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Section 7.1 Experiments, Sample Space, and Events
Terminology
Experiment is an activity with observable results.
Outcomes of the experiment are the results of the
experiment.
Sample Point is an outcome of an experiment.
Sample Space is the set of consisting of all possible
sample points of an experiment.
Event is a subset of a sample space of an experiment.
If E and F are two events, then, the union of two events,
the event E  F, the intersection of two events, the event
E  F, the complement of an event E, the event E C , and
the mutually exclusive events E and F if E  F =  .
Example 1:
Consider the experiment of casting a die and observing
the number that falls uppermost. Let S {1, 2, 3, 4, 5, 6}
denote the sample space of the experiment and E = {2, 4,
6} and F = {1, 3} be events of this experiment. Compute
a) E  F, b) E  F, and E C .
Example 2:
An experiment consists of tossing a coin three times and
observing the resulting sequence of “heads” and “tails”.
a) Describe the sample space S of the experiment.
b) Determine the event E that exactly two heads
appear
c) Determine the event F that at least one head
appears.
Example 3:
An experiment consists of casting a pair of dice and
observing the number that fall s uppermost each die.
a) Describe an appropriate sample space of S for this
Experiment.
b) Determine the events E1, E2, ......., E12 that the
sum of the numbers falling uppermost is 1, 2, 3, 4...
12, respectively.
Example 4
An experiment consists of studying the composition of a
three-child family in which children were born at
different times.
a) Describe an appropriate sample space S for this
experiment.
b) Describe the event E that there are two girls and a
boy in the family
c) Describe the event F that oldest child is a girl.
d) Describe the event G that oldest child is a girl and
the youngest child is a boy.
Example 5
The Ever-Brite Battery Company is developing high
amperage, high –capacity battery as a source for
powering electric cars. The battery is tested by
installing it in a prototype electric car and running the
car with a fully charged battery on a test track at a
constant speed of 55 mph until the car runs out of
power. The distance covered by the car is then
observed.
a) What is the sample space for this experiment?
b) Describe the event E that the driving rang under
test conditions is less than 150 miles
c) Describe the event F that the driving range is
between 200 and 250 miles, inclusive.
Section 7.2 Definition of Probability
If S is a finite sample space with n outcomes, which is S=
{ s1, s2 ... sn }, then, the events { s1 }, { s2 } ... { sn } which
consists of exactly one point, are called simple or
elementary, events of the experiment. Simple events are
mutually exclusive.
The probability Distribution:
Simple Event


{ s1 }



{ s2 }



_



_



_



{ sn }

Probability 

P ( s1 ) 

P ( s2 ) 


_



_



_


P ( sn ) 

The Probability Function:
1. 0<= P ( si ) <=1 (i=1, 2,...n)
2. P( s1 )+P( s2 )+...+P( sn )=1
3. P( { si }  { sj } )= P( si ) +P( sj ) i  j (i=1,2..n;j=1,2..n)
Uniform Sample Spaces:
If S = { s1, s2 ... sn } is the sample space for an experiment
which the outcomes are equally likely, then assign the
probabilities
P( s1 ) =P( s2 ) =...= P( sn )= 1n
to each of the simple events s1, s2 ... sn
Computing the probability of an event in a uniform
sample space:
Let S be a uniform sample space and let E be any event.
Then,
of favorable outcomes in E
n( E )

P (E) = Number
Number of favorable outcomes in S
n( S )
Example 1
A fair die is cast and the number that falls uppermost is
observed. Determine the probability distribution for the
experiment.
Example 2
Refer to the table below which is tests involving 200 car
test runs. Each run was made with fully charged battery.
Distance Covered in
Miles(x)
0<=x<=50
50<x<=100
100<x<=150
150<x<=200
200<x<=250
250>x
Frequency of
Occurrence
4
10
30
100
40
16
a) Describe an appropriate sample space for this
experiment
b) Find the probability distribution for this experiment.
Find a Probability of an Event E
1. Determine a sample space S associated with the
experiment.
2. Assign probabilities to the sample events of S.
3. If E = { s1, s2 ... sn } where { s1 }, { s2 } ... { sn } are
simple events, then,
P (E) = P ( s1 ) +P ( s2 ) +...+P ( sn )
If E is empty set, P (E) =0
Example 3
If a ball is selected at random from an urn containing three
red balls, two white balls and five blue balls what is the
probability that it will be white ball?
Example 4
A pair of fair dice is cast. What is the probability that
a) The sum of the numbers shown uppermost is less than
5?
b) At least one 6 is cast?
Example 5
Let S = { s1, s2 , s3, s4, s5, s6 } be the sample space associated with
an experiment having the following probability distribution
 Outcome



Probability

{ s1 }
{ s2 }
{ s3 }
{ s4 }
{ s5 }
1
12
1
4
1
12
1
6
1
3
Find the probability of the event
a) A = { s1, s2 }
b) B = { s3, s2 , s5, s6 }
c) C = S
{ s6 }

1 

12 
Section 7.3 Rules of Probability
Rules:
Let S be a sample space, and E and F be events
1. P(E)>=0, P(S) =1
2. If and F are mutually exclusive ( E  F =  ), then,
P ( E  F ) = P (E) + P (F)
3. Addition rule. If E and F are any two events of a
experiment, then
P ( E  F ) = P (E) + P (F) – P ( E  F )
4. Rules of Complements
P( E c ) = 1-P(E)
Computing the probability of an event in a uniform
sample space:
Let S be a uniform sample space and let E be any event.
Then,
of favorable outcomes in E
n( E )

P (E) = Number
Number of favorable outcomes in S
n( S )
Example 1:
A card is drawn from a well-shuffled deck of 52 playing
cards. What is the probability that it is an ace or a spade?
Example 2:
Let E and F be two events of an experiment with sample
space S. Suppose P(E) = .2 and P(F) = .1
1. Assume that P ( E  F ) = .05. Compute
i. P ( E  F )
ii. P( E c  F c )
iii. P( E c  F c )
2. Assume that P ( E  F ) = 
i. P ( E  F )
ii. P ( E  F )
iii. P( E c )
iv. P( E c  F c )
Example 3:
A={x | x is a graduate students who had learned Spanish
at least one year in the Department of Foreign
Languages}, B={x | x is a graduate students who had
learned French at least one year in the Department of
Foreign Languages}, C={x | x is x is a graduate students
who had learned German at least one year in the
Department of Foreign Languages}. n (A) =200, n (B) =
178, n(C) = 140, n ( A  B) = 33, n( A  C ) = 24, n(C  B)
= 18, and n( A  B  C) = 3
1. at least one year one of the three language:
n ( A  B  C)
=n(A)+n(B)+n(C) –
n( ( A  B)  n( A  C )  n(C  B)  n( A  B  C )
=200+178+140-33-24-18+3=446
P ( A  B  C ) =446/480=0.9291
2. at least one year exactly one of the three language
:
n( ( ( A  B
c
 C c )  ( A c  C  B c )  ( B  C c  A c )) =
146+130+101=377
P( ( ( A  B  C )  ( A
377/480=0.7854
c
c
c
 C  B c )  ( B  C c  A c )) =
U  480
A
3. None of these 30
brands: n(B ( A  B  C) ) =480-446=34
146
130
P( ( A  B  C) ) =34/480=0.0708
c
c
21
3
15
101
C
Example 4:
Refer to the table below which is tests involving 200 car
test runs. Each run was made with fully charged battery.
Distance Covered in
Frequency of
Miles(x)
0<=x<=50
50<x<=100
100<x<=150
150<x<=200
200<x<=250
250>x
Occurrence
4
10
30
100
40
16
c) Describe an appropriate sample space for this
experiment
d) Find the probability distribution for this experiment.
e) What is the probability if the car is selected at random
and the car’s running distance is less than 200 and
greater than 100?