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Chapter
2.
Probability
space
Sample space
In the study of statistics we are
concerned with the presentation and interpretation
of
chance outcomes
that
occur in a planned
study or scientific
investigation.
For example, we may record the number
of accidents that occur monthly at the
intersection of Driftwood Lane and Royal Oak
Drive, hoping to justify the
installation of a traffic light; we might classify
items coming of an assembly line as
is varied.
The
statistician
is
often
dealing
with
either
experimental
data,
representing
counts
or
measurements
,
or
perhaps
with
categorical
data that can be
classified according to some criterion.
We shall refer to any
recoding of information, whether it be numerical
or categorical, as an
observation.
The number 2, 0, 1 and 2, representing
the number of accidents that occurred for each
month from January through April during
the past year at the intersection of
Driftwood Lane and Royal Oak Drive, constitute a
set of observations.
The
categorical
data
N,
D,
N,
D
and
D,representing
the
items
found
to
be
defective
or
non
defective
when
five
items
are
inspected, are recorded
as observations.
experiment
Experiment
：
any process that generates a set of
data.
tossing of a coin is
a statistical experiment .There are only two
possible outcomes, heads or tails.
Another experiment might be the
launching of a missile and observing its velocity
at specified times.
The
opinions of voters concerning a new sales tax can
also be considered as observations of an
experiment.
particularly
interested in the observations obtained by
repeating the experiment several times.
example:
a coin is tossed repeatedly
?
cannot predict the result of a given toss
(i.e.
the
outcome depend on chance)
? know the
entire set of possibilities for each toss.
Sample space
Definition 2.1
The set of all possible outcomes of a
statistical experiment is called the
sample space
and is
represented by S.
Each
outcome in a sample space is called an
element
or a
sample point.
What's the possible
outcomes when a coin is tossed?
The sample space may be written
S
=
｛
H,
T
｝
H and T
corresponds to 'heads' and
'tails', respectively.
Example 2.1
Consider the experiment of tossing a
die. interested in the number that shows on the
top face
S
1
=
｛
S
2
=
｛
｝
｝
S
1
provides
more
information
than
S
2
If
we
know
which
elements
in
S
1
occurs,
we
can
tell
which
outcome
in
S
2
occurs;
however, a knowledge of what happens in
S
2
is of little help in determining which
element in
S
1
occurs.
Remarks:
? more
than one sample space can be used to describe
the
outcomes of an experiment.
? It is desirable to use a
sample space that gives the most information.
Example
2.3
Suppose
that
three
items
are
selected
at
random
from
a
manufacturing
process.
Each
item
is
inspected
and
classified defective, D, or non
defective, N.
List the
elements of the sample space providing the most
information.
The sample
space is
S
=
｛
DDD, DDN, DND, DNN, NDD,
NDN, NND, NNN
｝
.
sample space with a large
or infinite number of sample points
，
How to describe?
by a statement or rule
examples
? S
=
｛
x |
｝
?
all points (x,y) on the boundary or the interior
of a circle of radius 2 with center at the origin
sample space
with a large or infinite number of sample points
，
How to describe?
by a statement or rule
examples
? S
=
｛
x | x is a city with a
population over 1 million
｝
? all points (x,y) on the
boundary or the interior of a circle of
radius 2 with
center at the origin
｛
|
x
2
+
y
2
≤
｝
Event
For any
given experiment we may be interested in the
occurrence of certain
events
rather than in the outcome of a specific
element in the sample space.
event A:
the outcome (when a die is tossed) is
divisible by 3.
Event A will occur if the outcome is an
element of the set A =
｛
3,
6
｝
A
=
｛
3,
6
｝
is the subset
of the sample space
S
1
=
｛
｝
in
Example 2.1.
event B: the number of defectives is
greater than 1 in Example 2.3; this will occur if
the outcome is an element of the subset
B
=
｛
｝
Definition 2.2
An
event
is a subset of a
sample space.
Example 2.4.
Given the sample space S
=
｛
t|t
≥
0
｝
, where t is the life in
years of a certain electronic component.
Event A
is that the component fails before the
end of the fifth year.
subset A =
｛
t|0
≤
t < 5}
Two special
subset
a subset that includes the
entire sample space S
a subset
contains no elements at all, denoted by
?
, called
null
set.
｛
x|x
is an even factor of 7
｝
then B must be the null set, since the
only possible factors of 7 are odd numbers 1 and
7.
complement
Definition 2.3
The
complement
of
an event A with respect to S is the subset of all
elements of S that are not in A. We denote
the complement of A by the symbol
A
’
Example 2.5
Let R be the
event that a red card is selected from an ordinary
deck of 52 playing cards and let S be the entire
deck.
What is
R
’
?
R
’
is the event
that the card selected from the deck is not red
but a black card.
intersection
operations
with events result in the formation of
new events
Definition
2.4
The
intersection
of
two
events
A
and
B,
denoted
by
the
symbol
A
∩
B,
is
the
event
containing
all
elements that are common to A and B.
Example 2.7
Let P be the
event that a person selected at random while
dining at a popular cafeteria is a taxpayer, and
let Q be the event
that the person is over 65 years of
age.
Then the event
P
∩
Q is the set of all
taxpayers in the cafeteria who are over 65
years of age.
Example 2.8
Let M
=
｛
｝
and N
=
｛
r, s,
t
｝
; then it follows that
M
∩
?
That is, M
and N
have no elements in common
and, therefore, cannot both occur simultaneously.
Definition
2.5
Two events A and B are
mutually exclusive
, or
disjoint
if
A
∩
?
,
that is, if A and B have no elements in
common.
Union
Definition 2.6
The
union
of the
two events A and B, denote by symbol
A
∪
B, is the event containing
all the elements that
belong to A or B
or both.
Example 2.10
Let A
=
｛
a, b,
c
｝
A
∪
｛
｝
; then
｛
a, b, c, d,
e
｝
.
Example 2.11
Let
P
be the event that an
employee selected at random from an oil drilling
company smokes cigarettes.
Let
Q be the event that the
employee selected drinks alcoholic beverages. Then
the event P
∪
Q is the set of
all employees who
either drink or
smoke, or do both.
Example
2.12
If M
=
｛
x|3
< x < 9
｝
｛
x|5 < x
< 12
｝
then
M
∪
N
=
｛
x|3 < x <
12
｝
Several results
The
relationship
between
events
and
the
corresponding
sample
space
can
be
illustrated
graphically
by
means
of
Venn
diagrams
.
A
∩
?
?
，
A
∪
?
= A
A
∩
A
’
=
?
，
A
∪
A’
= S
S’=
?
?
’=S
（
A’
）
’=A
（
A∩B
）
’=A’
∪
B’
，
（
A
∪
B
）
’=A’∩B’
ng sample points
One
the
problems
that
the
statistician
must
consider
and
attempt
to
evaluate
is
the
element
of
chance
associated
with
the
occurrence of certain
events when an experiment is performed.
In many cases we shall be
able to solve a probability problem by counting
the number of points in the sample space without
actually listing each element.
Multiplication
rule
Theorem 2.1
If an operation can be performed in
n
1
ways, and if
for each of these a second operation can be
performed in
n
2
ways, then the
two operations can be performed together in
n
1
n
2
ways.
Example 2.13
How many sample points are
in the sample space when a pair of dice is thrown
once?
Solution
The first dice can land in any one
of
n
1
= 6 ways. For
each of these 6 ways the second die can also land
in
n
2
= 6
ways.
Therefore,
the pair of dice can land in
n
1
n
2
=
6×
6 = 36
possible ways.
The multiplication rule of Theorem 2.1
may be extended to cover any number of operations.
Theorem 2.2
If an operation can be performed
in
n
1
ways, and if
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 i n
3
ways, and so forth, then the sequence
of k operations
can be performed in
n
1
n
2
. .
n
k
ways.
Example
2.15
Sam is going to
assemble a computer by himself.
He has the choice of ordering chips
from two brands, a
hard drive from
four, memory from three and an accessory bundle
from five local stores.
How
many different ways can Sam
order the
parts?
Solution
Since
n
1
=
2,
n
2
=
4,
n
3
= 3,
and
n
4
= 5, there
are
n
1
×
n
2
×
n
3
×
n
4
= 2×
4×
3×
5 =
120
different ways to order the parts.
Permutation
How many different
arrangements are possible for sitting 6 people
around a table?
How many
different orders are possible for drawing 2
lottery tickets from a total of 20?
The different arrangements are called
permutation.
Definition 2.7
A permutation is an
arrangement of all or part of a set of objects.
Consider the
three letters a, b, and c.
The possible permutations are abc, acb,
bac, bca, cab, and cba.
There are 6 different arrangement.
Using Theorem 2.2 we could
arrive at the answer 6 without listing the
diferent orders.
There are
n
1
= 3 choices
for the first position, then
n
2
= 2 for
the second, leaving
only
n
3
= 1 choice
for the last position, giving a total of
n
1
×
n
2
×
n
3
= 3×
2×
1 = 6
permutations.
Theorem 2.3
The
number of permutations of n distinct objects is
n!.
n
! is read 'n factorial'
n! = n(n-1)(n-
2)…(3)(2)(1)
The number of
permutations of the four letters a, b, c, and d
will be 4! = 24.
What's the
number of permutations that are possible by taking
the four letters two at a time?
Using Theorem 2.1, we have
n
1
= 4 choices
for the first position and
n
2
= 3 for the
second.
A total of
< br>n
1
n
2
= 12 permutations.
In
general, n distinct objects taken r at a time can
be arranged in n(n-1)(n-
- r + 1) ways.
Theorem 2.4
The number of permutations of n
distinct objects taken r at a time is
Example 2.17
Three awards (research, teaching and
service) will be given one year for a class of 25
graduate students in a
statistic
department.
If each student
can receive at most one award, how many possible
selections are there?
Solution
Since
the awards are distinguishable, it is a
permutation problem.
The
total number of sample points is
circular
permutation:
permutation that occur by arranging
objects in a circle
Example.
4
people
are
playing
bridge,
we
do
not
have
a
new
permutation
if
they
all
move
one
position
in
a
clockwise
direction.
By
considering one person in a fixed position and
arranging the other three in 3! ways.
Theorem 2.5
The
number of permutations of n distinct objects
arranged in a circle is (n-1)!
combinations
combinations:
the number of ways of
selecting r objects from n without regard order.
The number of such combinations is
denoted by
C
r,n
Consider the four letters a, b, c, and
d.
? the ways of selecting
one letter from four? a
,
b, c, and d that is
? the ways of selecting two letters
from four? ab, ac, ad, bc, bd, cd; that is
? the ways of selecting three letters
from four?
abc, abd, acd, bcd; that is
Theorem 2.8
The number of combinations
of n distinct objects taken r at a time is
Question 1
: What's the
number of distinct permutations of n things of
which
n
1
of one
kind,
n
2
of a
second kind,...,
n
k
of a
k
th
kind?
hint: n
objects, n positions
Example
2.19
In
a
college
football
training
session,
the
defensive
coordinator
needs
to
have
10
player
standing
in
a
row.
Among these
10 players, there are 1 freshman, 2 sophomore, 4
juniors and 3 seniors, respectively.
How many different ways
can
they be arranged in a row if only their class
level will be distinguished?
Solution
The
total number of arrangements is
Question 2
:
What's the number of
arrangements of 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?
Example
2.20
In how many
ways can 7 scientists be assigned to one triple
and two double hotel rooms?
Solution
:
The total number of possible partitions
would be
4.
Probability of an Event
Perhaps it was
man's unquenchable thirst for gambling that led to
the early development of probability theory.
In an effort to
increase their winnings, gamblers
called upon mathematicians to provide optimum
strategies for various games of chance.
Some of the mathematicians
providing these strategies were Pascal, Leibniz,
Fermat, and James Bernoulli.
Probability
theory
has
branched
out
far
beyond
games
of
chance
to
encompass
many
other
fields
associated
with
chance
occurrences, such as
politics, business, weather forecasting, and
scientific research.
probability
The remainder of chapter 2 only
consider sample space contains a finite number of
elements
To every point in
the sample space we assign a probability such that
the sum of all probabilities is 1.
The probability of an event A is
denoted by P (A).
Definition 2.8
The
probability of an event A is the sum of the
weights of all sample points in A.
Therefore,
0≤P (A)≤1,
P (
?
)
= 0
P (S) = 1,
Furthermore, if
A
1
,
A
2
,
A
3
,
…
is
a sequence of mutually exclusive events, then
Example 2.22
A coin is tossed twice.
What is the probability that at least
one head occurs?
Solution
The sample space is S
=
｛
HH, HT, T H, T
T
｝
The coin is
balanced, each of these outcomes would be equally
likely to occur.
That is,
the probability of each sample point is
1/4.
event A:
at least one head
occurring.
A
=
｛
HH, HT, T
H
｝
Example 2.23
A
dice is loaded in such a way that an even number
is twice as likely to occur as an odd number.
If E is the
event that a number less than 4 occurs
on a single toss of the die, find P (E).
Solution
The sample space is S
=
｛
1, 2, 3, 4, 5,
6
｝
assign a
probability of
w
to each odd number, and 2 to each even
number, we have 9
w
= 1.
1/ 9 , and 2/9 are assigned
to each odd and even number, respectively.
Since E
=
｛
1, 2,
3
｝
Example 2.24
In Example 2.23 let A be the event that
an even number turns up and let B be the event
that a number divisible
by 3 occurs.
Find P (A
∪
B) and P
(A∩B)
Solution
We can get A =
｛
2,
4,
6
｝
A
∪
B
=
｛
2,
3
，
4,
6
｝
and
A∩ B
=
｛
6
｝
assign:
1/
9→each odd number,
2/9→ each even number,
we
have
If the sample space
contains N
elements, all of
which are equally likely to occur, We assign 1/N
to each element.
Event A containing n of
these N
sample points. P
(A)?
Theorem 2.9 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
Example 2.25
A statistic class for
engineers consists of 25 industrial, 10
mechanical, 10 electrical, and 8 civil engineering
students.
If a
person is randomly selected by the instructor to
answer a question, find the probability that the
student chosen is
(a) an industrial
engineering major, (b) a civil engineering or an
electrical engineering major.
Solution
Denote by I, M, E, and C the students
majoring in industrial, mechanical, electrical and
civil engineering, respectively.
students in the class:
equally to be selected;
the total
number:
53.
(a) Since 25 of the 53 students are
majoring in industrial engineering, the
probability of the event I is
(b) Since 18 of
the 53 students are civil and electrical
engineering majors, it follows that
.
｛
3,
6
｝
, therefore,
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