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Chapter 7
Probability
Definition of Probability
• What is probability? There seems to be no
agreement on the answer.
• There are two broad schools of thought:
frequency and nonfrequency.
• Even among the frequency school, there
are at least two definitions: a priori and a
posteriori.
• A priori is defined without any empirical data.
It is rather a statement about one’s state of
mind. It is the basis of theoretical
mathematical probability.
Definition of Probability
• A posteriori, or relative long-run frequency,
definition is empirical nature.
• It says that, in an actual series of tests,
probability is the ratio of the number of times an
event occurs to the total number of trials.
• A priori approach supplies a constitutive
definition of probability, whereas the a posteriori
approach supplies an operational definition of
probability.
Definition of Probability
• In nonfrequency approach, there are two
values: (1) the probability value itself, and
(2) the weight of evidence associated with
it. The weight of evidence is subjective.
• In summary, the long-run relative
frequency approach is the most prevalent
in behavioral science research.
Sample Space, Sample Points, and
Events
• The sample space includes all possible
outcomes of an “experiment” that are of interest
to the experimenter. The primary elements of U
are called elements or sample points.
• Table 7.1, Figure 7.1, 7.2
• An event is a subset of U. In standard usage,
events are more encompassing than points. All
points are events (subsets), but not all events
are points.
Determining Probabilities with
Coins
The probabilities of all the points in the
sample space must add up to 1.00.
• Figure 7.3. Each complete path of the tree
(from the start to the third toss) is a
sample point.
An Experiment with Dice
• Table 7.2
Some Formal Theory
• A weight is a positive number assigned to
each each element, x, in U, and written
w(x), such that the sum of all these
weights, w(x) , is equal to 1.
• We write m(A), meaning “The measure of
the set A.” This simply says the sum of the
weights of the elements in the set A. For
example,
1
m( A) 
 w( x)  2
x in A
Compound Events and Their
Probabilities
• An event is a set of possibilities; it is a
possible set of events; it is an outcome of
a probability “experiment.” A compound
event is the co-occurrence of two or more
single (or compound) events.
• The two set operations of intersection and
union—the operations of most interest to
us—imply compound events.
Compound Events and Their
Probabilities
• If there is no overlap between two sets,
• A  B  E , then the following equation
holds:
p ( A  B)  p ( A)  p ( B)
• If two sets are not disjoint, rather they
overlap, A  B  E , then the following
equation holds:
•
p( A  B)  p( A)  p( B)  p( A  B)
Independence, Mutual Exclusiveness, and
Exhaustiveness
• Exhaustiveness means that the subsets of
U use up all the sample space, or
• A  B  K  U , where A, B, …,K are
subsets of U, the sample space. In
probability language, this means:
p( A  B    K )  1.00
• If the events (sets) A, B, and C are
mutually exclusive, then
p( A  B  C )  p( A)  p( B)  P(C )
Independence, Mutual Exclusiveness, and
Exhaustiveness
• Two events, A and B, are statistically independent
if the following equation holds:
p( A  B) we
p( Arank
)  p( Border
)
• Suppose
examination papers and
then assign grades on the basis of these ranks,
the grades given by the rank-order method are
not independent.
• In any area of research, one cannot assume that
multiple observations of one subject are
independent.
Conditional Probability
• Definition of Conditional Probability
• Let A and B be events in the same space, U,
as usual. The conditional probability is
denoted: p(A︳B), which is read, “The
probability of A, given B.” The formula for the
conditional probability involving two events is:
p( A  B)
p( A B) 
p( B)
• The sample space has, through knowledge,
been reduced from U to B.
Conditional Probability
• When events are independent, p( A B)  p( A)
• Table 7.5
• The probability of success without any
other knowledge is a probability problem
on the whole sample space U. This
probability is 0.4. But given knowledge of
MAT score, the sample space is reduced
from U to a subset of U  65
• Figure 7.5
Bayes’ Theorem: Revising
Probabilities
• With Bayes’ Theorem, one could update or
revise current probabilities based on new
information or data.
p( H i A) 
p( H i ) p( A H i )
k
 p( H
j 1
j
) p( A H j )
Bayes’ Theorem: Revising
Probabilities
p( D)  0.1
p( ~ D)  0.95
p (~ D )  0.9
p( ~ D)  0.05
p( D)  0.91
p( D)  0.09
P( D) p( D)
0.91(0.10)
p( D ) 

 0.67
P( D) p( D)  P( ~ D) p(~ D) 0.91(0.10)  (0.05)(0.90)
P( D) p( D)
0.09(0.10)
p ( D ) 

 0.01
P( D) p( D)  P( ~ D) p(~ D) 0.09(0.10)  (0.95)(0.90)