Classic Probability Revisited (II):
Algebraic Operations of the Extended Probability Theory
YINGXU WANG
International Institute of Cognitive Informatics and Cognitive Computing (ICIC)
Laboratory for Computational Intelligence, Software Science, and Denotational Mathematics
Dept. of Electrical and Computer Engineering
Schulich School of Engineering and Hotchkiss Brain Institute
University of Calgary
2500 University Drive, NW, Calgary, Alberta, CANADA T2N 1N4
Abstract: - Part II of this paper presents a set of comprehensive algebraic operators on the extended mathematical
structures of the general probability theory. It is recognized that the classic probability theory is cyclically defined among a
small set of highly coupled operations. In order to solve this fundamental problem, a reductive framework of the general
probability theory is introduced. It is found that conditional probability operation on consecutive events is the key to
independently manipulate other probability operations. This leads to a revisited framework of rigorous manipulations on
general probabilities. It also provides a proof for a revisited Bayes’ law fitting in more general contexts of variant sample
spaces and complex event relations in fundamental probability theories. The revisited probability theory enables a rigorous
treatment of uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in
contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft
computing, and brain informatics.
Key-Words: - Denotational mathematics, probability theory, probability algebra, fuzzy probability, formal
inference, cognitive informatics, cognitive computing, computational intelligence, semantic computing, brain informatics,
cognitive systems
1. Introduction
Probability theory is a branch of mathematics that
deals with uncertainty and probabilistic norms of
random events and potential causations as well as
their algebraic manipulations. The development of
classic theories of probability can be traced back to
the work of Blaise Pascal (1623-1662) and Pierre de
Fermat (1601-1665) [Todhunter, 1865; Venn, 1888;
Hacking, 1975]. Many others such as Jacob
Bernoulli, Reverend T. Bayes, and Joseph Lagrange
had significantly contributed to probability theory.
Theories of probability in its modern form was
unified by Pierre Simon and Marquis de Laplace in
the 19th century [Kolmogorov, 1933; Whitworth,
1959; Hacking, 1975; Mosteller, 1987; Bender,
1996]. Set theories [Cantor, 1874; Zadeh, 1965,
1968, 1996, 2002; Artin, 1991; Ross, 1995; Pedrycz
& Gomide, 1998; Novak et al., 1999; Potter, 2004;
Gowers, 2008; BISC, 2013; Wang, 2007] provide an
expressive power for modeling the discourse and
axioms of probability theories. A theory of fuzzy
probability and its algebraic framework has been
presented in [Wang, 2015e].
The philosophy of probability theory is analogy-
based where large-enough experiments are required
for establishing prior probability estimations and
norms in a certain sample space. The main
methodology of classic probability theory is an
external or black box predication for a set of
uncertain phenomena of a complex system without
probing into its internal mechanisms. Although the
range of prior probability for any predicated event is
[0, 1], the range of posterior probability is merely
reduced to {0, 1} immediately after the given event
has realized in a certain probability space.
It is recognized that the classic probability theory
is cyclically defined among a set of highly coupled
operations where only logically conjunctive,
disjunctive, and conditional events are considered.
This paper presents a revisited theory of probability,
which extends classic probability theory to a
comprehensive set of probability operations. An
extended set of algebraic operators on the revisited
mathematical model of probability is rigorously
defined in Section 2, which extends the traditional
probability operations of addition, multiplication,
and condition to subtraction and division. The
conventional mutual-coupled probability operations
are independently separated in a deductive structure
on the basis of the refined model of conditional
probability. Formal properties of probability and
PROOF
DOI: 10.37394/232020.2022.2.11
Yingxu Wang
E-ISSN: 2732-9941
86
Volume 2, 2022
rules of algebraic operations on general probabilities
are summarized in Section 3. Proven theorems and
practical examples are provided throughout the
paper for elaborating each of the fundamental
definitions and operations in the general theory of
probability. The revisited theory of probability may
be used to solve a number of challenging problems
in classic probability theory such as complex
sequential, concurrent, and causal probabilities as
well as real-time probabilities under highly
restrictive timing constraints.
Due to its excessive length, this paper is
presented in two parts on: i) The mathematical
models of general probability; and ii) The algebraic
operations of the extended probability theory. This
paper is the second part of the general probability
theory on formal algebraic operators on the
extended mathematical structures of the general
probability theory.
2. Algebraic Operations on the
General Probability Models
The theoretical framework of general probability is
formally represented by the mathematical model and
the set of algebraic operators of probability as
outlined in Sections 3 and 4 of Part I. A set of six
probability operators are identified as those of
conditional, multiplication, division, addition,
subtraction, and complement operations, which stem
from a unified mathematical model of the
conditional probability. Each probability operator is
formally defined and elaborated in the following
subsections towards an algebraic framework of the
theory of general probability.
2.1 The Conditional Operator on
Consecutive Probabilities
The conditional operation of consecutive
probabilities deals with coupled influences between
related events in both invariant and variant sample
spaces. Because conditional probability forms the
foundation for all other operators in the algebraic
system of the general probability theory, it must be
rigorously analyzed in order to avoid the dilemma of
the cyclic definition as in classic probability theory.
The nature of conditional probability is
constrained by different contexts determined by
three control factors of the Cartesian product,
SRD
, as defined in Table 1 where S demotes
the sample space (variant/invariant),
R
relation of
events (joint/disjoint), and
D
dependency of events
(dependent/ independent/mutually-exclusive (ME)).
Therefore, the contexts of the general probability are
classified into four categories according to the
probability characteristics in the Cartesian product,
i.e.: a) invariant sample space and disjoint/ME-
dependent events, b) invariant sample space and
joint/independent events, c) variant sample space
and disjoint/independent events, and d) invariant
sample space and joint/dependent events.
Table 1. Contexts of Relations and Dependencies of
Events in the General Probability Theory
No,
Category Definition
()SRD
Sample
space
(S)
Events
Relation (R)Dependency (D)
i Disjoint/mutuall
y
-exclusive
(ME) events in invariant
sample space
SRD
'SS
XY
(),MEXY
ii Joint/independent events in
invariant sample space
SRD
XY
XY
iii Disjoint/independent events
in variant sample space
'SRD
'SS
XY
XY
iv Joint/dependent events in
variant sample space
'SRD
XY
(')
X
YY
Theorem 1. The conditional operator on
consecutive probabilities of an event b influenced
by that of a preceding event a in the sample space S
in U, (|)
P
ba, is determined by a ratio between the
changed sizes of sets of succeeding events B’ and of
the sample space S’ given , and 'aAS bBS ,
i.e.:
(|) ( )
) Invariant , unrelated , and ME-dependent :
0
) Invariant , related , and independent :
( )
) Variant ', unrelated , and independent : '
Pb a PA B
iSR DSRD
ii S R D S R D
Pb
ii S R D S R D
() ()
'( ) , ( )
1() ||
) Variant ', related and dependent : '
() ( ) ()
"( ) , ( )
1() ||
i
i
ii
i
Pb Pa
Pb Pa
Pa A
iv S R D S R D
Pb Pb Pb
Pb Pb
Pb B
(1)
Proof. Theorem 1 can be proven in each of the
four contexts as defined in Table 1 according to
Definition 10 in Part I as follows:
PROOF
DOI: 10.37394/232020.2022.2.11
Yingxu Wang
E-ISSN: 2732-9941
87
Volume 2, 2022
,, ,and b ',
|'|
( | ) ( ) ,
|\|
) ' ( ,ME),
||
0
||
) '
aba A S B S
B
Pb a PA B a A b B
Sa
iSSAB AB
S
ii S S A B A
,
||
( )
||
) '
B
BPb
S
iii S S A B A
,
|| () ()
= '( ), ( )
||11 () ||
) ',
() ( )
||1 ()
''( ), ( )
||1 1 () ||
i
i
ii
i
B
BPb Pa
Pb Pa
SPa A
iv SSAB A B
Pb Pb
B
Pb
PbPb
SPb B
(2)
It is noteworthy in Theorem 1 that S is variant in
general as constrained by Theorem 1 in Part I
because of the coupling of the conditional events a
and b. In other words, the general probability in an
invariant sample space is only a special case of that
of the general variant context.
Example 1. In an invariant sample space
1{ 0.68, 0.32}SH T== = as modeled in Example 2 in
Part I, the events head (H) and tail (T) are mutually
exclusive in a single toss of the coin. That is, both
events cannot happen simultaneously. Once a head
is observed, tail will certainly not appear in the same
trail, and vice versa. This is a typical context of
mutually exclusive (
HT or disjoint), and
dependent (
| or |TH HT ) events of
conditional probability according to Theorem 1(i)
where (| ) 0, and ( )PT H ifH T H T.
It is noteworthy that a pair of mutually exclusive
events and XY
are dependent because
() (|)0XY X Y PYX , due to the
interactive influence between the non-independent
events.
Example 2. Given a bag containing five black
balls (B) and five white balls (W) in
510
2
16
{(| )0.11, (| )0.09}
ii i i
ii
SPbbB PwwW
RR
==
=Î= Î=
as
modeled in Example 3 in Part I. Assume the ball
drawn from the bag will be returned to the bag
before the next trial, i.e., '
22
SS, it is a case of
invariant sample space, related and independent
events of conditional probability according to
Theorem 1(i) as follows:
(|) ()0.45
(| ) () 0.55
(|) ()0.45
(|) () 0.55
PW B PW
PBW PB
PW W PW
PB B PB
Example 3. Reconsider Example 2 in
510
'
2
16
{ ( | ) 0.11, ( | ) 0.09}
ii i i
ii
SPbbB PwwW
RR
==
=Î= Î=
where
the ball drawn from the bag will not be returned, i.e.,
'
22
SS
, it becomes a case of variant sample space,
disjoint /independent or joint/dependent events of
conditional probability according to Theorem 1(iii)
or 1(iv), respectively, as follows:
() ()
( | ) '( ) , ( ) 0.55 / 5 0.11
1() ||
0.45 0.45
0.51
1 0.11 0.89
() ( )
( | ) '( ) , ( ) 0.45 / 5 0.09
1() ||
0.55 0.55
0.60
1 0.09 0.91
() ()
( | ) ''( ) 1
PW P B
PW B P W Pb
i
Pb B
i
PB PW
PB W P B Pw
i
Pw W
i
PB Pb
i
PB B P B
== == =
-
===
-
== == =
-
===
-
-
== , ( ) 0.11
()
0.55 0.11 0.44
0.49
1 0.11 0.89
() ()
( | ) ''( ) , ( ) 0.09
1()
0.45 0.09 0.36
0.40
1 0.09 0.91
Pb
i
Pb
i
PW P w
i
PW W P W P w
i
Pw
i
=
-
-
===
-
-
== =
-
-
===
-
Contrasting the results obtained in Examples 2
and 3, it is noteworthy that the conditional
probabilities in Contexts (iii) and (iv) of Theorem 1
have increased or decreased, respectively, due to the
size shrinkages of sample spaces and/or the number
of events as a result of the conditional coupling. The
changes between the variant ( '
2
S) and invariant ( 2
S)
sample space can be analyzed as follows:
'( | ) ( | ) 0.51 0.45 0.06
'( | ) ( | ) 0.49 0.55 0.06
'( | ) ( | ) 0.60 0.55 0.05
'( | ) ( | ) 0.40 0.45 0.05
PW B PW B
PBB PBB
PBW PBW
PWW PWW
ì
ï-=-=
ï
í
ï-=-=-
ï
î
ì
ï-=-=
ï
í
ï-=-=-
ï
î
The results indicate that conditional probabilities
in the variant and invariant sample spaces may be
significantly different due to the increment or
decrement of coupled influences.
2.2 The Complement Operator on the
Context of Probability
Theorem 2. The complement of probability of an
event aAS
in U, ()
P
a, is determined by the
probability of all events in S excluding only that of
a, i.e.:
() 1 ()Pa Pa (3)
Proof. Theorem 2 can be proven according to
Definition 10 in Part I as follows:
PROOF
DOI: 10.37394/232020.2022.2.11
Yingxu Wang
E-ISSN: 2732-9941
88
Volume 2, 2022
and ,
() ( | )
|||\|||||
|| || ||||
1 ( )
aAS aAS
Pa PAa A S a A S
ASASA
SSSS
Pa
(4)
Example 4. On the basis of Example 4 in Part I,
the complement of probability in the sample space
2
1{ 0.46, 0.22, 0.22, 0.10}SHHHTTHTT== = = = can be
determined according to Theorem 2 as follows:
( ) 1 ( ) 1 0.46 0.54
( ) 1 ( ) 1 0.22 0.78
PHH PHH
PTH PTH
Corollary 1. The double complements of the
general probability of an event aAS in U,()
P
a,
results in an involution to the same probability, i.e.:
() 1 () 1 (1 ()) ()Pa Pa Pa Pa (5)
2.3 The Multiplication Operator on
Disjunctive Probabilities
Theorem 3. The multiplication of probabilities
of disjunctive events a and b in the sample space S
in U,()
P
ab, is determined by the product of the
probabilities of () and ( | )Pa Pb a given
and 'aAS bBS , i.e.:
()( )()(|)
) Invariant , unrelated , and ME-dependent :
0
) Invariant , related , and independent :
( ) ( )
) Variant ', unrelated , and indepen
Pa b PA B PaPb a
iSR DSRD
ii S R D S R D
PaPb
iii S R
dent : '
() ()
( ) '( ) ( ) , ( )
1() ||
) Variant ', related and dependent : '
() ( ) ()
( ) ''( ) ( ) , ( )
1() ||
i
i
ii
i
DS R D
Pb Pa
PaP b Pa Pa
Pa A
iv S R D S R D
Pb Pb Pb
PaP b Pa Pb
Pb B
(6)
Proof. Theorem 3 can be proven according to
Definition 10 in Part I and Theorem 1 as follows:
, , , and b ',
|\ |
||||
( ) ( ) ,
|| |||\ |
( ) ( | )
ii
i
aba A S B S
Ba
AB A
P
ab PAB a A
SSSa
PaPb a
) ' ( , ME),
0
) '
iSSAB AB
ii S S A B A
,
( ) ( )
) '
B
PaPb
iii S S A B A
,
() ()
( ) ( ) '( ), ( )
1() ||
) ',
() ( ) ()
( ) ( ) ''( ), ( )
1() ||
i
i
ii
i
B
Pb Pa
Pa PaP b Pa
Pa A
iv SSAB A B
Pb Pb
P
b
Pa PaP b Pb
P
bB
(7)
Example 5. Given an invariant sample space
1{0.68, 0.32}SH T== = as modeled in Example 2 in
Part I, i.e., '
11
SS
, the following disjunctive
probabilities for two consecutive tosses of the
uneven coin can be derived by a probability
multiplication according to Theorem 3(ii):
( ) ( ) ( ) 0.68 0.32 0.22
( ) ( ) ( ) 0.68 0.68 0.46
( ) ( ) ( ) 0.32 0.68 0.22
( ) ( ) ( ) 0.32 0.32 0.10
PH T PHPT
PH H PHPH
PT H PT PH
PT T PT PT
Example 6. Given a variant sample space
510
2
16
{(| )0.11, (| )0.09}
ii i i
ii
SPbbB PwwW
RR
==
=Î= Î=
as
modeled in Example 3 in Part I, i.e., '
22
SS
, the
following probability multiplications for two
consecutive draws of the uneven balls in the bag can
be obtained according to Theorem 3(iii) or 3(iv),
respectively:
()( )
( ) ( ) '( ) , ( ) 0.09
1()
0.68 0.32 0.22
0.24
1 0.09 0.91
()()
()()'() ,()0.11
1()
0.32 0.68 0.22
0.25
10.11 0.89()
( ) () ''() ()
i
i
i
i
PBPW
PB W PBP W Pw
Pw
PW P B
PW B PW P B Pb
Pb
PB P
PBBPBPBPB
´= = =
-
·
===
-
´= = =
-
·
===
-
-
´= =
()
,() 0.11
1()
0.68(0.68 0.11) 0.39
0.44
1 0.11 0.89
() ()
( ) ()''() () ,() 0.09
1()
0.32(0.32 0.09) 0.07
0.08
10.09 0.91
i
i
i
i
i
bPb
Pb
PW P w
PW W PW P W PW P w
Pbw
=
-
-
===
-
-
´= = =
-
-
===
-
Corollary 2. The revisited Bayes’ law of
probability can be rigorously derived based on
Theorem 3 as follows:
PROOF
DOI: 10.37394/232020.2022.2.11
Yingxu Wang
E-ISSN: 2732-9941
89
Volume 2, 2022
,, , ', and ' ,
( ) ( ) ( | ),
aba A S b B S S S
Pa b PaPb a A B A
( ) ( )
( ) ( | )
( )
(|) (|)
, '
() ()
B
PaPb
PbPa b
Pb a
Pb a Pa b SSAB A
Pb Pa
BB
(|) (|)
, Otherwise
() ()
A
Pb a Pa b
Pb Pa
(8)
Corollary 2 and Theorem 3 indicate that Bayes’
law in classic probability theory is a special case of
general probability multiplication, which may only
hold iff '|SSA BABB , i.e., when the
conditions for invariant sample space and related but
independent events are satisfied.
2.4 The Division Operator on Composite
Probabilities
The algebraic operation of probability division is an
inverse operation of probability multiplication,
which is not defined in traditional probability
theory.
Theorem 4. The division of probability of an
event b by that of another event a in the sample
space S in U, (/)
P
ba, is determined by the ratio of
their probabilities where and 'aAS bBS i.e.:
|| ()
(/) ( ) , 0 () ()
|| ()
) Invariant , unrelated , and ME dependent :
0
) Invariant , related , and independent :
()
()
) Variant ', unrelate
BPb
Pb a P Pa Pb
APa
iSR DSRD
ii S R D S R D
Pb
Pa
iii S
d , and independent : '
'( )
()
) Variant ', related and dependent : '
''( )
()
RDSRD
Pb
Pa
iv S R D S R D
Pb
Pa
(9)
Proof. Theorem 4 can be proven according to
Definition 10 in Part I as well as Theorems 1 and
Theorem 3 as follows:
,, , and ',
||
( / ) (| | / | |) ||
||/||
, 0 | | | |
||/||
()
, 0 ( ) ( )
()
) ' ( , ME),
||/|| () 0
||/|| () ()
aba A S b B S
B
Pb a P B A A
BS AB
AS
Pb Pa Pb
Pa
iSSAB AB
BSPb
A S Pa Pa
0, ( )
) '
AB
ii S S A B A
,
||/|| ()
||/|| ()
) '
B
BSPb
AS Pa
iii S S A B A
,
||/|'| '() 1 ()
,
||/|| () ()1 ()
) ' ',
() ( )
|'|/|'| "() 1
,
||/|| () ()1 ()
i
i
ii
i
B
BS Pb Pb aA
A S Pa Pa Pa
iv S S A B A B
Pb Pb
BSPb bB
AS Pa Pa Pb
(10)
Example 7. In the invariant sample space
1{ 0.68, 0.32}SH T== = as modeled in Example 2 in
Part I, the events head (H) and tail (T) are mutually
exclusive in a single toss of the unfair coin.
Therefore, the following probability divisions of
unrelated events can be obtained according to
Theorem 4(i), respectively:
(/)0
(/ ) 0
PH T
PT H
Example 8. Redo Example 5 with none-
mutually-exclusive events in 1{ 0.68, 0.32}SH T== = ,
the following probability divisions between those of
two consecutive tosses and the first toss can be
obtained according to Theorem 4(ii), respectively, as
follows:
( ) 0.22
(/) 0.32
( ) 0.68
( ) 0.22
(/) 0.69
( ) 0.32
( ) 0.46
(/) 0.68
( ) 0.68
( ) 0.10
(/) 0.31
() 0.32
PHT
PHT H PH
PTH
PTH T PT
PHH
PHH H PH
PTT
PTT T PT
It is noteworthy that, according to Theorem 4(ii),
the event of the divisor must not be mutually
exclusive to that of the dividend. Otherwise,
Theorem 4(i) should be applied such as in the cases
of ( /)0,( /)0,( /)0,PHH T PTT H PHT T
and
(/)0PTH H
in the given context.
PROOF
DOI: 10.37394/232020.2022.2.11
Yingxu Wang
E-ISSN: 2732-9941
90
Volume 2, 2022
Example 9. Given a variant sample space
2'
2{ 0.28, 0.27, 0.27, 0.18}SBWWBBBWW== = = =
as
modeled in Example 5 in Part I, i.e., 2' 2
22
SS
, the
following probability divisions between two draws
of the uneven balls in the bag can be obtained
according to Theorem 4(iii) or 4(iv), respectively:
'( ) 0.28
(/) 0.51
( ) 0.55
'( ) 0.27
(/) 0.60
( ) 0.45
"( ) 0.27
(/) 0.49
( ) 0.55
"( ) 0.18
(/) 0.40
() 0.45
PBW
PBW B PB
PWB
PWB W PW
PBB
PBB B PB
PWW
PWW W PW
===
===
===
===
The results obtained in Example 9 can be
verified by applying the multiplication rules given
in Eq. 9(iii) and 9(iv) as shown in the following
example. This approach is particularly useful when
the product probability is unknown.
Example 10. Redo Example 9 in
2'
2{ 0.28, 0.27, 0.27, 0.18}SBW WB BB WW== = = =
according to Eq. 9(iii) and 9(iv) obtaining the same
results as follows:
() 0.45 0.45
( / ) '( ) 0.51
1 ( ) 1 0.11 0.89
() 0.55 0.55
( / ) '( ) 0.60
1 ( ) 1 0.09 0.91
() ( ) 0.55 0.11 0.44
( / ) "( ) 0.49
1 ( ) 1 0.11 0.89
() () 0.45 0.
(/)"()1()
i
i
i
i
i
i
PW
PBW B P W Pb
PB
PWB W P B Pw
PB Pw
PBB B P B Pb
PW Pw
PWW W P W Pw
== = ==
--
== = ==
--
--
== = ==
--
--
== =
-
09 0.36 0.40
1 0.09 0.91
==
-
In probability theory, it is often interested in
predicating the odds of random outcomes about the
ratio of the probabilities of an event’s success and
failure.
Definition 1. An odd, ()e, is a ratio between
probabilities of an event e and its complement, or
that of its success e
s
and failure e
f
, i.e.:
,, ,
()
() ()
( ) 1() ()
()
ee
e
e
es f E S
P
s
Pe Pe
e
P
ePf
Pe
(11)
It is noteworthy that the value of odds is a
nonnegative real number, i.e., () 0e
, which may
be great than 1.0 according to Definition 1.
2.5 The Addition Operator on Conjunctive
Probabilities
Theorem 5. The addition of probabilities of two
conjunctive events a or b in the sample space S in U,
()Pa b, is determined by the sum of the
probabilities of () and ()Pa Pb excluding that of the
intersection ()
P
ab
given and 'aAS bBS , i.e.:
()( )()()()(|)
) Invariant , unrelated , and ME-dependent :
( ) ( )
) Invariant , related , and independent :
( ) ( ) ( ) ( )
) Variant
Pa b PA B Pa Pb PaPb a
iSR DSRD
Pa Pb
ii S R D S R D
Pa Pb PaPb
iii
', unrelated , and independent : '
( ) ( ) ( ) '( )
) Variant ', related and dependent : '
( ) ( ) ( ) ''( )
SR DSRD
Pa Pb PaP b
iv S R D S R D
Pa Pb PaP b
(12)
Proof. Theorem 5 can be proven according to
Definition 10 in Part I and Theorem 1 as follows:
, , , and b ',
||
( ) ( ) ||
||||| |
|| || ||
( ) ( ) ( ) ( | )
) ' ( , ME),
( ) ( )
) '
aba A S B S
AB
Pa b PA B S
ABAB
SS S
Pa Pb PaPb a
iSSAB AB
Pa Pb
ii S S A B A
,
( ) ( ) ( ) ( )
) '
B
Pa Pb PaPb
iii S S A B A
,
()() ()
( ) ( ) , ( )
1() ||
() () () '()
) ',
() ( ) ()
( ) ( ) ( ) , ( )
1() ||
( ) ( ) ( ) ''( )
i
i
ii
i
B
PaPb Pa
Pa Pb Pa
Pa A
Pa Pb PaP b
iv S S A B A B
Pb Pb
P
b
Pa Pb Pa Pb
P
bB
Pa Pb PaP b
(13)
Example 11. Suppose a system encompasses two
components C1 and C2 with estimated failure rates
as F1 = 0.7 and F2 = 0.3, respectively, in an invariant
sample space. The conjunctive probabilities for a
system failure of either C1 or C2 can be determined
according to Theorem 5(ii) as follows:
Example 12. Reuse the individual probabilities
obtained in Example 2 in Part I in the invariant
sample space 1{ 0.68, 0.32}SH T== = . The following
additions of conjunctive probabilities for expecting
some mixed head and tail of an unfair coin in two
tosses can be derived according to Theorem 5(i):
12 1 2 1 2
()()()()()
0.7 0.3 0.7 0.3
1.0 0.21 0.79
PF F PF PF PF PF
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( ) () () ()()
0.24 0.24 0 0.48
( )()()()()
0.36 0.16 0 0.52
P HT TH P HT P TH P HT P TH
P HH TT P HH P TT P HH P TT
Example 13. Consider the variant sample space
2'
2{ 0.28, 0.27, 0.27, 0.18}SBW WB BB WW== = = =
as
modeled in Example 5 in Part I where no ball will
be returned into the bag after a draw. The following
probability additions between two conjunctive
draws of the uneven balls in the bag can be obtained
according to Theorem 5(iii) or 5(iv), respectively:
()( )
()()()
1()
0.55 0.45 0.25
0.55 0.45 1 0.72
10.11 0.89
()()
()()()
1()
0.45 0.55 0.25
1 1 0.73
10.09 0.91
()(() ()
()()()
i
i
i
PBPW
PB W PB PW Pw
PW P B
PW B PW P B Pw
PB PB Pb
PB B PB PB
+= + -
-
·
=+- =- =
-
+= + -
-
·
=- =- =
-
-
+= + - )
1()
0.55(0.55 0.11) 0.24
1.1 1.1 0.83
1 0.11 0.89
()(() ())
( ) () () 1()
0.45(0.45 0.09) 0.16
0.9 0.9 0.72
1 0.09 0.91
i
i
i
Pb
PW PW P w
PW W PW PW Pw
-
-
=- =- =
-
-
+= + - -
-
=- =- =
-
2.6 The Subtraction Operator on
Decompositive Probabilities
The algebraic operation of probability subtraction is
an inverse operation of probability addition, which
is not defined in traditional probability theory.
Theorem 6. The subtraction of related
probability of an event b from that of a in the
sample spaces S in U, ()
P
ab, is determined by
the probability of event a excluding that of b given
and 'aAS bBS , i.e.:
()()()(|)
) Invariant , unrelated , and ME dependent :
( )
) Invariant , related , and independent :
( ) ( ) ( ) ( ) ( )
) Variant ', unrelated
Pa b Pa PaPb a
iSR DSRD
Pa
ii S R D S R D
Pa PaPb PaPb
iii S
, and independent : '
( ) ( ) '( ) ( ) '( )
) Variant ', related and dependent : '
( ) ( ) ''( ) ( ) ''( )
RDSRD
Pa PaP b PaP b
iv S R D S R D
Pa PaP b PaP b
where () 1 ()Pb Pb . (14)
Proof. Theorem 6 can be proven according to
Definition 10 in Part I and Theorem 1 as follows:
, , , and ',
|\|
( ) ( \ ) ||
||| |
|| ||
( ) ( | )
) ' ( , ME),
( )
) '
aba A S b B S
AB
Pa b PA B S
AAB
SS
Pa Pb a
iSSAB AB
Pa
ii S S A B A
,
( ) ( ) ( ) ( )(1 ( )) ( ) ( )
) '
B
Pa PaPb Pa Pb PaPb
iii S S A B A
,
()()
( ) ,
1()
()()
( )(1 ) ( )(1 '( ))
1()
( ) '( )
) ' ,
() ( )
( ) ( ) ,
1()
() ( )
( )(1
i
i
i
ii
i
i
B
PaPb
Pa a A
Pa
PaPb
Pa Pa P b
Pa
PaP b
iv S S A B A B
Pb Pb
Pa Pa b B
Pb
Pb Pb
Pa
) ( )(1 ''( ))
1()
( ) ''( )
i
Pa P b
Pb
PaP b
(15)
Example 14. Given the invariant sample space
1{0.68,0.32}SH T== = as modeled in Example 2 in
Part I, the following probability subtraction
operations on the unfair coin can be derived
according to Theorem 6(i) and 6(ii), respectively:
11 1
11 1
( ) ( ) ( ) 0.68 0 0.68 //
( ) ( ) ( ) 0.32 0 0.32 //
( ) ( ) ( ) 1 (1 0.68) 0.32 //
( ) ( ) ( ) 1 (1 0.32) 0.68 //
()()0
()()0
P
H T PH PHT ME
PT H PT PTH ME
P
SH PSPH HS
P
ST PSPT TS
PH H P
PT T P
Example 15. Consider the variant sample spaces
510
2
16
{ ( | ) 0.11, ( | ) 0.09}
ii i i
ii
SPbbB PwwW
RR
==
=Î= Î=
and 2'
2{ 0.28, 0.27, 0.27, 0.18}SBW WB BB WW== = = =,
respectively, as modeled in Examples 3 and 5 in Part
I. The following probability subtraction operations
on the uneven balls in the bag can be solved
according to Theorem 6(iii), respectively:
PROOF
DOI: 10.37394/232020.2022.2.11
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E-ISSN: 2732-9941
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()
()()'()()(1 )
1()
0.45
0.55(1 ) 0.55 • 0.49 0.27
10.11 ()
()()'()()(1 )
1()
0.55
0.45(1 ) 0.45 • 0.40 0.18
10.09 ()
()()"()()(1
1
PW
PBWPBPWPB Pb
i
PB
PWBPWPBPW Pw
i
PW
PBB W PBBP W PBB P
-= = -
-
=- = =
-
-= = -
-
=- = =
-
-= = -
-)
()
0.45
0.27(1 ) 0.27 • 0.49 0.13
10.11 ()
( )()"()()(1 )
1()
0.55
0.18(1 ) 0.18 • 0.40 0.07
10.09
bi
PB
PWW B PWW P B PWW Pw
i
=- = =
-
-= = -
-
=- = =
-
Example 16. Given the same layout as that of
Example 15, the following probability subtraction
operations on the uneven balls in the bag can be
solved according to Theorem 6(iv), respectively:
() ()
()()"()()(1 )
1()
0.55 0.11
0.28(1 ) 0.28 0.51 0.14
10.11 () ()
()()"()()(1 )
1()
0.45 0.09
0.27(1 ) 0.27 0.60 0.1
10.09
i
i
i
i
PB Pb
PBW B PBWP B PBW Pb
PW P w
PWB W PWB P W PWB Pw
-
-= = - -
-
=- =·=
-
-
-= = --
-
=- =·=
-
6
() ()
()()"()()(1 )
1()
0.55 0.11
0.27(1 ) 0.27 0.51 0.14
10.11 () ()
( )()'()()(1 )
1()
0.45 0.09
0.18(1 0.18 0.60 0.
10.09
i
i
i
i
PB Pb
PBB B PBBP B PBB Pb
PW P w
PWW W PWW P W PWW Pw
-
-= = - -
-
=- =·=
-
-
-= = --
-
=- =·=
-11
Table 2. Algebraic Rules of Probability Algebra
No. Rule
Invariant sample space
(' )SS
Variant
sample space
(' )SS
Unrelated events
()AB
Related events
()AB
1 Commutative (|) (|)Pb a Pa b
()()Pa b Pb a =
(/) (/)Pa b Pb a
()()
P
ab Pba =
()()Pa b Pb a
2 Associative ( | ( | )) (( | ) | )Pa b c P a b c
( ( )) (( ) )Pa b c P a b c =
(/(/)) ((/)/)Pa b c P a b c
( ( )) (( ) )Pa b c P a b c =
( ( )) (( ) )Pa b c P a b c
3 Distributive ( ( )) (( ) ( ))Pa b c P a b a c =
( ( )) (( ) ( ))Pa b c P a b a c =
(( ) / ) (( / ) ( / )), ( ) 0Pb c a Pba ca Pa =
(( ) / ) (( / ) ( / )), ( ) 0Pb c a Pba ca Pa =
4 Transitive () () () () () ()Pa Pb Pb Pc Pa Pc = =
5 Complement
( ) 1 ( ), ( ) ( ) 1
( ) 1, ( ) 0
( ) 0, ( ) 1
Pa Pa Pa Pa
PS P
PS P
6 Involution () ()Pa Pa
7 Idempotent
( ) (), ( ) ()
( / ) 1, ( ) 0
P
a a Pa Pa a Pa
Pa a Pa a
8 Identity
1
( ) ( ), ( ) 0
( / ) ( ), ( / ) ( )
( ) 1, ( ) ( )
( ) ( ) ( ), ( / ) 0
( ) ( ), ( ) 0
Pa S Pa Pa
Pa S Pa PS a P a
Pa S Pa Pa
PS a PS Pa P a
Pa Pa P a
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DOI: 10.37394/232020.2022.2.11
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Corollary 3. The complement of probability on
an event aAS in U, ()
P
a, is a special case of
probability subtraction, i.e.:
() 1 ()
( ) ( ) ( ),
Pa Pa
PS Pa PS a a E S
(16)
3. The Formal Properties and Rules of
the Extended Probability Theory
The mathematical model of general probability, the
framework of the revisited probability theory, and
the formal operators of probability algebra as
developed in Sections 2 and Part I of the paper
enable rigorous analyses of the nature, properties,
and rules of probabilities as well as their algebraic
operations. A set of 36 algebraic properties and
rules of the extended probability operations is
summarized in Table 2
Basic rules of probability algebra in the universe
of discourse of probability U can be expressed in
categories of the commutative, associative,
distributive, transitive, complement, involution,
idempotent, and identity rules. It is noteworthy that
it is unnecessary that each of the probability
operators obeys all the general algebraic rules. Each
algebraic rule on probability multiplication,
division, addition, subtraction, conditional, and
complement operations can be proven by applying
related definitions and arithmetic principles. The
algebraic rules of the probability theory may be
applied to derive and simply complex probability
operations in formal probability manipulations and
uncertainty reasoning by both humans and cognitive
systems. The framework of the revisited probability
theory reveals that classic probability theory is a
special case and subsystem of the revisited
probability theory in terms of both mathematical
models and probability operations.
4. Conclusion
As the second part of the revisited probability
theory, a general theory of probability has been
rigorously introduced as an extension of the classic
probability to deal with complicated variant sample
spaces as well as complex event relations and
dependencies. The revisited probability theory has
been formally described as a framework of
hyperstructures of dynamic probability and their
algebraic operations. Mathematical models and
formal operators of the general probability
framework have enabled rigorous analyses of the
nature, properties, and rules of probability theories
and their algebraic operations. It has been found that
the conditional probability played a centric role in
the framework of probability theories in order to
solve the highly coupled cyclic-definition problems
in traditional probability theories. It has been proven
that Bayes’ law may be revisited and validated
based on the properties of the variant sample spaces
as revealed in this paper. This work has also led to a
theory of fuzzy probability that further extends the
general probability theory to fuzzy probability
spaces and fuzzy algebraic operations.
Acknowledgment: The author would like to
acknowledge the support in part by a discovery fund
of the Natural Sciences and Engineering Research
Council of Canada (NSERC). The author would like
to thank the anonymous reviewers for their valuable
suggestions and comments on the previous version
of this paper.
References:
[1] Artin, M. (1991), Algebra, Prentice-Hall, Inc.,
Englewood Cliffs, USA.
[2] Bender, E.A. (1996), Mathematical Methods in
Artificial Intelligence, Los Alamitos, CA: IEEE CS
Press.
[3] BISC (2013), Internal Communications, University
of California, Berkeley, CA.
[4] Cantor, G. (1874), On a Characteristic Property of
all Real Algebraic Numbers, J. Reine Angew.
Math., 77, 258-262.
[5] di Finetti, B. (1970), Theory of Probability, John
Wiley & Sons, Inc., NY.
[6] Gowers, T. ed. (2008), The Princeton Companion to
Mathematics, Princeton University Press, NJ.
[7] Hacking, I. (1975), The Emergence of Probability:
A Philosophical Study of Early Ideas about
Probability, Inductions and Statistical
Inference, Cambridge University Press, Cambridge,
UK.
[8] Johnson, R.A. and G.K. Bhattacharyya (1996),
Statistics: Principles and Methods, 3rd ed., John
Wiley & Sons, Inc., NY.
[9] Kolmogorov, A.N. (1933), Foundations of the
Theory of Probability, 2nd English ed. (1956),
Chelsea Publishing Co., NY.
[10] Lipschutz, S. and M. Lipson (1997), Theory and
Problems of Discrete Mathematics, 2nd ed.,
McGraw-Hill, NY.
[11] Mosteller, F. (1987), Fifty Challenging Problems in
Probability, Dover Publishing, Inc., NY.
[12] Novak, V., I. Perfilieva, and J. Mockor (1999),
Mathematical Principles of Logic, Kluwer,
Boston/Dordrecht.
[13] Pedrycz, W. and F. Gomide (1998), Introduction to
Sets, Cambridge, MA: MIT Press.
PROOF
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Volume 2, 2022
[14] Potter, M. (2004), Set Theory and Its Philosophy: A
Critical Introduction, Oxford University Press.
[15] Ross, T.J. (1995), Logic with Engineering
Applications, McGraw-Hill Co., 1995.
[16] Todhunter, I. (1865), A History of the Theory of
Probability from the Time of Pascal to that of
Laplace, Reprinted 1965, Chelsea Publishing Co.,
NY.
[17] Venn, J. (1888), The Logic of Chance, 3rd ed.,
Reprinted 1962, Chelsea Publishing Co., NY.
[18] Wang, Y. (2007), Software Engineering
Foundations: A Software Science Perspective, CRC
Series in Software Engineering, Vol. II, Auerbach
Publications, NY, USA.
[19] Wang, Y. (2008a), On Contemporary Denotational
Mathematics for Computational Intelligence, Trans.
Computational Science, 2, 6-29.
[20] Wang, Y. (2008b), On the Big-R Notation for
Describing Iterative and Recursive Behaviors,
International Journal of Cognitive Informatics and
Natural Intelligence, 2(1), Jan., 17-28.
[21] Wang, Y. (2011), Inference Algebra (IA): A
Denotational Mathematics for Cognitive Computing
and Machine Reasoning (I), International Journal
of Cognitive Informatics and Natural Intelligence,
5(4), 61-82.
[22] Wang, Y. (2012a), In Search of Denotational
Mathematics: Novel Mathematical Means for
Contemporary Intelligence, Brain, and Knowledge
Sciences, Journal of Advanced Mathematics and
Applications, 1(1), 4-25.
[23] Wang, Y. (2012b), Formal Rules for Causal
Analyses and Inferences, International Journal on
Software Science and Computational Intelligence,
4(4) , 70-86.
[24] Wang, Y. (2012c), Inference Algebra (IA): A
Denotational Mathematics for Cognitive Computing
and Machine Reasoning (II), International Journal
of Cognitive Informatics and Natural Intelligence,
6(1), 21-46.
[25] Wang, Y. (2014a), The Theory of Arithmetic in
the Extended Domain of Numbers, Journal of
Advanced Mathematics and Applications, 3(2), 165-
175.
[26] Wang, Y. (2014b), Causal Inferences based on
Semantics of Concepts in Cognitive Computing,
WSEAS Transactions on Computers, 13, 430-441.
[27] Wang, Y. (2014c), Towards a Theory of
Probability for Cognitive Computing, Proc. 13th
IEEE International Conference on Cognitive
Informatics and Cognitive Computing (ICCI*CC
2014), London, UK, IEEE CS Press, Aug., pp. 21-
29.
[28] Wang, Y. (2014d), Keynote: Latest Advances in
Neuroinformatics and Systems, Proceedings of
2014 International Conference on Neural Networks
and Systems, Venice, Italy, March, pp. 14-15.
[29] Wang, Y. (2015a), Towards the Abstract System
Theory of System Science for Cognitive and
Intelligent Systems, Springer Journal of Complex
and Intelligent Systems, June, 1(3), 11-32.
[30] Wang, Y. (2015b), Formal Cognitive Models of
Data, Information, Knowledge, and Intelligence,
WSEAS Transactions on Computers, 14, 770-781.
[31] Wang, Y. (2015c), Keynote: Cognitive Robotics
and Mathematical Engineering, 14th IEEE
International Conference on Cognitive Informatics
and Cognitive Computing (ICCI*CC 2015),
Tsinghua University, Beijing, China, IEEE CS
Press, July, pp. 4-6.
[32] Wang, Y. (2015d), Towards a Logical Algebra
(FLA) for Formal Inferences in Cognitive
Computing and Cognitive Robotics, 14th IEEE
International Conference on Cognitive Informatics
and Cognitive Computing (ICCI*CC 2015),
Tsinghua University, Beijing, China, IEEE CS
Press, July, pp. 24-32.
[33] Wang, Y. (2015e), Fuzzy Probability Algebra
(FPA): A Theory of Fuzzy Probability for Fuzzy
Inference and Computational Intelligence, Journal
of Advanced Mathematics and Applications, 4(1),
38-55.
[34] Whitworth, W.A. (1959), Choice and Chance,
Hafner Publishing Co.
[35] Zadeh, L.A. (1965), Sets, Information and
Control, 8, 338-353.
[36] Zadeh, L.A. (1968), Probability Measures of
Events, Journal of Math. Analysis and Appl. 23,
421-427.
[37] Zadeh, L.A. (1996), Logic and the Calculi of
Rules and Graphs, Multiple-Valued Logic, 1, 1-38.
[38] Zadeh, L.A. (2002), Toward a Perception-based
Theory of Probabilistic Reasoning with Imprecise
Probabilities, Journal of Statistical Planning and
Inference, Elsevier Science, 105, 233-264.
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