Chapter 2
– Probability
1.
If an operation can be performed in n
1
ways, and for each of these a second operation can be performed in n
2
ways, and for each of
the first two a third operation can be performed in n
3
ways, and so forth, then the sequence of k operations can be performed in n
1
n
2
…n
k
ways.
2.
The number
of permutations of n objects is n!
3.
The number of permutations of n distinct objects taken r at a time is
n
P
r
=
n
!
(
n
!
r
)!
.
4.
The number of permutations of n
objects arranged in a circle is (n1)!.
5.
The number of distinct permutations of n things of which n
1
are of one kind, n
2
of a second kind,..., n
k
of a kth kind is
n
!
n
1
!
n
2
!
!
n
k
!
.
6.
The number of ways of partitioning a set of n objects into r cells with n
1
elements in the first cell, n
2
elements in the second, and so forth is
n
n
1
,
n
2
,...
n
r
!
"
#
$
%
&
=
n
!
n
1
!
n
2
!
!
n
r
!
.
7.
The number of combinations of n distinct objects taken r at a time is
n
r
!
"
#
$
%
&
=
n
!
r
!(
n
’
r
)!
.
8.
If an experiment can result in any
one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is
P
(
A
)
=
n
N
.
9.
The
conditional probability of B, given A, denoted
P
(
B

A
)
=
P
(
A
!
B
)
P
(
A
)
.
10.
Two events A and B are independent if and only if P(BA) = P(B) or P(AB) =
P(A).
12.
Two events A and B are independent if and only if
P
(
A
!
B
)
=
P
(
A
)
P
(
B
)
.
13.
If the events B
1
, B
2,
…, B
k
constitute a partition of the sample
space S such that P(B
i
)
≠
0 for i = 1,2,…,k, then for any event A of S,
P
(
A
)
=
P
(
B
i
!
A
)
=
P
(
B
i
)
P
(
A

B
i
)
i
=
1
k
"
i
=
1
k
"
.
14.
(Bayes Rule) If the events B
1
, B
2
,
…,B
k
constitute a partition of the sample space S such that P(B
i
)
≠
0 for i = 1,2,…,k, then for any event A in S such that P(A)
≠
0,
P
(
B
r

A
)
=
P
(
B
r
!
A
)
P
(
B
i
!
A
)
i
=
1
k
"
=
P
(
B
r
)
P
(
A

B
r
)
P
(
B
i
)
P
(
A

B
i
)
i
=
1
k
"
.

Chapter 3
– Random Variables and Probability Distributions
1.
The set of ordered pairs (x, f(x)) is a
probability function, probability mass function, or probability distribution of the discrete random variable X if, for each possible outcome x, a. f(x)
≥
0, b.
f
(
x
)
x
!
=
1
, c. P(X=x) = f(x).
2.
The cumulative distribution function F(x) of a discrete random variable X with probability distribution f(x) is
F
(
x
)
=
P
(
X
!
x
)
=
f
(
t
)
t
!
x
"
for 
∞
< x<
∞
.
3.
The function f(x) is a probability density function for the continuous random variable X, defined over the
set of real numbers R, if a.
f
(
x
)
!
0
"
x
#
R
, b.
f
(
x
)
dx
=
1
!"
"
#
, c.
P
(
a
<
X
<
b
)
=
f
(
x
)
dx
a
b
!
.
4.
The cumulative distribution function F(x) of a
continuous random variable X with density function f(x) is
F
(
x
)
=
P X
!
x
(
)
=
f
(
t
)
dt
"#
x
$
for 
∞
< x<
∞
.
5.
The function f(x,y) is a joint probability
distribution or probability mass function of the discrete random variables X and Y if a. f(x,y)
≥
0 for all (x,y), b.
f
(
x
,
y
)
y
!
=
1
x
!
, c. P(X=x,Y=y) =
f(x,y).
6.
For any region A in the xy plane, P[(X,Y)
∈
A] =
f
(
x
,
y
)
!
A
!
.
7.
The function f(x,y) is a joint density function of the continuous random
variables X and Y if a. f(x,y)
≥
0 for all (x,y), b.
f
(
x
,
y
)
dxdy
=
1
!"
"
#
!"
"
#
, c. P[(X,Y)
∈
A] =
f
(
x
,
y
)
dxdy
A
!!