets to the right of the tree diagram above.

16
CHAP. 11 SET THEORY
1.47.
The elements of S X T X W are the ordered triplets listed to the right of the tree diagram.
1.48. Each has 60 elements.
1.49. Both are true: A X A = (BX B) n(CX C) = (Bx C)n(CX B).
1.51.
1.52.
1.53. (i)no, (ii) no, (iii) yes, (iv) yes.

Chapter 2
Techniques of Counting
INTRODUCTION
In this chapter we develop some techniques for determining without direct enumeration
the number of possible outcomes of a particular experiment or the number of elements in
a particular set. Such techniques are sometimes referred to as combinatorial analysis.
FUNDAMENTAL PRINCIPLE OF COUNTING
We begin with the following basic principle.
Fundamental Principle of Counting: If some procedure can be performed in nl dif-
ferent ways, and if, following this procedure, a second procedure can be
performed in n 2 different ways, and if, following this second procedure, a
third procedure can be performed in n3different ways, and so forth; then the
number of ways the procedures can be performed in the order indicated is the
product n~ n
2 n 3 ....
Example 2.1: Suppose a license plate contains two distinct letters followed by three digits with
the first digit not zero. How many different license plates can be printed?
The first letter can be printed in 26 different ways, the second letter in 25 dif-
ferent ways (since the letter printed first cannot be chosen for the second letter),
the first digit in 9 ways and each of the other two digits in 10 ways. Hence
26 25 9 10 10 = 585,000
different plates can be printed.
FACTORIAL NOTATION
The product of the positive integers from 1to n. inclusive occurs very often in mathe-
matics and hence is denoted by the special symbol n ! (read “n factorial”):
~ 2 != 1 0 2 . 3 .- - * ( n - 2 ) ( n - l ) n
It is also convenient to define O ! = 1.
Example 2.2: 2! = 1 . 2 = 2, 3! = 1 . 2 0 3 = 6, 4! = 1 . 2 . 3 . 4 = 24,
5! = 5 * 4 ! = 5.24 = 120, 6! = 6 * 5 ! = 6.120 = 720
Example 2.3:
8 ! - 8 ~ 7 * 6 !
-
6! 6!
- - -- 8 . 7 = 56 12*11*10 =
1 2 - 1 1 * 1 0 * 9 ! -
-
9! 9!
PERMUTATIONS
An arrangement of a set of n objects in a given order is called a permutation of the
objects (taken all at a time). An arrangement of any r L n of these objects in a given
order is called an r-permutation or a permutation of the n objects taken r at a time.
Example 2.4: Consider the set of letters a, b, c and d. Then:
(i) bdca, dcba and acdb are permutations of the 4 letters (taken all at a time);
(ii) bad, adb, cbd and bca are permutations of the 4 letters taken 3 at a time;
(iii) ad, cb, d a and bd are permutations of the 4 letters taken 2 at a time.
16

17
CHAP. 21 TECHNIQUES OF COUNTING
The number of permutations of n objects taken r at a time will be denoted by
P(n,r )
Before we derive the general formula for P(n,r ) we consider a special case.
Example 25: Find the number of permutations of 6 objects, say a,b, c, d, e,f, taken three a t a
time. In other words, find the number of “three letter words” with distinct letters
that can be formed from the above six letterls.
Let the general three letter word be represented by three boxes:
Now the first letter can be chosen in 6 different ways; following this, the second
letter can be chosen in 5 different ways; and, following this, the last letter can be
chosen in 4 different ways. Write each number in its appropriate box as follows:
Thus by the fundamental principle of counting there are 6 5 4 = 120 possible
three letter words without repetitions from the six letters, or there are 120 permu-
tations of 6 objects taken 3 a t a time. That is,
P(6,3) = 120
The derivation of the formula for P(n,r ) follows the procedure in the preceding example.
The first element in an r-permutation of n-objects can be chosen in n different ways; follow-
ing this, the second element in the permutation can be chosen in n - 1 ways; and, following
this, the third element in the permutation can be chosen in n - 2 ways. Continuing in this
manner, we have that the rth (last)element in the r-permutation can be chosen in n - (r- 1)=
n -Y +1ways. Thus
n f
Theorem 2.1: P(n,r ) = n(n- l)(n- 2) .-(n-r +1) = -
-
( n - r ) !
The second part of the formula follows from the fact that
r )! - n !
-
n(n- l)(n- 2) - (n-Y +1) = n(n- I)@-2). -(n--r +1) (n-
( n - r ) ! (n-r )!
In the special case that r = n, we have
P(n,n) = n ( n - l ) ( n - 2 ) - 3 * 2 * 1 = n !
Namely,
Corollary 2.2: There are n ! permutations of n objects (taken all at a time).
Example 2.6: How many permutations are there of 3 objects, say, a,b and c?
By the above corollary there are 3! = 1 2 3 = 6 such permutations. These
are abc, acb, bac, bca, cab, cba.
PERMUTATIONS WITH REPETITIONS
Frequently we want to know the number of permutations of objects some of which are
alike, as illustrated below. The general formula follows.
Theorem 2.3: The number of permutations of n objects of which nl are alike, n
2 are
alike, ...,n+ are alike is
n !

18 TECHNIQUES O F COUNTING [CHAP. 2
We indicate the proof of the above theorem by a particular example. Suppose we want
to form all possible 5 letter words using the letters from the word DADDY. Now there
are 5! = 120 permutations of the objects DI,A,D2,D3, Y where the three D’s are dis-
tinguished. Observe that the following six permutations
produce the same word when the subscripts are removed. The 6 comes from the fact that
there are 3! = 3-201 = 6 different ways of placing the three D’s in the first three posi-
tions in the permutation. This is true for each of the other possible positions in which the
D’s appear. Accordingly there are
5 ! 120
_ - - = 20
3! - 6
different 5 letter words that can be formed using the letters from the word DADDY.
Example 2.7: How many different signals, each consisting of 8 flags hung in a vertical line,
can be formed from a set of 4 indistinguishable red flags, 3 indistinguishable white
flags, and a blue flag? We seek the number of permutations of 8 objects of which
4 are alike (the red flags) and 3 are alike (the white flags). By the above theorem,
there are
different signals.
ORDERED SAMPLES
Many problems in combinatorial analysis and, in particular, probability are concerned
with choosing a ball from an urn containing n balls (or a card from a deck, or a person
from a population). When we choose one ball after another from the urn, say r times, we
call the choice an ordered sample of size r. We consider two cases:
(i) SampZing with replacement. Here the ball is replaced in the urn before the next ball
is chosen. Now since there are n different ways to choose each ball, there are by the
fundamental principle of counting
r times
&
n * n - n - . . n = nr
different ordered samples with replacement of size r.
(ii) Sampling without replacement. Here the ball is not replaced in the urn before the next
ball is chosen. Thus there are no repetitions in the ordered sample. In other words,
an ordered sample of size r without replacement is simply an r-permutation of the
objects in the urn. Thus there are
P(n,r ) = n(n- l)(n-2)- .-(n-r +1) = -
n !
( n - r ) !
different ordered samples of size r without replacement from a population of n objects.
Example 2.8: In how many ways can one choose three cards in succession from a deck of 52 cards
(i) with replacement, (ii) without replacement? If each card is replaced in the
deck before the next card is chosen, then each card can be chosen in 52 different
ways. Hence there are
52052.52 = 523 = 140,608
different ordered samples of size 3 with replacement.

-
-
CHAP. 2
1 TECHNIQUES OF COUNTING 19
On the other hand if there is no replacement, then the first card can be chosen
in 52 different ways, the second card in 51 different ways, and the third and last
card in 60 different ways. Thus there are
52*51*50 = 132,600
different ordered samples of size 3 without replacement.
BINOMIAL COEFFICIENTS AND THEOREM
The symbol (
:
) ,read "nCr", where r and n are positive integers with r 6 n, is defined
as follows: '
(
:
) =
n(n- l)(n-2). ..(n- r +1)
1 2 3. *(r-l ) r
These numbers are called the binomial coefficientsin view of Theorem 2.5 below.
Observe that (:>has exactly r factors in both the numerator and denominator. Also,
(n-r +l)(n-r )! - n !
n(n- 1 ) .* .(n- r + 1) - n(n- 1)
(
;
) = 1 * 2 * 3 - . ( r - 1)r 1 * 2 * 3 -* ( r - l ) r ( n - r ) ! r !(n-r )!
Using this formula and the fact that n - ( n - r ) = r, we obtain the following important
relation.
Lemma 2.4: (
. E.
) = (y) or, in other words, if a +b := n then (
:
) = (:)*
1 0 * 9 * 8 * 7 * 6 * 5 . 4 120 or 1 0 . 9 . 8
~
ExampIe 2.10: (y)= 1 * 2 * 3 * 4 * 5 * 6 * 7 (tp> = (;") = 1
.
2
.
3= 120
Note that the second method saves both space and time.
Remark: Motivated by the second formula for
(3and the fact that O ! = 1, we define:
. .
n f O !
-= 1 and, in particular,
O ! O !
The Binomial Theorem, which is proved (Problem 2.18) by mathematical induction, gives
the general expression for the expansion of (a+b)".
Theorem 2.5 (Binomial Theorem):
Example 2.11: (a+b)5 = a
5 + 6a4b + 02 a3b2 + FAa2b3 + 5ab4 + b5
1 - 2
= a
5 + 5a4b + lOa3b2 + lOa2b3 + Sub4 + b5
(a+ b)6 = a
6 + 6a5b + g a 4 b 2 + -
' a3b3 + E a 2 b 4 + 6ab5 + be
1 . 2 . 3
= a
6 -I- 6a5b + 15aW + 2OaSb3 + 15a2b4 + 6ab5 + be

The following properties of the expansion of (a+b)" should be observed:
(i) There are n +1terms.
(ii) The sum of the exponents of a and b in each term is n.
(iii) The exponents of a decrease term by term from n to 0; the exponents of b increase
term by term from 0 to n.
(iv) The coefficient of any term is where 7c is the exponent of either a or b. (This fol-
lows from Lemma 2.4.)
(v) The coefficients of terms equidistant from the ends are equal.
We remark that the coefficients of the successive powers of a +b can be arranged in a
triangular array of numbers, called Pascal's triangle, as follows:
(a+ b)O = 1 1
(a+b)l = a + b 1 1
(a+b)2 = a
2 + 2ab + b2 1 2 1
( a + b)3 = a
3 + 3a2b + 3ab2 + b3 1 3 3 1
( a + b)4 = a
4 + 4a3b + 6aW + 4ab3 + b4 1
(a+ b)5 = a
5 + 5a4b + lOa3b2 + lOa2b3 + 5ab4 + b5
( a + b)6 = a
6 + 6a5b + 15a4b2 +20aW + 16a2b4 + 6ab5 + b6
........................ ........................................ ............................
Pascal's triangle has the following interesting properties.
(a) The first number and the last number in each row is 1.
(b) Every other number in the array can be obtained by adding the two numbers appearing
directly above it. For example: 10 = 4 +6, 15 = 5 + 10, 20 = 10 +10.
We note that property (b)above is equivalent to the following theorem about binomial
coefficients.
Theorem 2.6: ("Y)= (A)+ (
:
)
Now let nl, n2, ...,n, be nonnegative integers such that
)is defined as follows:
n
nl, n2, ...,n,
the expression
nl +n
2 + +n, = n. Then
n n !
nl, n2, ...,nr nl!n2!- n,!
For example,
7
(2,3,2) =
7!
2!3!2!
= 210 8
(4,2,2,0) =
8!
4!2!2!0!
= 420
These numbers are called the multinomial coefficientsin view of the following theorem
which generalizes the binomial theorem.
Theorem 2.7: (al+a
2 + - - +a,)" = n ayla?. .. a,".

21
CHAP. 21 TECHNIQUES O F COUNTING
COMBINATIONS
Suppose we have a collection of n objects. A combination of these n objects taken r at
a time, or an r-combination, is any subset of r elements. In other words, an r-combination
is any selection of r of the n objects where order does not count.
Example 2
.
1
2
: The combinations of the letters a,b,c,d taken 3 a t a time are
{a,b, c}, {a,b,d}, .[a,c,d},{b,c, d } or simply abc, abd, acd, bcd
Observe that the following combinations are equal:
abc, acb, bac, bca, cab, cba
That is, each denotes the same set {a,
b, c}.
The number of combinations of n objects taken r at a time will be denoted by
C(n,r )
Before we give the general formula for C(n,r), we consider a special case.
Combinations Permutations
abc abc, acb, bac, bca, cab, cba
abd abd, adb, bad, bda, dab, dba
acd acd, adc, cad, cda, dac, dca
bcd bcd, bdc, cbd, cdb, dbc, dcb
Thus the number of combinations multiplied by 3! equals the number of permu-
tations:
P(4,3)
C(4,3)*3! = P(4,3) or C(4,3) = -
3!
Now P(4,3) = 4 3 2 = 24 and 3! = 6; hence C(4,3) = 4 as noted above.
Since each combination of n objects taken r at a time determines r ! permutations of the
objects, we can conclude that
P(n,r) = r !C(n,r )
Thus we obtain
P(n,r) - n !
Theorem 2.8: C(n,r ) = -- r !( n - r ) !
r !
Recall that the binomial coefficient was defined to be r !(n
n !
-r )!,hence
We shall use C(n,r) and
(? interchangeably.

Example 2.14: How many committees of 3 can be formed from 8 people? Each committee is
essentially a combination of the 8 people taken 3 at a time. Thus
different committees can be formed.
ORDERED PARTITIONS
Suppose an urn A contains seven marbles numbered 1 through 7
. We compute the num-
ber of ways we can draw, first, 2 marbles from the urn, then 3 marbles from the urn, and
lastly 2 marbles from the urn. In other words, we want to compute the number of ordered
partitions
(AI,
A2,A3)
of the set of 7 marbles into cells A1 containing 2 marbles, APcontaining 3 marbles and As
containing 2 marbles. We call these ordered partitions since we distinguish between
each of which yields the same partition of A.
Since we begin with 7 marbles in the urn, there are
(3ways of drawing the first 2
marbles, i.e. of determining the first cell AI;
following this, there are 5 marbles left in the
urn and so there are
(3ways of drawing the 3 marbles, i.e. of determining A2;finally,
there are 2 marbles left in the urn and so there are ways of determining the last cell
As. Thus there are
7 5 2 7.6 5.4-3 2.1 210
(
2
)
(
3
)
(
2
) = ~ ' = ~ '
different ordered partitions of A into cells A1 containing 2 marbles, A2 containing 3 marbles,
and AS containing 2 marbles.
Now observe that
7! 5! 2! 7!
since each numerator after the first is cancelled by the second term in the denominator of
the previous factor. In a similar manner we prove (Problem 2.28)
Theorem 2.9: Let A contain n elements and let nl,n2,...,n,be positive integers with
n
l +n
2 + - +n,= n. Then there exist
n!
n1!n2!ns! n,!
different ordered partitions of A of the form (AI,
A2,...,A,)where AI con-
tains n
l elements, A2 contains n
2 elements, ...,and A, contains n,elements.
Example 2.15: In how many ways can 9 toys be divided between 4 children if the youngest child
is to receive 3 toys and each of the other children 2 toys?
We wish to find the number of ordered partitions of the 9 toys into 4 cells
containing 3,2, 2 and 2 toys respectively. By the above theorem, there are
9!
3!2!2!2! = 7560
such ordered partitions.

23
CHAP.21 TECHNIQUES OF COUNTING
TREE DIAGRAMS
A tree diagram is a device used to enumerate all the passible outcomes of a sequence of
experiments where each experiment can occur in a finite number of ways. The construction
of tree diagrams is illustrated in the following examples.
Example 2.16: Find the product set A X B X C where A =I {1,2}, B = {a,b, c } and C = {3,4}.
The tree diagram follows:
Observe that the tree is constructed from left to right, and that the number of
branches at each point corresponds to the number of possible outcomes of the next
experiment.
Example 2.17: Mark and Eric are to play a tennis tournament. The firstperson to win two games
in a row or who wins a total of three games wins the tournament. The following
diagram shows the possible outcomes of the tournament.
Observe that there are 10 endpoints which correspond to the 10 possible out-
comes of the tournament:
MM, MEMM, MEMEM, MEMEE, MEE, EMM, EMEMM, EMEME, EMEE, EE
The path from the beginning of the tree to the endpoint indicates who won which
game in the individual tournament.

Solved Problems
FACTORIAL
2.1. Compute 4!, 5!, 6!, 7! and 8!.
4! = 1.20304 = 24 7! = 7 * 6 ! = 7.720 = 5040
5! = 1.2.3.4.5 = 5 * 4 ! = 5-24 = 120 8! = 8 - 7 ! = 8.5040 = 40,320
6! = 1.203.4.5.6 = 6 * 5 ! = 6.120 = 720
13I 7!
2.2. Compute: (i)-
L (ii)m.
ll!’
13! -
- 1 3 ~ 1 2 ~ 1 1 * 1 0 * 9 * 8 * 7 ° 6 ~ 5 * 4 * 3 - 2 * 1
- 13.12 = 156
(i) i
g 1 1 ~ 1 0 . 9 . 8 * 7 ~ 6 . 5 * 4 - 3 * 2 * 1
-
13! - 13*12*11! -
- 13.12 = 156
or - -
1
1! 11!
7’ 1 -
1
7! -
- - -
(ii) A -
10! - 1 0 * 9 - 8 . 7 ! 1009.8 - 720
(n+2)!
.
2
.
3
. Simplify: (i)
(n-
n !
1)!’
(ii)I
n .
(i) -
n!
-
- n(n -l)(n-2). - - 3 2
* 3 2 1
1 n! - n(n-l)! -
= n or, simply, -- -- n
(n- l)! (n - l)(n-2). - (n- l)! (n- l)!
(n+2)! - (n+2)(n+ l)n(n- l ) ( n - 2 ) - - . 3 * 2 * 1
- 3 2
-
-
(n+2)(n+ 1) = n2 +3n + 2
(ii) -
n!
- n(n - l)(n- 2). - 1
or, simply, - - - ( n + N n + l ) * n ! = (n+2)(n+1) = n2 + 3 n + 2
n! n!
PERMUTATIONS, ORDERED SAMPLES
2.4. If repetitions are not permitted, (i) how many 3 digit numbers can be formed from
the six digits 2, 3, 5, 6, 7 and 9? (ii) How many of these are less than 400? (iii) How
many are even? (iv) How many are odd? (v) How many are multiples of 5?
In each case draw three boxes [-I nr
lto represent an arbitrary number, and then
write in each box the number of digits that can be placed there.
The box on the left can be filled in 6 ways; following this, the middle box can be filled in
5 ways; and, lastly, the box on the right can be filled in 4 ways: 1
6
1 Fl.Thus there
are 6 - 5.4 = 120 numbers.
The box on the left can be filled in only 2 ways, by 2 or 3, since each number must be less
than 400; the middle box can be filled in 5 ways; and, lastly, the box on the right can be filled
in 4 ways: Thus there are
P
I
. 2 5 4 = 40 numbers.
The box on the right can be filled in only 2 ways, by 2 or 6, since the numbers must be even;
the box on the left can then be filled in 6 ways; and, lastly, the middle box can be filled in
4 ways: r
l P
I
.
Thus there are 5 4 2 = 40 numbers.
The box on the right can be filled in only 4 ways, by 3, 5, 7 or 9, since the numbers must be
odd; the box on the left can then be filled in 5 ways; and, lastly, the box in the middle can
be filled in 4 ways: r
l F
l
.Thus there are 6 4 4 = 80 numbers.

25
CHAP. 2
1 TECHNIQUES O F COUNTING
(v) The box on the right can be filled in only 1 way, by 5, since the numbers must be multiples
of 5; the box on the left can then be filled in 5 ways; a.nd, lastly, the box in the middle can
be filled in 4 ways: 1
7
1 r
l
.
Thus there are 5 4 1 = 20 numbers.
2.5. In how many ways can a party of 7 persons arrange themselves (i) in a row of
7chairs? (ii) around a circular table?
(i) The seven persons can arrange themselves in a row in 7 6 5 4 3 2 1 = 7!ways.
(ii) One person can sit at any place in the circular table. The other six persons can then arrange
themselves in 6 5 4 3 2 1 = 6!ways around the table.
This is an example of a circular permutation. In general, n objects can be arranged in
a circle in (n-l)(n-2) * 3 2 1 = (n-l)! ways.
2.6. (i) In how many ways can 3 boys and 2 girls sit in a row? (ii) In how many ways
can they sit in a row if the boys and girls are each to sit together? (iii) In how
many ways can they sit in a row if just the girls are to sit together?
(i) The five persons can sit in a row in 5 4 3 2 1 = 5! := 120 ways.
(ii) There are 2 ways to distribute them according to sex: BBBGG or GGBBB. In each case the
boys can sit in 3 2 1 = 3! = 6 ways, and the girls can sit in 2 1 = 2! = 2 ways. Thus,
altogether, there are 2 3! 2! = 2 6 2 = 24 ways.
(iii) There are 4 ways to distribute them according to sex: GGBBB, BGGBB, BBGGB, BBBGG.
Note that each way corresponds to the number, 0,1, 2 or 3, of boys sitting to the left of the
girls. In each case, the boys can sit in 3! ways, and the girls in 2! ways. Thus, altogether,
there are 4*3!*2!
= 406.2 = 48 ways.
2.7. How many different signals, each consisting of 6 flags hung in a vertical line, can
be formed from 4 identical red flags and 2 identical blue flags?
This problem concerns permutations with repetitions. There are --
- 15 signals since
4!
6!
2!
there are 6 flags of which 4 are red and 2 are blue.
2.8. How many distinct permutations can be formed from all the letters of each word:
(i) them, (ii) unusual, (iii) sociological?
(i) 4! = 24, since there are 4 letters and no repetitions.
(ii) -
7! -
- 840, since there are 7 letters of which 3 are U.
3!
12!
(iii)
3!2!2!2! '
since there are 12 letters of which 3 are 0 , 2 are c, 2 are i, and 2 are 1.
2.9. (i) In how many ways can 3 Americans, 4 Frenchmen, 4 Danes and 2 Italians be
seated in a row so that those of the same nationality sit together?
(ii) Solve the same problem if they sit at a round table.
(i) The 4 nationalities can be arranged in a row in 4! ways. In each case the 3 Americans can
be seated in 3! ways, the 4 Frenchmen in 4! ways, the 4 Danes in 4! ways, and the 2 Italians
in 2! ways. Thus, altogether, there are 4! 3! 4!4!2! = 165,888arrangements.
(ii) The 4 nationalities can be arranged in a circle in 3! ways (see Problem 14.4 on circular
permutations). In each case the 3 Americans can be seated in 3! ways, the 4 Frenchmen in
4! ways, the 4 Danes in 4! ways, and the 2 Italians in 2! ways, Thus, altogether, there are
3!3!4! 4! 2! = 41,472arrangements.

2.10. Suppose an urn contains 8 balls. Find the number of ordered samples of size 3
(i) with replacement, (ii) without replacement.
'
(i) Each ball in the ordered sample can be chosen in 8 ways; hence there are 8.8 0 8= 83 = 512
samples with replacement.
(ii) The first ball in the ordered sample can be chosen in 8 ways, the next in 7 ways, and the last
in 6 ways. Thus there are 807.6 = 336 samples without replacement,
2.11. Find n if (i) P(n,2) = 72, (ii) P(n,4) = 42P(n, Z), (iii) ZP(n,2) +50 = P(2n,2).
(i) P(n,2) = n(n - 1)= n2-n; hence n2-n = 72 or n2 -n -72 = 0 or (n-9)(n+8) = 0.
Since n must be positive, the only answer is n = 9.
(ii) P(n,4) = n(n - l)(n- 2)(n-3) and P(n,2) = n(n - 1). Hence
n(n -l)(n-2)(n-3) = 42n(n- 1) or, if n # 0, # 1, (n-2)(n- 3) = 42
or n 2 - 5 n + 6 = 42 or n2-5n-36 = 0 or (n-9)(n+4) = 0
(iii) P(n, 2) = n(n - 1)= n2 -n and P(2n, 2) = Zn(2n-1)= 4n2 -2n. Hence
Z(n2- n) +50 = 4n2 - 2n or 2n2 - 2n + 50 = 4n2 - 2n or 50 = 2n2 or n2 = 25
2.12. Compute: (i) (136) , (ii) (142) (iii) (55).
Recall that there are as many factors in the numerator as in the denominator.
15 14 13012011
= 3003
(iii) (;">= 1 - 2 . 3 * 4 * 5
2
.
1
3
. Compute: (i) (:) , (ii) (
:
) , (iii) (y).
I
Note that 8 - 5 = 3; hence we could also compute as follows:
(58)
807.6
(;)
(58)
= G>
= (:) = 1
.
2
.
3 = 56
9.8
= 1.2 = 36.
(ii) Now 9 -7 = 2; hence
(iii) Now 10-6 = 4; hence
1 - 2 - 3 . 4
2.14. Expand and simplify: (2x +y2)5.

27
2.16. Prove: 24 = 16 = (t) + (;) + (i)+ (t) + (:) .
Expand (1+l ) 4 using the binomial theorem:
+ (i):w+
24 =: (1+1)* = ( : ) I 4 + (:)1311 ( : ) i l l 3 + (:)14
+ + + +
= (:) (14) (3 (i) (:>
2.17. Prove Theorem 2.6: (" '> = ( l) + (
:
) .
n! 72!
Now (l-nl) + (:) = (r - 1)! (n- 7" +1)! + -
-
r ! (12 -r )!* To obtain the same denomi-
n - r + l
nator in both fractions, multiply the first fraction by and the second fraction by
n - r + l '
Hence
r . n ! ( n - r + l ) - n !
(A) (:> =
+ r +
(r - I)! (n-r +I ) ! r ! (n- r +1) (n-r ) !
- r a n ! (n- r +1) n!
- + -
-
r ! ( n - r + l ) ! r ! ( n - r + l ) !
r - n ! + ( n - r + l ) * n ! __ [ r + ( n - r + l)] o n !
-
-
r ! ( n - r + l)!
.-
r ! (n-Y+ l ) !
- (n+l ) n ! - (n+1)!
-
r ! ( n - r + l ) ! - r ! (n--r+ l ) !
(a +b)" = 5(
:
)
2.18. Prove the Binomial Theorem 2
.
5
: an-rb'.
r=O
The theorem is true for n = 1, since
$ (:>d - r b r = (:>.lbO + ( : ) a o b l == U + b = (a+b)l
f=;O
We assume the theorem holds for (a4-b)n and prove it is true for (a+b)n+l.
(a+b)n+l = (a+b)(a+b)n
= ( a + b ) [ a n + + ' * . + ( r 1) u n - r + l br-1
+ (:>.n-r,r + * . . + ( : ) a 6 n - 1 + bn1
Now the term in the product which contains br is obtained from
(E(r - 1 )an-r+l,r-l] + a[(:)un-rbr] = (,cl ) ~ n - r + l b r + (:>..-r+lbr
= + (:)]un-r+lbr
[ ( , r : l )

But, by Theorem 2.6,
Note that (a3- b)(a+b)n is a polynomial of degree n +1 in b. Consequently,
(a+ b)n+' = (a+ b)(a+ b)n = y( ;
')an-..' b7
r=O
which was to be proved.
2.19. Compute the following multinomial coefficients:
10
(i) (& 1>' (,,,s,,0)' Pii) ( 5 ' 3 ' 2 , Z )
(5, 3f:,
(iii) The expression 2) has no meaning since 5 +3 +2 +2 # 10.
COMBINATIONS
2.20. In how many ways can a committee consisting of 3 men and 2 women be chosen
from 7 men and 5 women? /T
The 3 men can be chosen from the 7 men in
The 3 men can be chosen from the 7 men in ( ) ways, and the 2 women can be chosen from the
ways, and the 2 women can be chosen from the
- .
ways. Hence the committee can be chosen in
ways. Hence the committee can be chosen in
$/
(37) (:) =
(37) (:) = E= 350 ways.
E= 350 ways.
1 0 2 - 3
1 0 2 - 3
2.21. A delegation of 4 students is selected each year from a college to attend the National
Student Association annual meeting. (i) In how many ways can the delegation be
chosen if there are 12 eligible students? (ii) In how many ways if two of the
eligible students will not attend the meeting together? (iii) In how many ways if
two of the eligible students are married and will only attend the meeting together?
l2 l1 l0
(i) The 4 students can be chosen from the 12 students in = 1 - 2 . 3 - 4 = 495 ways.
(7)
(ii) Let A and B denote the students who will not attend the meeting together.
Method 1.
1 0 . 9 . 8 . 7 -
If neither A nor B is included, then the delegation can be chosen in
1 . 2 . 3 . 4 -
(y)=
210 ways. If either A or B, but not both, is included, then the delegation can be chosen in
1 0 0 9 - 8
2 (y) = 2 01.2.3 = 240 ways. Thus, altogether, the delegation can be chosen in
210+240 = 450 ways.
Method 2.
(3
If A and B are both included, then the other 2 members of the delegation can be chosen in
= 45 ways. Thus there are 495 - 45 = 450 ways the delegation can be chosen if
A and B are not both included.
(iii) Let C and D denote the married students. If C and D do not go, then the delegation can be
(y)=
chosen in 210 ways. If both C and D go, then the delegation can be chosen in
= 45 ways. Altogether, the delegation can be chosen in 210 +46 = 255 ways.

CHAP. 21 TECHNIQUES O F COUNTING 29
2.22. A student is to answer 8 out of 10 questions on an exam. (i) How many choices
has he? (ii) How many if he must answer the first 3 questions? (iii) How many
if he must answer at least 4 of the first 5 questions?
(i) The 8 questions can be selected in - 45 ways.
- y::
(ii) If he answers the first 3 questions, then he can choose the other 5 questions from the last
7 questions in
(i) = (i) = 1.2 = 21 ways.
7 . 6
(iii) If he answers all the first 5 questions, then he can choose the other 3 questions from the last
/ -
(i,)=
5 in 10 ways. On the other hand, if he answers only 4 of the first 6 questions, then
he can choose these 4 in (:) = (15) = 5 ways, and he can choose the other 4 questions
from the last 5 in (:) = (;>= 5 ways; hence he can choose the 8 questions in 6.5 =
25 ways. Thus he has a total of 35 choices.
2.23. Find the number of subsets of a set X containing n elements.
Method 1.
The number of subsets of X with T
+ + + n - 1 +
(0 (;) (1) + ... ( ) (:)
subsets of X. The above sum (Problem 2.61) is equal to 2n, i.e. there are 2n subsets of X.
Method 2.
There are two possibilities for each element of X either it belongs to the subset or it doesn't;
hence there are
n times
-
2 . 2 . . . . . 2 = 2n
ways to form a subset of X,i.e. there are 2n different subsets of X.
2.24. In how many ways can a teacher choose one or more students from six eligible
students?
Method 1.
By the preceding problem, there are 26 = 64 subsets of the set consisting of the six students.
However, the empty set must be deleted since one or more students are chosen. Accordingly there
are 26 - 1 = 64 - 1 = 63 ways to choose the students.
Method 2.
Either 1,2,3,4,5 or 6 students are chosen. Hence the number of choices is
(3+ (3+ (0+ (3+ (3+ (i) = 6 + 15 + 20 + 15 + 6 + 1 = 63
ORDERED AND UNORDERED PARTITIONS
2.25. In how many ways can 7 toys be divided among 3 children if the youngest gets
3 toys and each of the others gets 2?
We seek the number of ordered partitions of 7 objects into cells containing 3, 2 and 2 objects,
7'
respectively. By Theorem 2.9, there are jj-&jj = 210 such partitions.

2.26. There are 12 students in a class. In how many ways can the 12 students take
3 different tests if 4 students are to take each test?
Method 1.
We seek the number of ordered partitions of the 12 students into cells containing 4 students
each. By Theorem 2.9, there are -
12! = 34,650 such partitions.
4! 4! 4!
Method 2.
(:)There are (y)ways to choose 4 students to take the first test; following this, there are
ways to choose 4 students to take the second test. The remaining students take the third test.
Thus, altogether, there are (“> (8q) = 495 70 = 34,650 ways for the students to take the tests.
2.27. In how many ways can 12 students be partitioned into 3 teams, AI,A2 and As, so that
each team contains 4 students?
Method 1.
Observe that each partition {Al,Az,A3} of the students can be arranged in 3! = 6 ways
12’
as an ordered partition. Since (see the preceding problem) there are 4!4!4! = 34,650 such ordered
partitions, there are 34,650/6 = 6775 (unordered)partitions.
Method 2.
Let A denote one of the students. Then there are ways to choose 3 other students to be
on the same team as A. Now let B denote a student who is not on the same team as A; then there
are (i)ways to choose 3 students of the remaining students to be on the same team as B. The
remaining 4 students constitute the third team. Thus, altogether, there are
(;’>.(:) =
165 35 = 5775 ways to partition the students.
2.28. Prove Theorem 2.9: Let A contain n elements and let nl,n2, ...,n, be positive integers
with nl +n2 + +f i r = n. Then there exist
n !
nl!np!n3! nr!
different ordered partitions of A of the form (Al,A2,...,A,) where A1 contains nl
elements, A2 contains n
2 elements, ..., and Ar contains n, elements.
We begin with n elements in A; hence there are (
l
) ways of selecting the cell Al. Following
this, there are n - n, elements left, i.e. in A Al, and so there are (“i 2 n 1 ) ways of selecting
A2. Similarly, for i = 3, ...,T, there are
ni- I
ways of selecting Ai. Thus
there are
(n:>(n--2n1)(n-:3-n2) ... (“-“I- nr
* . * enr-1) (*)
different ordered partitions of A. Now (*) is equal to
n! (n- nl)!
.( n - n l - - n r - l ) !
a
- . a
nl! (n-n,)! - n2! (n-nl -n2)! n,! ( n - n l - -nr)!
n !
But this is equal to nl!n2!., .n r ! since each numerator after the first is cancelled by the second
- -nr)! = O! = 1. Thus the theorem is proved.
term in the denominator and since (n-nl- *

31
TREE DIAGRAMS
2.29. Construct the tree diagram for the number of permutations of (a,b, c}.
C bac
a bca
b cab
a cba
The six permutations are listed on the right of the diagram.
2.30. A man has time to play roulette at most five
times. At each play he wins or loses a dollar.
The man begins with one dollar and will stop
playing before the five times if he loses all his
money or if he wins three dollars, i.e. if he has
four dollars. Find the number of ways that the
betting can occur.
The tree diagram on the right describes the way the
betting can occur. Each number in the diagram denotes
the number of dollars the man has at that point. Observe 3
-
2
that the betting can occur in 1
1 different ways. Note
that he will stop betting before the five times are up in
only three of the cases.
Supplementary Problems
FACTORIAL
2.31. Compute: (i) 9!, (ii) lO!, (iii) 11!
16! 8’ 10!
2.32. Compute: (i) z, 14!
(iii) j
$ , (iv) E.
(ii)
2
.
3
3
. Simplify: (i) 7 nl (n-l)! (n-r +l)!
-
(n+I)! , (ii) (n-2)!’ (iii) -
( n + 2 ) ! ’ (iv) (n-r-l)!.
PERMUTATIONS
2.34. (i) How many automobile license plates can be made if each plate contains 2 different letters
followed by 3 different digits? (ii) Solve the problem if the first digit cannot be 0.
2
.
3
5
. There are 6 roads between A and B and 4 roads between El and C.
(i) In how many ways can one drive from A to C by way of B?
(ii) In how many ways can one drive roundtrip from A to C by way of B?
(iii) In how many ways can one drive roundtrip from A to C without using the same road more
than once?

32 TECHNIQUES OF COUNTING [CHAP. 2
2.36. Find the number of ways in which 6 people can ride a toboggan if one of three must drive.
2.37. (i) Find the number of ways in which five persons can sit in a row.
(ii) How many ways are there if two of the persons insist on sitting next to one another?
2.38. Solve the preceding problem if they sit around a circular table.
2.39. (i) Find the number of four letter words that can be formed from the letters of the word HISTORY.
(ii) How many of them contain only consonants? (iii) How many of them begin and end in a
consonant? (iv) How many of them begin with a vowel? (v) How many contain the letter Y?
(vi) How many begin with T and end in a vowel? (vii) How many begin with T and also contain S?
(viii) How many contain both vowels?
2.40. How many different signals, each consisting of 8 flags hung in a vertical line, can be formed from
4 red flags, 2 blue flags and 2 green flags?
2.41. Find the number of permutations that can be formed from all the letters of each word: (i) queue,
(ii) committee, (iii) proposition, (iv) baseball.
2.42. (i) Find the number of ways in which 4 boys and 4 girls can be seated in a row if the boys and
girls are to have alternate seats.
(ii) Find the number of ways if they sit alternately and if one boy and one girl are to sit in
adjacent seats.
(iii) Find the number of ways if they sit alternately and if one boy and one girl must not sit in
adjacent seats.
2.43. Solve the preceding problem if they sit around a circular table.
2.44. An urn contains 10 balls. Find the number of ordered samples (i) of size 3 with replacement,
(ii) of size 3 without replacement, (iii) of size 4 with replacement, (iv) of size 5 without replacement.
2.45. Find the number of ways in which 5 large books, 4 medium-size books and 3 small books can be
placed on a shelf so that all books of the same size are together.
2.46. Consider all positive integers with 3 different digits. (Note that 0 cannot be the first digit.)
(i) How many are greater than 700? (ii) How many are odd? (iii) How many are even? (iv) How
many are divisible by 5?
2.47. (i) Find the number of distinct permutations that can be formed from all of the letters of the word
ELEVEN. (ii) How many of them begin and end with E? (iii) How many of them have the 3 E
’
s
together? (iv) How many begin with E and end with N?
2.48.
6
9
2.49. Compute: (i) (3, ;,
1
) , (8(3, 2p2, (iii) (2, 2, 1, 1, 0)
2.50. Expand and simplify: (i) (2%+2/95, (ii) (x2-3 ~ ) ~ ~
(iii) (&a+26)5, (iv) (2a2- b)6.
2.51. Showthat
(
:
) + (2) + (:>+ (i)+ ... + (z) = 2%.
2.52. Showthat
(
:
) - (y) + (3- (i)+ (3
-.. 2 = 0.
2.53. Find the term in the expansion of (2x2-.+@3)8which contains x*.
2.54. Find the term in the expansion of ( 3 2 ~ 2
-22)‘ which contains g6.

33
CHAP. 21 TECHNIQUES OF COUNTING
COMBINATIONS
2.55. A class contains 9 boys and 3 girls. (i) In how many ways can the teacher choose a committee of 4?
(ii) How many of them will contain at least one girl? (iii) How many of them will contain exactly
one girl?
2.56. A woman has 1
1 close friends. (i) In how many ways can she invite 6 of them to dinner? (ii) In
how many ways if two of the friends are married and will not attend separately? (iii) In how
many ways if two of them are not on speaking terms and will not attend together?
2.57. There are 10 points A,B, ... in a plane, no three on the same line. (i) How many lines are
determined by the points? (ii) How many of these lines do not pass through A or B? (iii) How
many triangles are determined by the points? (iv) How many of these triangles contain the
point A? (v) How many of these triangles contain the side AB?
2.58. A student is to answer 10 out of 13 questions on an exam. (i) How many choices has he? (ii) How
many if he must answer the first two questions? (iii) How many if he must answer the first or
second question but not both? (iv) How many if he must answer exactly 3 of the first 6 questions?
(v) How many if he must answer at least 3 of the first 5 questions?
2.59. A man is dealt a poker hand (5 cards) from an ordinary playing deck. In how many ways can he
be dealt (i) a straight flush, (ii) four of a kind, (iii) a straight, (iv) a pair of aces, (v) two of a
kind (a pair)?
2.60. The English alphabet has 26 letters of which 6 are vowels.
(i) How many 6 letter words containing 3 different consonants
formed ?
(ii) How many of them contain the letter b?
(iii) How many of them contain the letters b and c?
(iv) How many of them begin with b and contain the letter c?
(v) How many of them begin with b and end with c?
(vi) How many of them contain the letters a and b?
(vii) How many of them begin with a and contain b?
(viii) How many of them begin with b and contain a?
(ix) How many of them begin with a and end with b?
(x) How many of them contain the letters a, b and c?
ORDERED AND UNORDERED PARTITIONS
2.61. In how many ways can 9 toys be divided evenly among 3 children?
and 2 different vowels can be
2.62. In how many ways can 9 students be evenly divided into three teams?
2.63. In how many ways can 10 students be divided into three teams, one containing 4 students and
the others 3?
2.64. There are 12 balls in an urn, In how many ways can 3 balls be drawn from the urn, four times
in succession, all without replacement?
2.65. In how many ways can a club with 12 members be partitioned into three committees containing
5, 4 and 3 members respectively?
2.66. In how many ways can n students be partitioned into two teams containing at least one student?
2.67. In how many ways can 14 men be partitioned into 6 committees where 2 of the committees contain
3 men and the others 2?
TREE DIAGRAMS
2.68. Construct the tree diagram for the number af permutations of {a,
b, C, d).
2.69. Find the product set {1,2,3) X {2,4) X {2,3,4) by constructing the appropriate tree diagram.

2.70. Teams A and B play in a basketball tournament. The first team that wins two games in a row or a
total of four games wins the tournament. Find the number of ways the tournament can occur.
2.71. A man has time to play roulette five times. He wins or loses a dollar at each play. The man begins
with two dollars and will stop playing before the five times if he loses all his money or wins three
dollars (i.e. has five dollars). Find the number of ways the playing can occur.
2.72. A man is at the origin on the x-axis and takes a unit step either to the left or to the right. He
stops after 5 steps or if he reaches 3 or -2. Construct the tree diagram to describe all possible
paths the man can travel.
2.73. In the following diagram let A , B , ...,F denote islands, and the lines connecting them bridges.
A man begins at A and walks from island to island. He stops for lunch when he cannot continue
to walk without crossing the same bridge twice. Find the number of ways that he can take his walk
before eating lunch.
2.74. Consider the adjacent diagram with nine points A, B, C, R, S, T,X, A-B-C
Y,Z. A man begins at X and is allowed to move horizontally or ver-
tically, one step at a time. He stops when he cannot continue to walk I l l
without reaching the same point more than once. Find the number of
'ways he can take his walk, if he first moves from X to R. (By sym-
metry, the total number of ways is twice this.)
Answers to Supplementary Problems
2.31. (i) 362,880 (ii) 3,628,800 (iii) 39,916,800
2.32. (i) 240 (ii) 2184 (iii) 1/90 (iv) U1716
2.33. (i) n + 1 (ii) n(n-1) = n2-n (iii) l/[n(n4-l)(n+2)] (iv) (n-r)(n-r f 1)
25 9 9 8 = 421,200
2.34. (i) 26 25 10 9 8 = 468,000 (ii) 26
4 3 5 = 360
4 4 6 = 24 24 = 576 (iii) 6
4 = 24 (ii) 6
2.35. (i) 6
2.36. 3 . 5 . 4 . 3 . 2 . 1 = 360
2.37. (i) 5! = 120 (ii) 4 * 2 !* 3 ! = 48
2.38. (i) 4! = 24 (ii) 2! 3! = 12
2.39. (i) 7 6 5 4 = 840 (iii) 5 5 4 4 400 (v) 4 * 6 * 5 * 4 = 4 8 0 (vii) 1 3 5 4 = 60
3 5 4 = 240
6 5 4 = 240 (vi) 1 . 5 . 4 . 2 = 4 0 (viii) 4
(ii) 5 4 3 2 = 120 (iv) 2
8!
2.40. -- 420
4 ! 2 ! 2 ! -
5! 11! 8 !
2.41. (i) 2
!
2
!= 30 (ii)
9!
= 45,360 (iii) = 1,663,200 (iv) --
2 ! 2 ! 2 ! - 5040

35
9
2.42. (i) 2*4! 4! =
I 1152 (ii) 2 7*3!-3! = 504 (iii) 1152-504 = 648
2.43. (i) 3!*4! = 144 (ii) 2*3!*3!= 72 (iii) 144-72 = 72
2.44. (i) 10*10*10= 1000 (iii) 10 10 10 10 = 10,000
8 = 720 (iv) . 1 0 * 9 * 8 * 7 * 6
= 30,240
(ii) 10
2.45. 3!5!4!3! = 103,680
2.46. (i) 309.8 = 216 (ii) 8 . 8 0 5
(iii) 9.8.1 = 72 end in 0, and
72+256 = 328 are even.
(iv) 9 -8.1 := 72 end in 0, and
divisible by 5.
= 320
808.4 = 256 end in the other even digits; hence, altogether,
8 - 8 01 = 64 end in 5; h.ence, altogether, 72 +64 = 136 are
41
2.47. (i) - 120 (ii) 4! = 24 (iii) 4
61
3! = 24 (iv) $= 12
3! -
2.48. (i) 10 (ii) 35 (iii) 91 (iv) 15 (v) 1140 (vi) 816
2.49. (i) 504 (ii) 210 (iii) 180
2.50.
2.51.
(i) 8x3 +12x2342 +6x114 +346
(ii) x8 - 12x62/+54x4~2
- 108x2~3
+81344
(iii) aV32 +5aW8 + 5a3b2 +2OdW +40ab4 +32b5
(iv) 64a12 - 192alob +240a8b2- 160aeb3+60a4b4 -
Hint. Expand (1+1)n. 2.53. 7Ox8gl2
l2a2b5 C b6
2.52. Hint. Expand (1-l)n. 2.54. 945x33469
2.55. (i) (142) (49) (3
(y) 4!
= 495, = 369, (iii) 3
(ii) (">- = 252
2.56. (i) (y) = 462, (ii) (39) + (:) = 210, (iii) ("5> +2*(:) = 378
2.57. 45, 28, 120, (iv) = 36, (v) 8
(i) (:">= (ii) (3= (iii) (y)= (2")
(ii) == ('3")
2.58. (i) = 286
(ii) z= = 165 + + = 276
(y) (f) (v) (;)(;) (3(D (:)(:)
(iii) 2 (y)= 2 (">= 110
2.59. (i) 4.10 = 40, (ii) 13.48 = 624, (iii) 10 045- 40 = 10,200. (We subtract the number of straight
4 12
(:)(:)
flushes.) (iv) (2)( 3 ) 43 = 84,480, (v) 13 d3 = 1,098,240
21 5
2.60. (i) (3)(2) - 5 ! = 1,596,000 (v) 19. (E) * 3 ! = 1140 (ix) 4 . (y)- 3 ' = 4560
5! = 91,200 (x) 4*19*5! = 9120
("2")
(ii) (:)(:) 5! = 228,000 (vi) 4
= 18,240
(iii) 19 (:)*Ei! = 22,800 (vii) 4
= 4560 (viii) 18,240 (same a:3 (vii))
(iv) 19 (i).4!

36 TECHNIQUES OF COUNTING [CHAP. 2
- -
9!
2.61.
3!3!3! - 1680
2.62. 1680/3! = 280 or
C
)
(
:
) = 280
(140)(;)
2.63. - 0 10! -
1 - 2100 or = 2100
4!3!3! 2! -
12! 2.66. 2n-1 - 1
2.64.
3!3!3!3! -
- 369,600
- - 14! 0 -
2.65. 12!
5!4!3! - 27,720 2*67' 3!3!2!2!2!2! 2!4! -
- 3,153,150
2.69.
2
2 3
4
4
The eighteen elements of the product set are listed to the right of the tree diagram.
2.70. 14 ways
2.71. 20 ways (as seen in the following diagram):

37
2.72. Hint. The tree is essentially the same as the tree of the preceding problem.
2
.
7
3
. The appropriate tree diagram follows:
There are eleven ways to take his walk. Observe that he must eat his lunch at either B, D or E.
2.74. The appropriate tree diagram follows:
There are 10 different trips. (Note that in only 4 of them are all nine points covered.)

Chapter 3
Introduction to Probability
INTRODUCTION
Probability is the study of random or nondeterministic experiments. If a die is tossed
in the air, then it is certain that the die will come down, but it is not certain that, say,
a 6 will appear. However, suppose we repeat this experiment of tossing a die; let s be
the number of successes, i.e. the number of times a 6 appears, and let n be the number of
tosses. Then it has been empirically observed that the ratio f = s/n, called the relative
frequency, becomes stable in the long run, i.e. approaches a limit. This stability is the
basis of probability theory.
In probability theory, we define a mathematical model of the above phenomenon by
assigning “probabilities” (or: the limit values of the relative frequencies) to the “events”
copnected with an experiment. Naturally, the reliability of our mathematical model for a
given experiment depends upon the closeness of the assigned probabilities to the actual
relative frequency. This then gives rise to problems of testing and reliability which form
the subject matter of statistics.
Historically, probability theory began with the study of games of chance, such as
roulette and cards. The probability p of an event A was defined as follows: if A can occur
in s ways out of a total of n equally likely ways, then
p = P(A) = -
s
n
For example, in tossing a die an even number can occur in 3 ways out of 6 “equally likely”
ways; hence p = = &. This classical definition of probability is essentially circular since
the idea of “equally likely” is the same as that of “with equal probability” which has not
been defined. The modern treatment of probability theory is purely axiomatic. This means
that the probabilities of our events can be perfectly arbitrary, except that they must satisfy
certain axioms listed below. The classical theory will correspond to the special case of
so-called equiprobable spaces.
SAMPLE SPACE AND EVENTS
The set S of all possible outcomes of some given experiment is called the sample space.
A particular outcome, i.e. an element in S, is called a sample point or sample. An event A
is a set of outcomes or, in other words, a subset of the sample space S. The event {a}con-
sisting of a single sample a E S is called an elementary event. The empty set @ and S
itself are events; is sometimes called the impossible event, and Sthe certain or sure event.
We can combine events to form new events using the various set operations:
(i) A U B is the event that occurs iff A occurs or B occurs (or both);
(ii) A nB is the event that occurs iff A occurs and B occurs;
(iii) A“,the complement of A, is the event that occurs iff A does not occur.
38

39
CHAP. 31 INTRODUCTION TO PROBABILITY
Two events A and B are called mutuubby excEusive if they are disjoint, i.e. if A nB = Q>.
In other words, A and B are mutually exclusive if they cannot occur simultaneously.
Example 3.1: Experiment: Toss a die and observe the number that appears on top. Then the
sample space consists of the six possible numbers:
S = (1,2,3,4,5, 6)
Let A be the event that an even number occurs, B that an odd number occurs and
C that a prime number occurs:
A = {2,4,6}, B = {1,3,5}, C = {2,3,5}
Then:
A UC = {2,3,4, 5,6} is the event that an even or a prime number occurs;
BnC = (3, 5} is the event that an odd prime number occurs;
C c = (1,4,6} is the event that a prime number does not occur.
Note that A and B are mutually exclusive: A nB = @; in other words, an even
number and an odd number cannot occur simultaneously.
Example 3.2: Experiment: Toss a coin 3 times and observe the sequence of heads (H) and
tails (T) that appears. The sample space S consists of eight elements:
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Let A be the event that two or more heads appear consecutively, and B that all
the tosses are the same:
A = {HHH,HHT,THH} and B = {HHH,TTT}
Then A nB = {HHH} is the elementary event in which only heads appear. The
event that 5 heads appear is the empty set (3.
Example 3.3: Experiment: Toss a coin until a head appears and then count the number of times
the coin was tossed. The sample space of this experiment is S = {1,2,3, ...,a
)
.
Here 03 refers to the case when a head never appears and so the coin is tossed an
infinite number of times. This is an example of a sample space which is countably
infinite.
Example 3.4: Experiment: Let a pencil drop, head first, into a
rectangular box and note the point on the bottom
of the box that the pencil first touches. Here S
consists of all the points on the bottom of the box.
Let the rectangular area on the right represent
these points. Let A and B be the events that the
pencil drops into the corresponding areas illus-
trated on the right. This is an example of a sam-
ple space which is not finite nor even countably
infinite, i.e. which is uncountable.
Remark: If the sample space S is finite or countably infinite, then every subset of S is
an event. On the other hand, if S is uncountable, as in Example 3.4, then for
technical reasons (which lie beyond the scope of this text) some subsets of S
cannot be events. However, in all cases the events shall form a a-algebra & of
subsets of S.

40 INTRODUCTION TO PROBABILITY [CHAP. 3
AXIOMS OF PROBABILITY
Let Sbe a sample space, let & be the class of events, and let P be a real-valued function
defined on 6. Then P is called a probability function, and P(A) is called the probability of
the event A if the following axioms hold:
[PI] For every event A, 0 4 P(A)L 1.
[P2] P(S)= 1.
[P3] If A and B are mutually exclusive events, then
P(AUB) = P(A)+P(B)
[P4] If Al,A2, ... is a sequence of mutually exclusive events, then
*
P(AiUA2U a ) = P(A1)+P(A2)+
The following remarks concerning the axioms [P3] and [P4] are in order. First of all,
using [P3] and mathematical induction we can prove that for any mutually exclusive events
A1,A2, ...,An,
P(A1UAzU - * * UA,) = P(A1)+P(A2)+ . - +P(A,) (*)
We emphasize that [P4]does not follow from [P3]even though (*) holds for every positive
integer n. However, if the sample space S is finite, then clearly the axiom [P4]is superfluous.
We now prove a number of theorems which follow directly from our axioms.
Theorem 3.1: If tz, is the empty set, then P ( p )= 0.
Proof: Let A be any set; then A and tz, are disjoint and A u P = A. By [PSI,
P(A) = P(AUp) = P(A)+P(@)
Subtracting P(A)from both sides gives our result.
Theorem 3.2: If A" is the complement of an event A, then P(Ac)= 1 - P(A).
Proof: The sample space S can be decomposed into the mutually exclusive events A and
Ac;that is, S= A uAc. By [Pz] and [P3] we obtain
1 = P(S) = P(AUAc) = P(A)+P(Ac)
from which our result follows.
Theorem 3.3: If AcB, then P(A)6 P(B).
Proof. If AcB, then B can be decomposed into the mutually
exclusive events A and B

A (as illustrated on the right).
Thus
P(B) = P(A)+P(BA)
The result now follows from the fact that P(BA) 0.
Theorem 3.4: If A and B are any two events, then
P(AB) = P(A)- P(AnB)
Proof. Now A can be decomposed into the mutually ex-
clusive events A B and A nB; that is, A = ( A B)U( AnB).
Thus by [PSI,
P(A) = P(AB) +P(AnB)
from which our result follows.

41
CHAP. 3
1 INTRODUCTION TO PROBABILITY
Theorem 3
.
5
: If A and B are any two events, then
P(AUB) = P(A)+P(B)- P(AnB)
Proof. Note that A UB can be decomposed into the mutually
exclusive events A B and B;that is, A UB = ( AB)UB. Thus
by [PSIand Theorem 3.4,
P(AUB) = P(AB) +P(B)
= P(A)- P ( AnB)+P(B)
= P(A)+P(B)- P(AnB)
which is the desired result.
Applying the above theorem twice (Problem 3.23) we obtain
Corollary 3
.
6
: For any events A, B and C,
P(AUBUC) = P(A)+P(B)
+P(C)- P(AnB) - P(AnC)- P(BnC)+P(AnBnC)
FINITE PROBABILITY SPACES
Let S be a finite sample space; say, S = {UI,
up,...,a,}.A finite probability space is
obtained by assigning to each point a+ E S a real number pi, called the probability of ai,
satisfying the following properties:
(i) each pi is nonnegative, pi 2 0
(ii) the sum of the pi is one, PI +pz + - - - +p, = 1.
The probability P(A) of any event A, is then defined to be the sum of the probabilities of
the points in A. For notational convenience we write P(a) for P({a+}).
Example 3.5: Let three coins be tossed and the number of heads observed; then the sample space
is S = {0,1,2,3}. We obtain a probability space by the following assignment
P(0) = 4, P(1) = 8, P(2) = 8 and P(3) = Q
since each probability is nonnegative and the sum of the probabilities is 1
. Let A
be the event that at least one head appears and let B be the event that all heads
or all tails appear:
A = {1,2, 3) and B = (0, 3)
Then, by definition,
P(A) = P(1)+P(2)+P(3) = 8 +8 +Q = f
and P(B) = P(0)+P(3) = Q +Q = 4
Example 3.6: Three horses A, B and C are in a race; A is twice as likely to win as 3 and B is
twice as likely to win as C. What are their respective probabilities of winning,
i.e. P(A),P(B) and P(C)?
Let P(C)= p; since B is twice as likely to win as C, P(B) = 2p; and since
A is twice as likely to win as B, P(A)= 2P(B)= 2(2p) = 413. Now the sum of the
probabilities must be 1; hence
p + 2 p + 4 p = 1 or 7 p = 1 or p = 3
Accordingly,
P(A) = 4p = 4, P(B) = 2p = +, P(C) = p = +
Question: What is the probability that B or C wins, i.e. P({B,C})? By definition
P({B,C}) = P ( B ) + P ( C ) = + + += $

FINITE EQUIPROBABLE SPACES
Frequently, the physical characteristics of an experiment suggest that the various
outcomes of the sample space be assigned equal probabilities. Such a finite probability
space S, where each sample point has the same probability, will be called an equiprobable
or uniform space. In particular, if S contains n points then the probability of each point
is lln. Furthermore, if an event A contains r points then its probability is r ;
1
=
r
;
. In
other words,
number of elements in A
P(A) =
number of elements in S
number of ways that the event A can occur
or P(A) =
number of ways that the sample space Scan occur
We emphasize that the above formula for P(A) can only be used with respect to an
equiprobable space, and cannot be used in general.
The expression “at random” will be used only with respect to an equiprobable space;
formally, the statement “choose a point at random from a set S” shall mean that S is an
equiprobable space, i.e. that each sample point in S has the same probability.
Example 3.7: Let a card be selected a t random from an ordinary deck of 52 cards. Let
A = {the card is a spade}
and B = {the card is a face card, i.e. a jack, queen or king}
We compute P(A),P(B) and P(A nB). Since we have an equiprobable space,
number of face cards 12 3
- 52 - 13
number of cards
P(B) =
number of spades - 13 2
number of cards - 52 - 4
P(A)=
number of spade face cards - 3
P(AnB) = number of cards - 52
Example 3.8: Let 2 items be chosen at random from a lot containing 12 items of which 4 are
defective. Let
A = (both items are defective) and B = {both items are non-defective)
Find P(A) and P(B). Now
S can occur in (y) =
A can occur in (i) =
B can occur in (l)
=
Accordingly, P(A)=4=
66 ways, the number of ways that 2 items can be
chosen from 12 items;
6 ways, the number of ways that 2 defective items
can be chosen from 4 defective items;
28 ways, the number of ways that 2 non-defective
items can be chosen from 8 non-defective items.
and P(B)= =g.
Question: What is the probability that at least one item is defective? Now
C = (at least one item is defective)
is the complement of B; that is, C = Bc. Thus by Theorem 3
.
2
,
P(C) = P(BC) = 1 - P(B) = 1 -g = g
The odds that an event with probability p occurs is defined to be the ratio p : (1-p).
Thus the odds that a t least one item is defective is g: or 19:14 which is read
“19 to 14”.

CHAP. 31 INTRODUCTION TO PROBABILITY 43
Example 3.9: (Classical Birthday Problem.) We seek the probability p that n people have dis-
tinct birthdays. In solving this problem, we ignore leap years and assume that a
person’s birthday can fall on any day with the same probability.
Since there are n people and 365 different days, there are 365n ways in which
the n people can have their birthdays. On the other hand, if the n persons are to
have distinct birthdays, then the first person can be born on any of the 365 days,
the second person can be born on the remaining 364 days, the third person can be
364 363 * - (365 -n 4-1)
born on the remaining 363 days, etc. Thus there are 365
ways the n persons can have distinct birthdays. Accordingly,
365 364 -*-.-.
363 - - - (365 - n +1) - 365 364 363 .. 365 - n + 1
P = 365n - 365 365 365 365
It can be shown that for n 2 23, p < -&;
in other words, amongst 23 or more
people it is more likely that at least two of them have the same birthday than
that they all have distinct birthdays.
INFINITE SAMPLE SPACES
Now suppose S is a countably infinite sample space; say S = {al,u2, ...}. As in the
finite case, we obtain a probability space by assigning to each uiE S a real number Pi, called
its probability, such that
CO
(i) pi 2 0 and (ii) p1 + p 2 + ... = C p i = 1
i= 1
The probability P(A)of any event A is then the sum of the probabilities of its points.
Example 3.10: Consider the sample space S = (1,2,3, ...,a) of the experiment of tossing a
coin till a head appears; here n denotes the number of times the coin is tossed.
A probability space is obtained by setting
p ( 1 ) = 4, p(2) = i, ..., p(n) = l/Zn, ...) 1.403) = o
The only uncountable sample spaces S which we will consider here are those with some
finite geometrical measurement m(S) such as length, area or volume, and in which a point
is selected at random. The probability of an event A, i.e. that the selected point belongs
to A, is then the ratio of m(A)to m(S);that is,
length of A area of A volume of A
= lengthofs Or = area of S or = volume of^
Such a probability space is said to be uniform.
Example 3
.
1
1
: On the real line R,points a and b are selected at random such that -2 L b 4 0
and 0 4 a f 3, as shown below. Find the probability p that the distance d be-
tween a and b is greater than 3.
- Y d + -
I I I
-2 b 0 a 3
The sample space S consists of the or-
dered pairs (a,b) and so forms the rectangu-
lar region shown in the adjacent diagram.
On the other hand, the set A of points (a,
b)
for which d = a- b > 3 consists of those
points of S which lie below the line x -y = 3,
and hence forms the shaded area in the
diagram. Thus
-
area of A - z- I
p = P(A) =
area of S - 6 - 3
Remark: A finite or countably infinite probability space is said to be discrete, and an
uncountable space is said to be nondiscrete.

Solved Problems
SAMPLE SPACES AND EVENTS
3.1. Let A and B be events. Find an expression and exhibit the Venn diagram for the
event that: (i) A but not B occurs, i.e. only A occurs; (ii) either A or B, but not
both, occurs, i.e. exactly one of the two events occurs.
(i) Since A but not B occurs, shade the area of A outside of B as in Figure (a)below. Note
that Be, the complement of B, occurs since B does not occur; hence A and Bc occurs. In other
words, the event is A nBc.
(ii) Since A or B but not both occurs, shade the area of A and B except where they intersect as
in Figure (b) above. The event is equivalent to A but not B occurs or B but not A occurs.
Now, as in (i), A but not B is the event A nBc, and B but not A is the event B nAc. Thus
the given event is (A nBc) U (BnAc).
3.2. Let A, B and C be events. Find an expression and exhibit the Venn diagram for
the event that (i) A and B but not C occurs, (ii) only A occurs.
(i) Since A and B but not C occurs, shade the intersection of A and B which lies outside of C,
as in Figure (a) below. The event is A nB nC C .
(ii) Since only A is to occur, shade the area of A which lies outside of B and of C, as in Figure (b)
above. The event is A nBcnC C .
33. Let a coin and a die be tossed; let the sample space S consist of the twelve elements:
S = {Hl,
H2,H
3
,H4,H5,H
6
,Tl,T2,T3,T4,T5,T6}
(i) Express explicitly the following events: A = {heads and an even number ap-
pear}, B = {aprime number appears}, C = {tails and an odd number appear}.
(ii) Express explicitly the event that: (a) A or B occurs, (b) B and C occurs,
(c) only B occurs.
(iii) Which of the events A, B and C are mutually exclusive?

(i) To obtain A, choose those elements of S consisting of an H and an even number: A =
{H2, H4, H6).
To obtain B, choose those points in S consisting of a prime number: B = (H2, H3, H5, T2, T3, T
5
)
.
To obtain C, choose those points in S consisting of a T and an odd number: C = (Tl, T3, T5).
(ii) (a) A or B = A u B = {H2,H4, H6, H3, H5, T2, T3, T5)
(b) B and C=BnC={T3,T5}
(c) Choose those elements of B which do not lie in A or C: BnAcnCC = (H3, H5, T2).
(iii) A and C are mutually exclusive since A nC = @.
FINITE PROBABILITY SPACES
3.4. Suppose a sample space S consists of 4 elements: S = {al,a2, a3, a4>. Which func-
tion defines a probability space on S?
(i) P(a1)= 4, P(a2)= Q, P(a3)= &, P(a4)= 3.
(ii) P(a1)= 8, P(a2)= $, P(a3)= -$, P(a4)= +.
(iii) P(a1)= 4, P(a2)= &, P(a3)= Q, P(a4)= Q.
(iv) P(a1)= 8, P(a2)= &, P(a3)= &, P(a4)= 0.
77
(i) Since the sum of the values on the sample points is greater than one, 8+Q +& +5= a,
the function does not define a probability space on S.
(ii) Since P(a3)= -&, a negative number, the function does not define a probability space on S.
(iii) Since each value is nonnegative, and the sum of the values is one, 6+& +Q +Q = 1, the
function does define a probability space on S.
(iv) The values are nonnegative and add up to one; hence the function does define a probability
space on S.
3.5. Let S = {al,a2, a3, a4}, and let P be a probability function on S.
(i) Find P(a1)if P(a2)= Q, P(a3)= 9, P(a4)= Q.
(ii) Find P(a1)and P(a2)if P(a3)= P(a4)= & and P(u1)= 2P(a2).
(iii) Find P(a1)if P((a2,as})= 8, P((a2,a4})= 4 and P(a2)= Q.
(i) Let P(al)= p. Then for P to be a probability function, the sum of the probabilities on the
sample points must be one: p +Q +Q +$ = 1 or p = 5.
(ii) Let P(a2)= p, then P(al)= 2p. Hence 2 p + p + &+& = 1 or p = 9. Thus P(a2)= Q and
P(a1) ==Q.
(iii) Let P(aJ = p. P(a3) = P({az,a3N -P(a2) = Q -Q = -g
W 4 ) = Wa2, a
4
N -P(a,) = +-Q = Q
Then p +Q +Q +Q = 1 or p = 9,that is, P(al)= Q.
3.6. A coin is weighted so that heads is twice as likely to appear as tails. Find P(T)
and P(H).
Let P(T) = p; then P(H) = 2p. Now set the sum of the probabilities equal to one: p +2p = 1
or p = 9. Thus P(T) = p = Q and P(H)
= 2 p = 6.

Two men, ml and m2, and three women, WI,w zand w3,are in a chess tournament.
Those of the same sex have equal probabilities of winning, but each man is twice
as likely to win as any woman. (i) Find the probability that a woman wins the
tournament. (ii) If ml and WI are married, find the probability that one of them
wins the tournament.
Set P(wl)
= p ; then P(w2)
= P(w3)
= p and P(ml) = P(m2)= 2p. Next set the sum of the
probabilities of the five sample points equal to one: p +p +p +2p +2 p = 1 or p = 3.
We seek (i) P({wl,w2,w3})and (ii) P({ml,wl}).Then by definition,
P({Wl,W2,
w3})= P(Wl)
+P ( W 2 ) +P(w3) = 3+ 3+ 3 = +
= P(m,) +P(Wl)= 3+3 = $
3.8. Let a die be weighted so that the probability of a number appearing when the die
is tossed is proportional to the given number (e.g. 6 has twice the probability of
appearing as 3). Let A = {even number}, B = {primenumber}, C = (odd number).
(i) Describe the probability space, i.e. find the probability of each sample point.
(ii) Find P(A),P(B)and P(C).
(iii) Find the probability that: (a) an even or prime number occurs; (6) an odd
prime number occurs; (c) A but not B occurs.
(i) Let P(l) = p . Then P ( 2 ) = 2p, P(3) = 3p, P(4) = 4p, P(5) = 5 p and P ( 6 ) = 6p. Since the sum
of the probabilities must be one, we obtain p +2 p +3p +4 p +5p +6 p = 1 or p = 1/21. Thus
P(1) = $, P ( 2 ) = 6, P(3) = 3, P(4) = 2, P ( 5 ) = &, P ( 6 ) = 3
(ii) P(A) = P ( { 2 , 4 , 6 } ) = 3, P(B) = P({2,3,5}) = E, P(C) = P({1,3,5}) = 3.
(iii) (a) The event that an even or prime number occurs is A UB = { 2 , 4 , 6 , 3 , 5 } , or that 1 does
20
not occur. Thus P(A UB) = 1 - P(l) = h.
(b) The event that an odd prime number occurs is BnC = {3,5}. Thus P(BnC) =
8
P({3,5H = 5.
(c) The event that A but not B occurs is A nBc = {4,6}. Hence P(AnBc) = P({4,6}) =g.
FINITE EQUIPROBABLE SPACES
3.9. Determine the probability p of each event:
(i) an even number appears in the toss of a fair die;
(ii) a king appears in drawing a single card from an ordinary deck of 52 cards;
(iii) at least one tail appears in the toss of three fair coins;
(iv) a white marble appears in drawing a single marble from an urn containing
4 white, 3 red and 5 blue marbles.
(i) The event can occur in three ways (a 2 , 4 or 6 ) out of 6 equally likely cases; hence p = = f.
1
(iij There are 4 kings among the 52 cards; hence p =
4
= B.
(iii) If we consider the coins distinguished, then there are 8 equally likely cases: HHH,HHT,
HTH,HTT,THH,THT,TTH,TTT.
Only the first case is not favorable to the given event;
7
hence p =5.
(iv) There are 4 +3 +5 = 12 marbles, of which 4 are white; hence p = = $.

3
.
1
0
. Two cards are drawn at random from an ordinary deck of 52 cards. Find the proba-
bility p that (i) both are spades, (ii) one is a spade and one is a heart.
There are (",") = 1326 ways to draw 2 cards from 52 cards.
(i) There are (123) = 78 ways to draw 2 spades from 13 spades; hence
number of ways 2 spades can be dr;= - 78 - -
1
= number of ways 2 cards can be dra.wn 1326 17
(ii) Since there are 13 spades and 13 hearts, there are 13.13 = 169 ways to draw a spade and a
heart; herice p = = 2
102-
3
.
1
1
. Three light bulbs are chosen at random from 15 bulbs of which 5 are defective.
Find the probability p that (i) none is defective, (ii) exactly one is defective, (iii)at
least one is defective.
There are (135) = 455 ways to choose 3 bulbs from the 15 bu.lbs.
(i) Since there are 15 - 5 = 10 nondefective bulbs, there are ('i)= 120 ways to choose 3 non-
defective bulbs. Thus p = 3= %.
(ii) There are 5 defective bulbs and (y)= 45 different paiirs of nondefective bulbs; hence there
are 5 225 45
45 = 225 ways to choose 3 bulbs of which one is d.efective. Thus p = 455 = z.
(iii) The event that at least one is defective is the complement of the event that none are defective
which has, by (i),probability E. Hence p = 1-3= E.
3
.
1
2
. Two cards are selected at random from 10 cards numbered 1 to 10. Find the proba-
bility p that the sum is odd if (i) the two cards are drawn together, (ii) the two
cards are drawn one after the other without replacement, (iii) the two cards are
drawn one after the other with replacement.
(i) There are (y)= 45 ways to select 2 cards out of 10. The sum is odd if one number is odd
and the other is even. There are 5 even numbers and 5 odd numbers; hence there are
5.5 = 25 ways of choosing an even and an odd number. Thus p = = i.
(ii) There are 10.9 = 90 ways to draw two cards one after the other without replacement.
5 = 25
There are 5 5 = 25 ways to draw an even number and then an odd number, and 5
ways to draw an odd number and then an even number; hence p = 9
= 2 = :.
(iii) There are 10.10 = 100 ways to draw two cards one :after the other with replacement. As
in (ii), there are 5.5 = 25 ways to draw an even number and then an odd number, and
25+25 50
5.5 = 25 ways to draw an odd number and then an even number; hence p = 1
0
0 = 100 = p
1
3.13. Six married couples are standing in a room.
(i) If 2 people are chosen at random, find the probability p that (a) they are
married, (b)one is male and one is female.
(ii) If 4 people are chosen at random, find the probability p that (a)2 married
couples are chosen, (b) no married couple is among the 4, (c) exactly one
married couple is among the 4.
(iii) If the 12 people are divided into six pairs, find the probability p that (a) each
pair is married, (b)each pair contains a male and a female.

(i) There are (y)= 66 ways to choose 2 people from the 12 people.
(a) There are 6 married couples; hence p = &=A.
6.6
(b) There are 6 ways to choose a male and 6 ways to choose a female; hence p = gs = fi.
(ii) There are (y)= 495 ways to choose 4 people from the 12 people.
15 1
(a) There are (i)= 15 ways to choose 2 couples from the 6 couples; hence p = 495 = 33.
6
(b) The 4 persons come from 4 different couples. There are t4) = 15 ways to choose 4 couples
from the 6 couples, and there are 2 ways to choose one person from each couple. Hence
2.2-2.2.15
p = - - 49s - 3
3
'
(c) This event is mutually disjoint from the preceding two events (which are also mutually
disjoint) and at least one of these events must occur. Hence p +&+ = 1 or p =Is
33
12!
(iii) There are 2
!
2
1
i
:
A
!
2
!
2
! = 26 ways to partition the 12 people into 6 ordered cells with 2 people
in each.
(a) The 6 couples can be placed into the 6 ordered cells in 6! ways. Hence p = 12!/26 = 10,395
(b) The six men can be placed one each into the 6 cells in 6! ways, and the 6 women can be
placed one each into the 6 cells in 6! ways. Hence p = 8!6!
12!/26-
- 231 *
3
.
1
4
. A class contains 10 men and 20 women of which half the men and half the women
have brown eyes. Find the probability p that a person chosen at random is a man
or has brown eyes.
Let A = {person is a man} and B = {person has brown eyes}. We seek P(A UB).
Then P(A)= = f, P(B) = = i, P(AnB) = 6= $. Thus by Theorem 3.5,
p = P ( A u B ) = P(A) +P(B)- P(AnB) = g + +- Q = Q
UNCOUNTABLE UNIFORM SPACES
3
.
1
5
. A point is selected at random inside a circle. Find the
probability p that the point is closer to the center of the
circle than to its circumference.
Let S denote the set of points inside the circle with radius r,
and let A denote the set of points inside the concentric circle of
radius Qr. (Thus A consists precisely of those points of S which
are closer to its center than to its circumference.) Accordingly,
area ofA - ~ ( Q T ) ~ 1
-
p = P(A) = - - - -
area of S rr2 4
3
.
1
6
. Consider the Cartesian plane R2,and let X denote the subset of points for which
both coordinates are integers. A coin of diameter 4 is tossed randomly onto the
plane. Find the probability p that the coin covers a point of X.
Let S denote the set of points inside a square with corners
(m,n), (m,n+l), (m+l,n), ( m + l , n + l ) E X
Let A denote the set of points in S with distance less than $ from
any corner point. (Observe that the area of A is equal to the area
inside a circle of radius &.) Thus a coin whose center falls in S will
cover a point of X if and only if its center falls in a point of A.
Accordingly,
area of A - ~($1' - T
1
p = P(A) =
area of S
- - - - - x
16 '2
Note. We cannot take S to be all of R
2 because the latter has
infinite area.

49
CHAP. 31 INTRODUCTION TO PROBABILITY
3
.
1
7
. Three points a, b and c are selected at random from the
circumference of a circle. Find the probability p that the
points lie on a semicircle.
Suppose the length of the circumference is 2s. Let x denote
the clockwise arc length from a to b, and let y denote the clockwise
arc length from U to c. Thus
O < x < 2 s and O < y < 2 s (*)
Let S denote the set of points in R2 for which condition (*) holds.
Let A denote the subset of S for which any of the following condi-
tions holds:
(i) x,y< s (iii) x < s and y - x > s
(ii) x,y > s (iv) y < s and x- y > s
Then A consists of those points for which a, b and c lie on a semi-
circle. Thus
area of A -
- 3s2 3
P = area of s 4.92 - 4
MISCELLANEOUS PROBLEMS
3
.
1
8
. Let A and B be events with P(A)= Q, P(B)= 4 and P(AnB) = -S;. Find (i) P(AUB),
P(Ac)and P(Bc), (iii) P(AcnBC),(iv) P ( A c u B c ) , (v) P(An B c ) , (vi) P(BnAC).
P(AuB) = P(A)+P(B)- P(AnB) = Q ++- $ = Q
P(Ac) = 1 - P ( A ) = 1 - Q = Q and P(Bc) = 1-P(B) = 1- -& = 4
Using De Morgan’s Law, (A uB)c = ACnBC, we have
P(AcnBC) = P((AuB)c) = 1 - P(AuB) = 1 - Q = Q
Using De Morgan’s Law, ( AnB)c = ACUBC, we have
P(AcuBc) = P((AnB)c) = 1- P(AnB) = 1- -S; = 3
Equivalently,
’
P(AcuBc) = P(Ac)+P(Bc)-P(AcnBC) = Q +4- Q = $
P(AnBC) = P(AB) = P(A)- P(AnB) = 8 - & = Q
P(BnAC) = P ( B ) - P ( A n B ) = &--& = &
3
.
1
9
. Let A and B be events with P(AuB) = 3, P(Ac)= Q and P(AnR) = &. Find
(i) P(A), (ii) P(B), (iii) P(AnBc).
(i) P(A) = 1 -P(Ac) = 1 - Q = +
(ii) Substitute in P(AUB) = P(A) +P(B)- P(AnB) to obtain 3 = Q +P(B)- & or P(B) = -&
(iii) P(AnBC) = P(A)- P(AnB) = Q - & = &
3.20. Find the probability p of an event if the odds that it will occur are a:b, that is,
“a to b”.
The odds that an event with probability p occurs is the ratio p : (1- p ) . Hence
a
l - - p - b
P -
-
U
or b p = a - a p or a p + b p = a or p = -a + b
3.21. Find the probability p of an event if the odds that it will occur are “3 to 2”.
p - 3
l--p - 5 from which p = j . We can also use the formula of the preceding problem to obtain
the answer directly: p = &=&= g.

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