ECON 4630
ECON 5630
TOPIC #3: PROBABILITY THEORY
I.
What is Probability?
A.
Definition: Probability is the relative frequency as the sample size becomes
infinitely large. Alternatively, probability is the number of favorable outcomes
divided by the total number of possible outcomes.
B.
Examples
2
C.
II.
Objective vs. Subjective Probability
Probabilities of More Complex Events
A.
Probability Trees in General
B.
An Example: Gender of Children
3
Outcomes
Prob1
Prob2 outcome
BBBB
.0625
.0731
e1
BBBG
.0625
.0675
e2
BBGB
.0625
.0675
e3
BBGG
.0625
.0623
e4
BGBB
.0625
.0675
e5
BGBG
.0625
.0623
e6
BGGB
.0625
.0623
e7
BGGG
.0625
.0575
e8
GBBB
.0625
.0675
e9
GBBG
.0625
.0623
e10
GBGB
.0625
.0623
e11
GBGG
.0625
.0575
e12
GGBB
.0625
.0623
e13
GGBG
.0625
.0575
e14
GGGB
.0625
.0575
e15
GGGG
.0625
.0531
e16
B
B
G
B
B
G
B
B
G
B
G
G
G
B
G
B
B
G
B
G
G
B
G
G
B
B
G
G
B
G
4
C.
The Special Rule of Multiplication: Assuming each outcome is independent of
every other (that is, the occurrence of one outcome has no effect on the
occurrence or non-occurrence of any other outcome), then
D.
Outcome Sets and Events
1.
Outcome Set
Definition: The outcome set S is the collection of all possible outcomes.
Venn Diagram:
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
e14
e15
e16
5
2.
Event
Definition: An event is a combination of outcomes. That is, an event E is a
subset of S
Example: Suppose E: at least 3 girls
So E = {
}
Venn Diagram:
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
e14
e15
e16
3.
Special Rule of Addition: If outcomes are mutually exclusive, the
probability of an event occurring is the sum of the probabilities of each
event occurring. That is, P( E)   P(e) .
6
Example: Suppose F: exactly 2 boys
So F = {
}
Venn Diagram:
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e11
e12
e13
e14
e15
e16
7
E.
Combinations of Events
1.
Union
Example: Suppose a couple would be sorry if they had fewer than 2 boys
OR if all 4 kids were of the same gender.
Let:
G: fewer than 2 boys
J: all same gender
Venn Diagram:
G={
}
J= {
}
GJ={
}
P( G  J ) =
8
2.
Intersection
Example: Suppose a couple would be sorry if they had fewer than 2 boys
AND if all 4 kids were of the same gender.
Let:
G: fewer than 2 boys
J: all same gender
Venn Diagram:
G={
}
J= {
}
GJ = {
}
P( G  J ) =
9
3.
The General Rule of Addition:
Recall the Special Rule of Addition:
4.
Complement
Definition: The complement of E, E , is all the points that are not in E.
10
F.
Conditional Probability
Definition: Conditional probability is the probability of some event occurring
given that some other event has occurred.
Notation:
Example: Suppose we know that G (fewer than 2 boys) has occurred. Given this,
what is the probability that J (all same gender) will occur?
The General Rule of Multiplication:
11
G.
Review Example
Suppose a restaurant finds that 75% of all customers use chili sauce, 80% use salt,
and 65% use both.
1.
What is the probability that a particular customer uses at least one of these
two condiments?
2.
What is the probability that a salt user uses chili sauce? Equivalently, what
is the probability that a customer will use chili sauce given that he or she
uses salt?
12
H.
Independence
Definition: If the occurrence of event A is unaffected by the occurrence or nonoccurrence of event B, A and B are independent of each other.
Note: Generally speaking, independence must be proved mathematically.
Method of Proving Independence #1:
Method of Proving Independence #2:
13
Example #1: Suppose we gather data on the performance of students in ECON
4630 and which hand each student writes with. Suppose our survey results are as
follows:
Region
Good: A or B
Less Good: C
and below
LeftHanded
0.100
Right
Handed
0.700
0.025
0.175
Is one’s performance independent of which hand one writes with?
14
Example #2: Suppose we ask Americans whether or not they believe Jon Gosselin
is to blame for the breakup of the family depicted in the TV series Jon and Kate
plus 8. Suppose our survey results are as follows:
Respondent’s
Gender
JG to blame
JG not to
blame
Male
0.078
0.250
Female
0.621
0.051
Is one’s opinion on this matter independent of one’s gender?
15
Example #3:
A personnel officer for a firm that employs many part-time salespersons tries out a new
sales aptitude test on several hundred applicants. Because the test is unproven, results are
not used in hiring. 40% of all applicants show high aptitude on the test and 12% of those
hired show both high aptitude and achieve good sales records. The firm’s experience is that
30% of all salespersons achieve good sales.
Let A be the event “shows high aptitude.”
Let B be the event “achieves good sales.”
a.
Find P(A), P(B), P(AB), and P(BA).
b.
Are A and B independent? Prove this mathematically.
16
I.
Combinations of Random Variables
1.
Definition: A random variable is a real-valued set function whose value is
determined by the outcome of an experiment.
2.
Examples:
3.
Notation:
4.
Probability Distributions in General
Outcome
P(X=0)
Probability
outcomes
P(X=1)
P(X=2)
P(X=3)
P(X=4)
5.
Calculating the mean:
X
P(x)
0
0.0731
1
0.2700
2
0.3738
3
0.2300
4
0.0531
xP(x)
17
6.
Calculating the variance:
X
P(x)
0
0.0731
1
0.2700
2
0.3738
3
0.2300
4
0.0531
(x-)
(x-)2
(x-)2P(x)
x2P(x)
18
7.
Rules for Transforming Random Variables
8.
x
1
Example: Let X = number of dots that turn up on a die
P(x)
xP(x)
(x-x)2P(x)
x2P(x)
2
3
4
5
6
19
Suppose this game has a payoff that is a linear function of X:, Specifically,
suppose Y = 2X + 8.
y
P(y)
yP(y)
(y-x)2P(y)
y2P(y)
20
9.
Joint Probability Distributions
Definition: Joint probability is the probability that two or more events will
occur at the same time.
Example 1: Consider rolling a pair of dice. Let X = the number of threes
and Y = the number of fives.
There are 36 possible outcomes:
1st die
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
2nd die
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
X
0
0
1
0
0
0
0
0
1
0
0
0
1
1
2
1
1
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
Y
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
1
1
1
2
1
0
0
0
0
1
0
21
0
1
2
X
Y
0
1
2
Example 2: Suppose we survey Denton residents regarding their
satisfaction with restaurant choices in Denton. Let X measure satisfaction
with restaurant choices (with 4 being very satisfied) and Y measure length
of residency (1 = 5 or fewer years, 2 = 6 or more). Perhaps our survey
comes up with the following:
1
2
3
4
1
0.04
0.14
0.23
0.07
2
0.07
0.17
0.23
0.05
X
Y
22
10.
Independence
11.
Conditional Probability
23
12.
Covariance: how variables vary together
Method of calculating covariance #1:
 XY   ( x   x )( y   y ) P( x, y)
X
Y
Method of calculating covariance #2:
 XY   xyP( x, y)   X Y
Example: Diameter and Usable Height of Trees (in feet)
20
25
P(d)
1
0.16
0.09
0.25
1.25
0.15
0.30
0.45
1.5
0.03
0.17
0.20
1.75
0.00
0.10
0.10
P(h)
0.34
0.66
1.00
H
D 
Marginal Distributions:
24
Mean and Variance of D and H:
d
H
P(d,h)
1
20
0.16
1
25
0.09
1.25
20
0.15
1.25
25
0.30
1.5
20
0.03
1.5
25
0.17
1.75
20
0.00
1.75
25
0.10
dhP ( d , h)
(d   d )( h   h ) P(d , h)
25
What does covariance tell us?
What does covariance not tell us?
13.
Correlation
Example: Diameter and Usable Height of Trees (in inches)
240
300
P(d)
12
0.16
0.09
0.25
15
0.15
0.30
0.45
18
0.03
0.17
0.20
21
0.00
0.10
0.10
P(h)
0.34
0.66
1.00
H
D 
26
Marginal Distributions:
Mean and Variance of D and H:
D
H
P(d,h)
12
240
0.16
12
300
0.09
15
240
0.15
15
300
0.30
18
240
0.03
18
300
0.17
21
240
0.00
21
300
0.10
dhP ( d , h)
(d   d )( h   h ) P(d , h)
27
Correlation coefficient:
28
NOTES:
29