Classical Probability Theory Versus Quantifiable Matinga Probability Law

In any set diagram, consider that every set has nothing in common with each other except that they are all part of of the universal set. Then the universal set is the intersection (mathematics) or the inner-join (information technology). The outcome is that intersection is each set added together and distributed across all sets adhering to conservation of information.

That means not increasing or decreasing the information count in the set count. For all set logic or set elements to be included in each set, it is must be divided by the number of sets "n", before distributing by multiplying the intersection or inner-join by the set population. That is all illustrated in the examples below.

The logic can be applied at subset level where each subset is treated similarly, at its own sub-universal set level to how the universal set was logically assigned above.

The upper diagram is a Venn Diagram

Its Union formula is:

(1) ε = A + B,

where A and B must be dependent with, intersected with each other or relational to each other.

A ∪ B = A + B - A ∩ B;

B ∩ A stands as the sole remaining intersection or inner-join.

(2) A ∪ B ∪ C = A + B + C - A ∩ B

- A ∩ C - B ∩ C

- A ∩ B ∩ C - A ∩ C ∩ B;

B ∩ A ∩ C remains the final remaining intersection or inner-join.

The lower diagram is a Matinga Set Diagram

Matinga Set Proof

Its union formula is: (3) A ∪ B = ( A + B )/2

For more than 2 sets: N [1] ∪ N [2] ∪ N [3] ∪ ... ∪ N [n]

= ( N [1] + N [2] + N [3] + ... + N [n] )/n

Test

The Unity Limit Rule is such that probability of each set is not permitted to breach the maximum upper magnitude of 1.

For 9 sets to be of probability 1/2 except for 1 extra set of probability 0, the union probability is:

(4) The classical set theory wouldplace the probability would be as below.

P [ A 10 ] = 0 ,

P [ A 1 ] + P [ A 2 ] + P [ A 3 ] + . . . + P [ A 10 ]

- P [ A 1 ] * P [ A 2 ] * P [ A 3 ] * . . . * P [ A 10 ]

= 1/2 + 1/2 + 1/2 + 1/2 + 1/2  + 1/2 + 1/2  + 1/2 + 1/2

+ 0 - 0

= 9 * (1/2) - 0

= 9/2 (FALSE) The theory fails in this example.

(5) The Matinga Probability Proof would yield the following.

P [ A 10 ] = 0 ,

( P [ A 1 ] + P [ A 2 ] + P [ A 3 ] + . . . + P [ A 10 ] ) / 10

= ( 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2

- 0 ) / 10

= ( 9 * (1/2) )/10

= 9/20

Substitute the 1/2 value for 1 in each set in the 2 examples above.

(6) from (4) = 9/10

(7) from (5) = 9/10

Substitute the value 1/2, in (4) and (5) for a/b. Substitute P [ A 10 ] with 1 - a/b.

(8) from (4) = 9 (a/b) - ( a ^ 9/b ^ 9 ) * ( 1 - a/b ) (INELEGENT)

(9) from (5) = ( 8 a/b + 1 )/10 (ELEGENT)

(10) There are 2 dependent events. They are the probability of a coin toss being heads and the other being tails.

P ( H ) = 3/7

P ( T ) = 1 - P ( H )

= 4/7

(11) from (10) = P ( H ) + P ( T ) - P ( H ) * P ( T )

= 3/7 + 4/7 - 3/7 * 4/7

= 3/7 + 4/7 - 12/49

= 37/49

(12) from (10) = ( P ( H ) + P ( T ) )/2

= 1 event presented from 2 possible events/2 events + 1 event presented from 2 possible events/2 events

= (1/2)/2 + (1/2)/2

= 1/2

or the expectation of head 'or' tails, where 'or' or 'alternatively' is signified by '+' is 1/2 in either instance. (TRUE)

Matinga Probability Dependancy Law

The union is the sum of all sets (the universal set), divided by their count.

Explanation:

Multiple sets of logic are dependent when they share commonality known as an intersection. They are completely dependent when they contain the full complement of the greater set’s logic in each subset of logic. That array must have its distribution respected as equal to all of the sets. The logical conclusion will see all sets other than the universal set added and then divided by the count (intersection) of sets in the distribution.

∪ = ε/∩,

union = ( universal set ) / intersection

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