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## A Mathematical Foundation For Politics And Law

By
Jamil Kazoun
(Jamil Talaat Malak Sukkarieh Kazoun)
Version 1.89
First draft, extremely rough, unedited, due to severe health limitations that may not allow the
work to be completed. I will revise it for a possible first edition if possible. Importance of the
material overrides the abundance of spelling mistakes and incompleteness, etc. The
importance of the subject is laying a solid mathematical foundation to subjects that had none.

Preface……………………………………………………………………………………………………………………………………………………..4
Study prerequisites…………………………………………………………………………………………………………………………………….4
Introduction………………………………………………………………………………………………………………………………………………5
Voting Mathematics…………………………………………………………………………………………………………………………………..5
Introduction: Using Proper Language………………………………………………………………………………………………………..5
Formalism ……………………………………………………………………………………………………………………………………………13
Introducing Targets And The Three Critical Points………………………………………………………………………………………..13
Accuracy And Reliability ……………………………………………………………………………………………………………………………17
Reliability Calculations:………………………………………………………………………………………………………………………….30
Overview of Accuracy, Reliability and Intelligence ……………………………………………………………………………………….32
Using The Normal Distribution For Randomness, Accuracy and Reliability…………………………………………………..34
A probability as a single value, as an interval, or as a distribution ………………………………………………………………….39
A Probability Point Stretched To A Distribution ………………………………………………………………………………………..51
The Political Error …………………………………………………………………………………………………………………………………….54
A single voter is equivalent to a coin…………………………………………………………………………………………………………..56
Poll or Group Guess requirements………………………………………………………………………………………………………….57
Warning …………………………………………………………………………………………………………………………………………..58
A Mental Experiment In Guessing: Humans Versus Coin Machines………………………………………………………………..60
Intelligence Versus Randomness or Dumbness : Unrepeatable Experiment……………………………………………..60
Intelligence Versus Randomness or Dumbness: Repeatable But Not Identical Experiment ……………………….61
Ramifications of the experiment …………………………………………………………………………………………………………
69
Regarding Intelligence ……………………………………………………………………………………………………………………….
69
Case 1: The Trial Of Jesus Christ ………………………………………………………………………………………………………….
70
Case 2: The Trial Of Socrates ………………………………………………………………………………………………………………
71
Degree Of Freedom, and Freedom ……………………………………………………………………………………………………………..
72
Freedom Concept In Mathematics ………………………………………………………………………………………………………
72
Group Power As Mob Coercion Power …………………………………………………………………………………………………….
86
A Vaccine For ‘Coercive Do-Gooders’ ………………………………………………………………………………………………………
87
Possible ways to prevent choice reduction ………………………………………………………………………………………………
88
The Threat Index ……………………………………………………………………………………………………………………………….
91
Opposition To Freedom ……………………………………………………………………………………………………………………..
92 Total system versus single event or variable probability
…………………………………………………………………………….. 102
Total Serial probability of multiple events………………………………………………………………………………………………
102
Total Parallel Probability ………………………………………………………………………………………………………………………
111
Definition Two events are in parallel if they are independent of each other. …………………………………………
111
Combined Serial And Parallel probabilities …………………………………………………………………………………………
114
Error versus missed opportunity …………………………………………………………………………………………………………..
116
Grading systems for students or employees and others ………………………………………………………………………………
119
Chapter: Candidate Selection …………………………………………………………………………………………………………………..
121
Introduction: Current Public Voting Systems With Multi Candidates Options Are Mathematically Flawed ……
121
Example Of Current Flaws In Some Polls: ……………………………………………………………………………………………….
123
Poll Design Exercise:…………………………………………………………………………………………………………………………….
124
Chapter: Mathematics In Law …………………………………………………………………………………………………………………..
125
Errors in the law-makers decisions: ……………………………………………………………………………………………………….
125
Exercise: …………………………………………………………………………………………………………………………………………….
125
Errors in the courts decisions: ………………………………………………………………………………………………………………
126
The Upper Bound Limit Of Court Accuracy ……………………………………………………………………………………………..
129
Exercise: …………………………………………………………………………………………………………………………………………
131
Exercise: …………………………………………………………………………………………………………………………………………
131
Exercise: Design a court decision making system …………………………………………………………………………………
131
Chapter: Applications And Strategies ………………………………………………………………………………………………………..
133
Example: How freedom of choice can increase. Mathematical strategy. ……………………………………………………
133
Exercise: How to weaken a cause ………………………………………………………………………………………………………….
134
Reference: ……………………………………………………………………………………………………………………………………………..
135
Preface
January 7, 2021
As of this date, I am not aware of any mathematical foundation being available or in use for
university education in the fields of politics and law. This book aims to remedy this problem.
As of today, I would venture to guess that every university that has a department for teaching
politics has their department named as “Political Science department”. But a better name for
these departments may be “Political Arts Department” or “Political Philosophy Department”.
Using the word “science” in a name implies use of mathematics, where the entire subject, from
A to Z is taught based on mathematics. That is not the case now. Not by any stretch of the
imagination.

Study prerequisites
An introductory course on probability and statistics. Or equivalent. Or support preparation material
given alongside this book. Algebra. Calculus are not necessary, but useful to cover advanced
mathematical material.
Introduction
Whenever few people are gathered and reside in a physical location, there seems to be a
tendency for some of them to want to form organizations to meet some of their needs. These
needs may be for security such as having a police force, or military for external security, or social
or cultural perceived needs. These efforts may be led by motivated people for personal gains or
well intentioned desire. Well-motivated as these efforts may be, they tend to be founded on mere
ideological thoughts that are not subject to mathematical analysis. Modern societies, almost
universally, use voting as the method of deciding, selecting individuals for government or
organizations, and for creating laws, and for adjudicating the laws, and therefore voting must be
analyzed mathematically to be applied properly. The human knowledge of voting as of the year
2020 is extremely flawed as I show.
Politics depend on voting. Voting depends on knowledge of probability mathematics. Knowledge
of probability and statistics are essential for making politics a science, that is, a subject founded
purely on mathematics.

Voting Mathematics
Introduction: Using Proper Language

If a person is a gambler on sports teams, with 60% correct guessing which team will win in a
competition, it may be normal to think or focus on this 60% correct guessing. So when this person
makes 100 bets, each for one dollar, he on average, wins 60 bets, or 60 dollars. This is the normal way
people talk about voting in an election, saying: the candidate won 60% of the votes, or this law passed
congress with 60% of the votes. What is missing in such talk is the other part of the equation. When a
person wins 60 bets out of 100, 60%, this also implies he lost 40 bets of this 100 bets, 40% or \$40.
Winning \$60 and losing \$40 for every 100 bets. This means this person actually wins \$20 for every 100
bets. Win = win proportion minus loss proportion= \$60 – \$40 = \$20. Therefore, his win or net win is not
60%, but is 20%. The 60% is not his “win” but his “win proportion”. A proportion is only one portion of a
100% whole, the other proportion is the loss, so that

Win Proportion + Loss Proportion = 100%
Therefore, a proper English language term and definition for the 60% is Win Proportion, and Loss
Proportion for the 40%. The Win, or easier to understand Net Win or Total Win is 20%.
A proportion is a portion or part of 100%.
In voting, there is typically the (In Favor, Opposed) pair of proportions or the (Yes, No) pair of
proportions.

Definition: Win Proportion is the percent of support in a poll
Definition: Loss Proportion is the percent of opposition in a poll
Definition Total Win = Win Proportion – Loss Proportion
Example: A betting person has a 55% Win Proportion, his implied Loss Proportion of 45% = 10%. When
this person spends \$30 on \$1 bets, his total win is \$3. In this case, the unit of the win is “\$”.

Example: An office candidate has a 55% Win Proportion, his implied Loss Proportion of 45% = 10%.
When this person faced 30 total voters in the election poll, his total win is 3 voters. In this case, the unit
of the win is “voter”.

Example: A senator has a 55% Win Proportion on his proposed law, his implied Loss Proportion of 45% =
10%. When this person faced 100 senators voting in the poll, his total win is 10 voters.

Example: A person buys a box of apples that has 100 apples and that has a 55% good apples Proportion
per box, his implied Loss Proportion of 45% = 10%. When this person spends \$1 per apple, total of \$100,
is his total win 10% of the apples or 55% apples? Does he have 10 good apples to eat or 55 apples to
eat?
Question is: How is this example different from the betting and voting examples?
Answer: A box of Apples represents a single bet not 100 bets. The bet unit is “box of 100 apples” not
“apple”. If the apples are bought individually, one apple at a time, then this would be equivalent to 100
bets with each bet having 55% chance of success for a good apple. If on first bet he wins a good apple,
and next bet loses the bet, he gives back the good apple. Now he has zero apples. The unit being
exchanged on a win or a loss is the same, it is a good apple. But giving a good apple on a win bet and
giving bad apple on a loss bet produces two different units in the value of a Win formula: a unit of “good
apple” , and a unit of “bad apple”. To subtract two items, they should be of the same unit.

Definition:
Total Win Value = Win Proportion * Win unit value – Loss Proportion * Loss unit value
From this equation, we can see that the Win unit value must be the same as the Loss unit value, such as
a good apple, or one dollar, or one vote, but not a mix of units, unless convertible.
Since win unit value = loss unit value
Total Win Value = (Win Proportion – Loss Proportion) * unit value
If we define the term “poll” as a “group vote”, with the unit = “vote”, we can see that:
Definition:
Total Win Value of a Poll = (Win Proportion – Loss Proportion) * vote

Example: A company board of directors holds a poll on building a water plant. The results of the poll are
Win Proportion 80% and Loss Proportion 20%.
Win Value of the Poll = (80% – 20%) * vote = 60% vote or 60% of the votes.
A Win can be defined as a Success or as Correctness. Example: On an exam, you won (succeeded with,
or correctly guessed or correctly answered) 60% of the questions and lost 40%. In life, in many
situations, as in politics, when it is difficult to create a mathematical definition of the subject of a poll,
and thus, it is not possible to define correctness qualitatively, we have to revert to a quantitative
definition using a statistical group guess. In such a case, the poll serves this purpose. And correctness
becomes defined merely as observed “success”, and incorrectness as observed “failure” as the binomial
probability does.
The terms (win, loss) can be easily exchanged with the terms (success, failure) which are typically used
in binomial probability terminology, or exchanged with (correct, error), (correct, incorrect) or (correct,
not correct). A good terminology to use sometimes in voting mathematics is (Correct, error) or ( correct,
incorrect) or best mathematically is (correct, not correct) since they are clear and clearly opposite as
defined by the “not” logical operator which puts these values in opposite direction, making them
vectors, and also allows arithmetic operations on them since they are the same unit of “correct” or
“correctness”. The opposite direction to a positive is a negative direction, which use the + or – symbols,
such as +1 and -1 written in shorthand as 1 and -1, or can be positive also but expressed by the angle
such as 1 with angle 0 degrees and 1 with angle 180 degrees. Both representations can be used. But
using (1 , -1) is simplest, so long as you do not forget that in voting -1 is not a number but a vector,
which means it is the number 1 pointing in the opposite direction of the vector 1, and adding the two
opposed vectors results in 0. Example: in a poll of 8 people, 5 voted in support and 3 opposed. 5 voters *
(correct) + 3 voters * not (correct) = (5 + (-3)) correct = (5 – 3) correct = 2 correct.
These terms (correct , not correct) can also be equivalent to saying (Yes, meaning the choice is correct,
and No, meaning the choice is not correct) or the shorthand (Yes, No), and the value of Yes is the vector
1, and the value of No is the vector -1. The notation for a vector is the arrow on top p as in

?⃗ ,
Here is a table of this progression in use of terms to represent probability and a choice:
progression probability Probability
compliment
comment
mathematical
labels
success failure Binomial
probability
Success error
Correct Error Engineering
Correct Incorrect
Correct Not Correct Mathematical
terms and
logic
operators
In Favor Opposed Public
language
I Support I Do Not
Support
Public
language
Yes No Public
language.
Excellent
shorthand for
public use
P P’ Mathematical
language. P
and its
compliment.
Only one
variable used
Poll Yes
mean
Mathematical
mean of the
Yes
proportion of
the poll. The
poll is the
entire
population.
Therefore,
the
proportion is
the
probability
Yes
sample
mean
In group
voting,
sample mean
is population
mean
Yes
population
//
mean
Yes
proportion
//

It is extremely important to understand and be thoroughly familiar and comfortable with using any of
these terms interchangeably.
Fundamental Equations Of Voting

Since vector analysis is not a prerequisite to this course, I will define a vector.
A vector is a scalar with a direction
Therefore, a vector is a number that has a direction it points to. In voting, the direction of a person
voting Yes is the number with value of 1 pointing one way, while a No vote is the number with value 1
pointing to the opposite direction. Yes and No can be the arrows ( ,  ) or (  ,  ) with arrow length
equal to 1. The ???⃗ , ?????⃗ ??⃗ together can be up and down or at any angle.
Example: ? denotes a vector. Therefore, when talking about a person’s vote, it is a vector of (1
? 1⃖),?⃗ ? (1 ?⃗ ? −1 )
A person vote or guess is a vector of 1 or −1
This is a fundamental basic data unit of a poll, a vote by a person. It is essential to understand voting.
Definition: yes = correct = 1, as in “this choice is correct”
Definition: no = not correct = -1, as in “this choice is not correct”
To get a total vote count, sum all the votes
Vote Total = sum of the individual personal votes = = sum of yes vector votes + sum of no
Since a No vector is a unit of – 1 , we have
Vote Total = (Yes total – No total) ; with the unit = vote or correctness

Since a Yes vote = a unit of 1 correctness, and a No vote = a unit of -1 correctness, we have Vote Total =
Net Correctness.
Using the term Poll, we have
Poll Total = Yes total – No total
Net Correctness = Yes total – No total
We assume that a poll with less than 50% yes support is dismissed as a reject. This simplifies
calculations for the moment by avoiding Poll Total and Net Correctness from being negative numbers.
We could take the absolute value, or magnitude of these equations, to maintain generalization, but we
will find out later that this is not needed, since we will prove that the poll voting scale begins at 50%.
The poll voting scale should not begin at 0%.
Example∶ a committee of 40 individuals had a poll, and the results were 30 Yes and 10 No, therefore :
Poll Total is Yes count + No count = 30 *(1 yes vote)+ 10*(−1 yes vote) = 20 yes votes
Poll Accuracy = Yes count + No count = 30 *(1 correctness)+ 10*(−1 correctness) = 20 correctness Exercise
prove that
Poll Total in percent = Yes total in percent – No total in percent
And if
YES is a label for Yes total in percent
And
NO is a label for No total in percent
Poll Total or Poll Result in percent = YES − NO
This is the proper total that should be reported as the poll result. This is a fundamental and basic
equation of voting and vote counting.
Example: Vote Total =0% at 50% Yes to 50% No, etc….
Here is a table of Vote Total or Vote Result:
YES NO Vote Total
50% 50% 0%
60% 40% 20%
70% 30% 40%
80% 20% 60%
90% 10% 80%
100% 0% 100%

Graph of the Vote Total or Vote Result equation

This equation shows why:
The voting scale begins at 50% YES to 50% NO, with its value being 0%

Formalism

In traditional books on probability and statistics, there is common use of some labels in formulas. I will
assume you are familiar with these:
For an event or space with N possibilities, the probability of a possibility is 1/N
Example : An ideal coin typically has 2 possibilities, Head and Tail, and when tossed, the probability of
Head landing is 1/2, and probability of Tail landing is 1/2 p : denotes the probability of an event
Example : p = 0.5 for a fair coin toss landing Head q
: denotes remaining probability, so that p + q = 1
Example : For a coin, p + q is 0.5 + 0.5 = 1
Also, p and q are in the interval [0, 1]
n, lower case, is the variable used to indicate the number of trials, or sample size. If a coin is tossed 30
times, the number of trials n=30. If 40 coins are tossed in a single toss, the sample size n=40.
A single coin tossed 30 times can also be considered a single experiment. Also tossing 40 coins at the
same time can be considered a single experiment. An experiment produces a mean. When an
experiment is repeated, then we can average the means of the all the experiments, and this is typically
referred to as u. In this book, u and p are considered the same, because our sample is the full
population, therefore, our sample mean is the population mean. And u and p are the same.
k, lower case, is the degrees of freedom variable = n – 1
The voting we will discuss in this book is typically for a vote limited to two choices which are (yes, no).
We will use the simplest form of proportion probability and its established equations, as applicable.
Also, for this, The Central Limit Theorem is assumed to apply.
The (Yes, No) choices of a vote are mutually exclusive.

Introducing Targets And The Three Critical Points
In guess probability, we can introduce the concept of a target. A target for a political candidate, is 100% support
in a poll. For a Supreme Court judge or group of them, the target is 100% support for the argument when a poll
among the court judges is held in a trial, etc. On the other side of human guessing, the opposing humans to a
candidate or an issue, the target is 0%, that is, their target is the failure of the candidate or issue.
With the introduction of a target, the idea of error becomes concrete and measurable. Error becomes the
distance away in percent of a result from the target. If a candidate gets 70% support in a poll, then the error is
100% – 70% = 30%. However, in polls, there are two parts, and both parts need to be entered in the error
equation. One part is those persons supporting the issue, and their target is 100% success and the other part are
those opposing the issue and their target is the 0% side of the interval, that is, they want 100% opposition to the
issue, so it would fail. so the errors are the distance of those supporting the issue from 100% target, plus the
error of those opposing the issue from the 0% target (see illustration. Proof is in next chapter.):
Error = |( target 1 – support amount )| + |(target 2 – opposition amount)| = |1-0.7| + |0-0.3|= 0.6 or 60%
This is the same result we get if we used our Accuracy formula, by setting Error=1-Accuracy
Using targets greatly simplifies the concept of probability error and its computation, or accuracy and its
computation. And it applies in competitive issues between two human opposed groups. But what happens when
one human or group has no opposition in a guessing situation, such as a student taking a multiple-choice exam
questions. We can still use the same targets of success and failure of 1 and 0, but it is extremely important to
introduce a third target, or point, 0.5, which is the point of Perfect Randomness. 1 and 0 indicate perfect
intelligence, with 0 being perfect intelligence with wrong data fed into the intelligence evaluation system, such as
“feeding a correct computer wrong data and getting wrong results” such as a patient with pain in his arm walking
to a doctor and miss-speaks saying “I have pain in my foot” and the doctor says “take this patch and put it on
your foot” . The doctor is intelligent, and gave the wrong answer because the patient did not feed him correct
data. In political issues, the opposition to a candidate may win 100% support against the candidate in a poll,
which produces a poll with 100% at the 0 end of the probability interval. In this case, the opposition seems to be
intelligent as evident by their 100% win.
For randomness related computations, 0.5 or 50% is the point of perfect randomness and can be used as a
target, because at a probability of 0.5, randomness is 100%, as Accuracy, and Bernoulli Distribution Entropy and
other equations show. The 0.5 point is the exact opposite of the 1 point that represents intelligence. Intelligence
implies certainty about output, while randomness implies uncertainty about the output. Therefore, the three
points 0, 0.5, and 1 are points of perfect certainty or uncertainty. 0.5 represents an inanimate or unintelligent
random machine, while 1 and 0 represents the opposite, that is an intelligent machine.
Example 1: A single fair coin tossing machine, is a machine that produces a random output of probability of 0.5.
This machine, if given an exam of 100 two-choice (yes, no) questions will answer 50% of the questions (50
questions) correctly. This is an extremely important issue to understand, because it implies, that a human
answering such exam questions and answering 50% correctly, is demonstrating the same intelligence ability as a
coin, which is 0% intelligence.
Example 2: Suppose you were driving a car and came to a fork in the road, one road to the left and one road to
Assume that there are one hundred law-makers standing at the fork of the road, and you ask them: Which way
do I go? Left or right? They vote, and fifty percent say go right, and fifty percent say go left. The poll result is
useless because it gives no information, guidance, for deciding which way to go. The poll result or accuracy is 50%
– 50% = 0% . Zero guidance or zero certainty on how to proceed. The poll result has a value of 0 if we wanted to
use it to make a decision on choosing left or right. If you decide at this point, your decision would be based on
pure luck and not on intelligence, and as if you are tossing a coin to decide on an important matter.
If you remember nothing from this book, you should try to remember this:
50% or 0.5 is the point of perfect randomness on the probability interval. Getting 50% support in a poll or
exam, means total randomness and 0% intelligence.
0.5 is the opposite of 1 in human guessing. Therefore, the Accuracy, or intelligence
interval can be viewed as limited to the interval [0.5 , 1] with the other side of the
interval being a mirror image.
With 0.5 being the Perfect Randomness Point, we can begin to use it as a fixed reference in our computations to
solve probability problems.
Example: If we look at the example above of a candidate getting 70% support in a poll, without any computation,
we can see that 70% is closer to 50% point of Perfect Randomness than 100%, the Point Of Perfect Intelligence.
Therefore, it can be dismissed as more of a random event than an intelligent event. As a matter of fact, when we
look at our reliability equation and graph, we see that reliability only starts at the 0.75% point, wich is the half
point of the [0.5 , 1] interval.
Let us look at some randomness functions of a probability. The first function is the variance function. Variance is
randomness. The second function is variance scaled to a 100% range interval, by multiplying it by 4, since the
maximum variance of YES is 0.25. The third function is the entropy of a binary choice probability of success or
failure, which is the basis of the Bernoulli Distribution, which is a special case of the binomial distribution. The
Variance function is: Yes x No = YES – YES^2
Variance percent function= 4xV
Bernoulli Entropy function = -YES x log2(YES) -NO x log2(NO) = -YES x log2(YES) -(1-YES) x log2(1-YES). Where the
Bernoulli Distribution equation is YES^YES x NO^NO.
Note that the slight difference between the Bernoulli Entropy function and Scaled Variance function. Error
calculations can be defined as differences, or as ratios or as logs of differences or ratios, etc. which may produce
slight differences in error computation results.
These three points, 0, 0.5, 1 are defining points on the probability interval.
Accuracy And Reliability
We saw previously that Net Correctness = Yes total – No total = YES – NO
If I were to use Binomial Theory terminology, then instead of the term net correctness, I would use net
success = success – failure= Accuracy .
Normal people talk in terms of accuracy, usually, and computation of accuracy is more easily understood.
Net Correctness is a proper definition of Accuracy, because it is the total amount of correctness in a poll.
Ideal Poll Accuracy = Net Correctness = Yes total – No total
YES + NO = 1 OR 100%
YES – NO = YES – (1 – YES) = 2 * YES – 1
Ideal Poll Accuracy= YES – NO = 2 YES – 1
Here is a graphical proof of Ideal Poll Accuracy In
Proof: Let 100% be the desired accuracy Target for any guess, Yes = 100%. The distance away from the
target is the amount of error. Since 50% represents Perfect Randomness, it is the 0% accuracy point, or
the start of the accuracy scale. Therefore, the accuracy scale interval is [0.5, 1]which we can map to [0,
1] by subtracting 0.5 from the result and multiplying it by 2.
Accuracy = (YES – ½) × 2 = 2 YES – 1 = YES – NO, which is the previous result.
Therefore, if YES = 60%, Poll Accuracy = (60% – 50%) × 2 = 20%
Proof: Another way to look at it is 100% being Target Of YES voters and 0% is the Target Of NO voters.
The distance away from the target is the amount of error
Accuracy + inaccuracy = 100%
Inaccuracy = distance from YES target + distance from NO Target
Accuracy = 1 – inaccuracy using p and q terms:
Inaccuracy = error of p + error of q error of p =
distance from p target = 1 – p error of q =
distance from q target = q – 0 inaccuracy=(1 – p)
+ (q – 0)= (1- p) + (1 – p) = 2 – 2p
Accuracy = 1 – inaccuracy = 1 – (2 – 2p) = 2p – 1
End of proofs
YES Ideal Accuracy = Yes Proportion of Ideal Poll Accuracy = YES * Ideal Poll
Accuracy
NO Ideal Accuracy = No Proportion of Ideal Poll Accuracy = NO *Ideal Poll Accuracy

When reporting poll results, it is appropriate to report only YES Ideal Accuracy
Example: A poll had 60% Yes proportion. Therefore, Yes Ideal Accuracy is 60% (60% – 40%) = 12%
Exercise: calculate all the remaining accuracies of the example above.
YES Ideal Accuracy = YES ( YES – NO) = YES (2 YES – 1) = 2 YES^2 – YES
This is a quadratic equation. Here is the graph
Notice that the YES Accuracy starts at YES = 0.5
Definition: Correctness is the status of a single event being correct or not
Definition: Accuracy is the sum of many event’s correctness
Let A represent Ideal Accuracy, (number of correct events – number of incorrect events) then:
A = Ideal Accuracy = YES – NO
A’ = Inaccuracy = 1 – A
It does not seem proper to have a vote where probability of correctness is less than the probability of
incorrectness. Therefore,
Not good to have A < A’ for V
A = A’ when A = 50%
Since
A = YES – NO = YES – (1 – YES) = 2YES – 1
A’ = error = 1 – (2YES – 1) = -2YES + 2 = 2NO
A’ = 2NO
Setting A to 50% or 0.5
0.5 = 2YES – 1
YES = 0.75 = 75% = Yes
Therefore YES = 75% gives A = 0% for V.
Stated in words:
A YES vote of 75% yields Net Accuracy of 0% for the vote.
This is clearly a low level value for a vote’s accuracy.
If Reliability of a tool for its intended use is defined as Net Accuracy
R = A – A’
Since A = 2YES – 1, and A’= 1 – R
R= 2YES – 1 – (1 – (2YES – 1))
=2YES -1 -1 + 2YES – 1
R = 4YES – 3
When R = R’, R= 50%, and R – R’ =0
Reliability of zero seems like it should be the lowest reliability level that is permissible.
Setting R to 50%
50% = 4YES -3 = 0.5
YES= (0.5 +3)/4= 0.875 = 87.5%
A YES vote of 87.5% yields Net Reliability of 0% for the vote.
This is clearly a low level value for a vote’s reliability, and can be considered the starting scale for polls results.
However, we still need to account for the Group Size Error, and focus on the YES proportion. R = 4YES – 3
The YES proportion is YES x R = YES(4YES – 3)
If YES(4YES – 3) = 4YES^2 -3YES = 0.5
YES = 0.89 or 89%. At this point, Net Reliability of YES is 0, and serves as the upgraded starting point for
considering a YES poll.
If I define Usability as a short term variable, convenient for representation but to not to use after, as the
net of net reliability. It is better to say “Net of Net Reliability” in order to stay connected directly with
the factors or variables we are measuring:
U = R – R’ =
U’ = 1 -U
U = 4YES – 3 – (1 – (4YES – 3)) = 4YES +4YES – 3 – 1 -3
U = 8YES -7 YES = (U + 7)/8
When U = U’, U= 50%, and U – U’ = 0
Setting U= 50% = 0.5, I get YES= (0.5
+ 7) / 8 = 0.9375 = 93.75%
Usability of zero seems like it should be the lowest Usability level that is permissible.
Therefore, at YES= 0.9375, Usability is 0%
This clearly is a low level of Usability.
Therefore, for a YES, there is A, R, U with these critical cutoff values:
C = 0 YES = 0.5, or 50%
A = 0 YES = 0.75, or 75%
R = 0 YES = 0.875% or 87.5%
U = 0 YES = 0. 9375 or 93.75%
For n = ∞, we have these formulas:
n=∞ Formula Formula
value =
At YES =
Accuracy A = YES – NO
= 2YES – 1
0 0.5
Accuracy of
YES
YES(2YES-1) 0 0.5
Net Accuracy,
Reliability
4YES-3 0 0.75
Reliability of
YES
YES(4YES-3) 0 0.75
Net Reliability 8YES-7 0 0.875
Net Reliability
of YES
YES(8YES-7) 0 0.875
Since the Standard Error is the Standard Deviation divided by square root of n, we can see the dramatic effect the
group size has on error. At n=1, the Standard Error equals the Standard Deviation. At n equal infinity, SE is 0.
Therefore, have a large group size is a must when considering accuracy in important subjects. A group using a poll
is a computation device. If this device is fundamentally flaw, how can we accept to use it, or allow it to exist?
Imagine a sample size or group size of 1 as in most courts, or a group size of only 9 in a supreme court.
Mathematically, we can see that these would not be allowed to exist as a mathematical construction for judging
and evaluation. Yet they exist in reality.
For these formulas, we assume n=1, substitute YES – SE for YES
n=1 ,
substitute
YES – SE for
YES in this
table
Formula Formula
value =
At YES =
Accuracy A = YES – NO =
2YES – 1
– 2SD/√n
0 0.8536
Accuracy of
YES
(YES-0.5/√n)(2YES
– 1
– Etc.
)
0 0.8536
Net
Accuracy,
Reliability
4YES-3 0 0.9557
Reliability of
YES
YES(4YES-3) 0 0.9557
Net
Reliability
8YES-7 0 0.9872
Net
Reliability of
YES
YES(8YES-7) 0 0.9872
Now we simply these formulas by simplifying the SE function.
SE = SD/√n. If we set p to 0.5 for humans in a poll, giving each person equal chance to be on either side of the poll,
support or oppose (details about this assumption in next chapter), then SD = 0.5, and SE =0.5/√n. we can use this
in our equations.
Define: Group Size Standard Error (GSE)= 0.5/√n
and
Group Size Accuracy (GSA) = 1 – Group Size Standard Error
The entire graph shifts with each value of n.
The entire graph shifts with each value of n.
Now we have:
n=1 ,
YES= YES – SE
Formula Formula
value =
At YES =
Accuracy A = YES – NO =
2YES – 1
– 1/√n
Accuracy of
YES
(YES –
0.5/√n)(
2YES – 1
– 1/√n
)
Net
Accuracy,
Reliability
4YES-3- 2/√n
Reliability of
YES
(YES – 0.5/√n)
(4YES-3- 2/√n)
Net
Reliability
8YES-7- 4/√n 0
Net
Reliability of
YES
(YES0.5/√n)(8YES74/√n)
0
Let us focus on the two relevant equations, Accuracy of YES, and Net Reliability Of YES.
Accuracy of YES =A = (YES – 0.5/√n)(2YES – 1 – 1/√n)
We use it to calculate:
1. We can calculate Accuracy of a poll, given group size n, and Yes poll result.
2. Given an accuracy threshold the group wants to set as a standard of Accuracy, and a group size n,
we can compute the YES threshold cutoff point below which all poll results would not be accepted.
Substitute YES for x in the image as solution. Yes= 0.25(2√n+n+(8An^2+n^2)^0.5)/n
3. Given a YES threshold and desired accuracy, we can compute the needed group size.
Example 1: The parliament took a poll on a proposed idea. Poll YES results were 83%. Their group size is 240.
What is the Accuracy of YES result?
Answer: A = (YES – 0.5/√n)(2YES – 1 – 1/√n) = 0.475 or 47%
Example 2: The citizens of a city decided that all decisions, including elections of a candidate, will be done by
direct public voting, and must have Poll YES Accuracy minimum of 98%. The city voting population is 30000. What
is the threshold cutoff point for the Poll YES results that must be set?
Answer: A = 0.98, n= 30000 , YES = 0.25(2√n+n+(8An^2+n^2)^0.5)/n = 0.996 or 99.6%
Example 3: A court wanted to know what jury group size is needed to achieve Poll YES accuracy of 99.9% in a
verdict. Find answer for Poll YES Accuracy of 95% , and 90%. And for 90% Accuracy for 98% jury support.
Answer: solving the Accuracy equation for n gives: n is 2.25 million person voting unanimously with YES = 100%
For 95% Accuracy in a verdict with unanimous support of YES = 100%, the jury size needed is 880 persons.
For 90% Accuracy in a verdict with unanimous support of YES = 100%, the jury size needed is 215 persons.
For 90% Accuracy in a verdict with support of YES = 98%, the jury size needed is 1256 persons.
Exercise: What is the maximum accuracy that a parliament with 100 members can attain? What is the YES Poll
result threshold needed to attain it.
Exercise: What is the maximum accuracy that a court with 5 judges as jury can attain? What is the YES Poll result
threshold needed to attain it? If a court want its verdicts to have 99% accuracy, how many judges must be the
group size?
Exercise: Tally the polls taken by your city hall or province or country parliament in the past year. What was the
average accuracy of all the polls?
Reliability Calculations:
Net Reliability Of YES = (YES-0.5/√n)(8YES-7-4/√n)
Net Reliability provides accuracy, and reliability, which can be equivalent to Intelligence
Therefore:
Intelligence: Net Reliability Of YES
Unintelligent: 1- Intelligence
Note: The Net Reliability and Net Reliability of YES provide very near results, so that the simpler equation can be
used in less demanding situations.
The advantages in using the Net Reliability formula over the Accuracy formula is that the Net Reliability formula
provides for higher standards. Accuracy starts at 50% while Net reliability start at 87.5%. Net reliability provides
accuracy, and reliability, which can be equivalent to Intelligence. Important decisions must have built in
reliability, and as such, the reliability formula is more appropriate to use, unless we use the Accuracy formula and
use the high threshold cutoff points that the Net Reliability formula provides! Even then, it is wise to calculate the
Net Reliability of A Poll YES as an integral part of result evaluation. Net Reliability of YES can serve as a single poll
result number.
Example: what is the reliability of a public poll by a country of 250 million voting adults, where the president is
elected with 68% YES support? What if elected with 88% YES support?
For 68% YES support, Net Reliability of the poll YES is less than 0.
For 88% YES support, Net Reliability of the poll YES is 3.5%.
Exercise: A Congress of 400 members wants all its decisions to meet a 98% Accuracy threshold. What is the YES
level to be set as the threshold point for all its polls?
Defining Precision
Precision is a measuring instrument maximum amount of measurement resolution. A one meter measuring tape
or stick may have typically 1 millimeter as its smallest markings, or precision. If the item is smaller than 1
millimeter, this instrument cannot measure it. But a caliper, another measuring instrument, can have 10 times
higher precision and is able to measure items that are 1/10th of a millimeter. This makes a caliper more precise
than this tape or stick ruler for measurement, and thus, this caliper has higher precision..
The Scale Of An Instrument
The scale of an instrument can be the system used for its units. The scale on a measuring tape or stick can be in
meter scale or feet scale, or centimeter scale or inches scale. A thermometer may have centigrade or Fahrenheit.
A meter scale and a centimeter belong to the same metric system, but a measuring stick limited to meter marks
or division has less precision than a stick with centimeter marks or divisions.
Defining The Probability Percent Scale
An interval between 0 an 1 that represents a percentage interval. To be a probability interval, it must also have a
minimum of two divisions, which correspond to a possibility space of two possibilities, because a possibility space
of only one possibility is not a probability space. The points 0 and 1 are not a probability, but a certainty. I leave
this issue for others to deal with. It is best to include the perfect randomness point of 0.5 on the interval as the
opposite of points 0 and 1. 0 and 1 have 100% certainty and 0.5 has 100% uncertainty. Therefore, 0.5 can be
considered also as the starting point of the probability interval. The precision of the probability interval scale is
the size of the smallest division according to this formula:
Probability Percent Scale Precision = Minimum(YES , NO)
If YES is ¾ and NO is ¼, then ¼ is the precision of the scale. Therefore, the precision of the probability scale is
specific to the problem being considered. Another factor effecting precision is the group size n. As n increases,
precision increases as per the standard error formula. SE = SD/√n.
Overview of Accuracy, Reliability and Intelligence
We saw that Randomness can be defined by the Entropy function of the Bernoulli Distribution.
H= Randomness = -YES* Log2(YES) – (1-YES) Log2(1-YES), (1st graph in Illustration, Red graph).
The compliment of randomness is Certainty or Accuracy =1 – H. (1st graph in Illustration, green graph).

If we subtract randomness from Accuracy, we get Net Accuracy, or Reliability (2nd graph in Illustration, blue
graph).
Similarly, we formulated Net Reliability = YES(8*YES-7), (3rd graph in Illustration, blue and orange graphs).
Important: Note how well our equation matches the accuracy and reliability equation based on Entropy.
Using The Normal Distribution For Randomness, Accuracy and Reliability
Let us take a quick and general view of how our equations compare with models based on the Normal
Distribution.
The Normal Distribution is defined here as in the illustration. I have taken a Normal Distribution, spread over the
x-axis, and scaled it by a factor c=?^2 to fit a normal probability interval [0,1]. This should greatly simplify the
Normal Distribution graph, and give it a meaning readily understood, based on probability and not Standard
Deviation or other complicated factors. By Scaling the Normal Distribution domain to fit the probability interval
[0,1], and scaling the Normal Distribution range to be inside [0%, 100%] with its maximum value at 100%, we now
have a percent probability density function (PPDF). We start with red (Normal Distribut5ion), and then map it
to blue, then to Orange, then to orange again to have the final PPDF.
Percent Probability Density Function (PPDF) is the last graph in Orange.
Note: Scaling of the graph causes some distortions, and the equation is best used for complete accuracy when
needed.
I have shown that the probability point 0.5 is the maximum randomness point, and is perfect randomness. With a
Normal Distribution having a mean or a probability of 0.5, we now have the full distribution representation of this
perfect randomness, in the PPDF graph. Therefore, the Normal Distribution of 0.5 represented on a percentile
interval (PPDF) represents the Perfect Randomness Distribution. The function can be used visually or
computationally to measure Randomness, or more importantly, its compliment = (1- Norm PPDF) = Accuracy,
which can be defined as Intelligence. We also want reliability along with the accuracy, then our equation of
intelligence is more complete. We use the same method we developed to formulate our Net Intelligence
equations.
PPDF= e^(-0.5c(x-u)^2/V , c=?2 , ? ?⃗ ?????? ?????⃗ ?, V = variance
Perfect Randomness = PPDF
Intelligence or Accuracy (based on PPDF)= 1- PPDF
Now I use the PPDF for accuracy and reliability to graph the equations, to show similarity to our simple equation.
Using the PPDF function:
Net Intelligence = Intelligence – Perfect Randomness = 1- 2 x PPDF
Net Reliability of Intelligence = 8 x Intelligence -7
See the graph bellow:

Now we can measure Accuracy and Reliability, that is intelligence of a Poll. We can also set threshold for needed
accuracy, and set the minimum group size requirements.
Example: a poll result are 90% YES. Look up the value of the 90% probability on the Net Reliability graph or in the
equation to find out the Intelligence level of the poll is 18%.
Example: a poll result are 98% YES. Look up the value of the 98% probability on the Net Reliability graph or in the
equation to find out the Intelligence level of the poll is 82%.
Example : AN official is given a test and he scores 95% correct answers, what is his intelligence as demonstrated
by the exam? 57%
A probability as a single value, as an interval, or as
a distribution
Let us take a look at graphs and illustrations to visually get acquainted with Accuracy and reliability. 5 gun bullets
are shot at a target. They can be enclosed in a circle whose center is at r = 0.7. The bullet circle radius is 0.2,
which is the dispersion of the bullets from this circle center:

Variance in data, and its associated linear interval on the x-axis, for our purpose is randomness, and is
undesirable, and is equivalent to error in important matters such as law making and court verdicts. In such
situations, variance is equivalent to uncertainty, which is the opposite of certainty. When dealing with law making
or court verdicts or important decisions, we should have a goal to have zero uncertainty in the data. This means,
that any area of data with variance is undesirable. The goal should be to have the linear variance interval equal to
0, or near 0, which implies 100% confidence in YES. This happens when YES is 1 or its opposite, 0, and 0.5, and /
or, for all the other points when n is equal to infinity.
Example: YES is 100% in a poll result on “be educated”. This means the Certainty of YES is 100% of support for
“be educated”. In this poll, NO is 0%. This means the Certainty of NO is 100% of no opposition for “be educated”.
If YES was 50%, it would mean 100% Uncertainty, or equivalently, 100% certainty of uncertainty.
Graph of ideal p, meaning group size n is infinite, which has an interval of size zero on the p axis:

Graph of YES, YES is 0.7 and n less than infinity which has an interval of size greater than zero on the p axis
(interval is blue area around p):
Graph bellow of Standard Error (± SE interval bracketing YES) as n = 1, 2, 4, 10, 100, 1000 :
SE as n of sample size
increases
SE as N of possibilities
increases from center
of randomness N =2
n = 1, 2, 4, 10, 100,
1000
N= 2, …,
Variance decreases

Graph of one experiment for p, which has a interval of size ±SD/√n which is an interval of size 2SD/√n on the p
axis.
Effect of group size n on YES, and its error and accuracy represented on a horizontal probability interval

It is important to see from these graphs the dramatic impact n and N have on accuracy of a poll. With a small
group size n, variance is large so as to make the data acquired from this group useless in terms of accuracy and
reliability. If the group polled had only one person, i.e., n=1, then the SD interval is [0.5-0.5 , 0.5+0.5] =[0 , 1]
which is the entire probability interval, or 100% of the probability interval.
If variance is randomness of the data in this interval, and randomness implies uncertainty, then we see that the
error, or specifically, the standard error interval, is 100% of the probability interval. Therefore, the entire
probability interval is not usable. Only one student in the entire poll, surveying a teacher’s quality, is not a usable
poll. It will not have reliability, because the DATA quantity is not enough. But if the number of students polled
were 40, 100, or 1000, we can begin to use the data, since the error from the group size is acceptable according
to desired standards. Since maximum variance or error occurs when YES is 0.5, using YES as 0.5 is the safest, and
most conservative standard for maximum accuracy of polls.
Assigning YES = 0.5 to a group guess means we are assuming total randomness, or 0 bias, or zero knowledge or
intelligence in the group. This is the highest standard for group size requirement, and is the most demanding
in requirement, because it forces the poll to require the highest minimum of group size. Example: SE = √V/√n
= SD/√n
If we want the contribution of the Group Size Error n to a be limited to a maximum of 3%, and,
Assume the most conservative assumptions that subtracts completely the group size error interval:
P = 0.5 for safest poll or guess assumptions in group size error calculations. We do not favor a
person or a group over the other, or assume it more intelligent than the other, and that every
poll is independent of previous polls.
n=( SD/ 3%)^2 p=0.5
SD = √(0.5 x 0.5) =0.5 n
= 278 persons
Therefore, a congress or a group involved in a poll needs a minimum size of 278 persons to ensure that the error
contribution from Group Size factor is no more 3%. This Group Size Error acts as an upper bound on accuracy of a
group. In other words, the maximum accuracy of a poll by 278 people is GSA =(100% – 3%)=97%. This happens if
the other factors involved do not lower the accuracy further.

A Probability Point Stretched To A Distribution
We saw above how a probability point stretches to become an interval, based on the group size. This
interval is a linear probability interval containing YES and it is computed using the SE formula, so that
YES as a single value is replaced with YES ± SE to become an interval.
This linear probability interval on the x-axis, also represents a set of data above it, a distribution of
numbers typically centered around YES, that may be a Bernoulli Distribution, a Binomial Distribution, a
Normal Distribution or a Chi Distribution, etc. Same for NO. Therefore: YES + NO = 1 can be looked at as
a sum of two dependent probability distributions. We do not need to know all the details of the
distribution, but need to be acquainted in general with it. (See graph bellow).
Example : YES = 0.5 implies a proportion distribution where the graph of YES looks like a bell or quadratic
curve. Similarly for NO. At YES = NO = 0.5, the distributions are identical and on top of each other.

Example: a fair coin has YES =0.5, NO = 0.5. Therefore, in an experiment of repeatedly tossing 100 coins
at the same time, each toss is most likely to produce 50 Heads and 50 Tails, but other possibilities can
occur, with decreasing likelihood of more Heads than Tails etc.
Graph of YES or NO with YES = 50%.
Note that at YES =0.6 for Heads, the chance is lower. Lower for YES = 0.7, etc. Chance of the 100 coins
repeatedly all landing Head (YES is 1 or 100%) is zero.

Exercise: must do exercise and project: 1. Get 100 fair coins, cents or centos etc., and toss 35 times.
Record the number of heads in each toss. Place the data in a horizontal table with two rows: bottom
row has 100 cells with values 1 to 100, and the top row has the number of tosses that had the amount in
the bottom row. Example: if 2 tosses had 98 heads, then put the value 2 above the bottom cell that has
98 as a value, as illustrated bellow:
2
1 2 3 … …

98 99 100

2. Draw the table as a graph with the bottom row as x-axis labeled “number of heads”, and top row as yaxis labeled “number of tosses that had this many heads”
3. Does the graph look like normal distribution graph? Is it almost centered around the 50 heads points
on the x-axis?
4. If we call each single 100 coins toss “an experiment”, and the 100 coins are called the sample size, what
is the mean of first single experiment (the first toss) on the x-axis? And second toss mean, and third
toss mean? Hint: It should simply be the corresponding point on the x-axis.
5. Redraw the graph dividing the x and y axis scale by 100. Since the x-axis is the mean, we use the mean
as probability, and give the x-axis YES label. Do the same for the y-axis and give it the frequency label,
and convert frequency numbers to percentages. Now the graph represents the frequency or density of
the probability 0.6 . Here is the table sample:

2/35
0.01 0.02 0.03 … … … 0.98 0.99 1

6. The graph will represent the probability versus frequency of occurrence or density of occurrence.

Observations:
Notice that the probability of heads in a 100 fair coins toss is 50%. This is the most likely to happen. The
frequency or density of YES = 60% is less, and for YES =70% is less, etc. Therefore, even if YES is 50% for
a coin landing head, it is normal to have other values, and these values form a normal distribution. What
is important to notice is that the frequency of these probabilities is decreasing as we move away from
YES = 0.5.

Note: It is important not to proceed further in the text without a complete understanding of the normal
distribution graph and what it represents as a coin tossing experiment. Your understanding of voting
mathematics will depend on this understanding.
Example: Congress in a poll on a proposed law gets 50% YES support? Is this significant? If the poll
results were instead 55% or 60%, would it be significant? Visually looking at the Distribution graph
above, at what point does the randomness become low, and therefore, the results may become
acceptable?
Answer: Getting 50% YES support in a poll, has 0% significance, because it is a purely random result,
because YES = 0.5, the point of Perfect Randomness. AT 0.5, the Normal Distribution Density graph has
its highest randomness point value, and at 0.55 has an equally high randomness value. This is from a
Normal Distribution point of View.
From a horizontal linear interval perspective, getting 55% YES support is not very far from the 0.5 point,
so it is not a significantly different result.
Same for the 60% YES poll result.
Visually looking at the distribution graph, at YES = 90% seem to be a point with low randomness value,
and the corresponding accuracy may be come acceptable.
Exercise: Using our Accuracy equation A. Calculate the complement of A’ equation, which represents the
Inaccuracy equation or the Randomness equation of YES. What is A – A’= ? What does this graph
represent? What is the A – A’ graph intersection point with the x-axis of YES? Draw A and A’, and the
point of intersection. What does this point of intersection represent?

The Political Error
The Political Error I define as an error that effects an entire population, instead of a single individual.
In private life, when a person buys a product, he buys it freely without coercion, to consume it. If there is error in
the product, the damage can be assumed to effect only the consumer of this product, even though
generalizations can be made to expend the damage to a radius of people, such as the enter family suffering from
an oven that had error in its design, and it exploded causing damage to the entire household members. The
damage amount, or the cost of damage can vary. But nonetheless, the damage can be assumed to be limited to 1
consumer for our analysis, and that this consumer, and others, are likely to stop buying the product, if they so
choose. And this should stop the damage or control it according to how much each consumer wants risk in his
life. Therefore a Consumer Error can be assumed to be limited to 1 person, or Private error population size = 1.
Private Error = Error percent in one product × 1 person
Example: Buying a car with error in design, and this error effects 1 car in 100 cars. Therefore the product error
rate is 1%.
But in politics, and its basic instrument, the law, a law is a product that is not freely consumed. It is a product
forced on the entire population whether they approve of it or not. Therefore, a law, as a forced product on the
entire population of a country, province or city, etc. Let us limit ourselves to laws at the country level. Therefore I
define:
Political Error = Error rate per person of one law ( or decision) × population size
Several countries have populations of more than 100 million persons. Some have populations near a billion. So if
a law has in it a design error rate of 1% or 1 in one hundred persons will suffer wrongly from this law, then the
Political Error is = 1% × population size. Therefore, a Political Error is an error that is magnified dramatically. If we
assume a population size of 100 million persons, then this 1% error is magnified 100 million times.
Political Error = 1% × 100 million = 0.01 × 100,000,000= 10^(-2) × 10^(8)= 10^(6) = 1,000,000 persons effected.
1% error translated to 1 million full errors, or 1 million persons likely to suffer the full effect of the design error.
Example: Congress produces a new law mandating death penalty for an act. If a trial is held, and a witness lied (or
whatever) and wrongly caused a conviction, this results in the death of an innocent person because of this law. If
we call this person a victim, in effect, killed by the law makers and/or the court, then the number of victims of
this law expected to be wrongly killed by congress over the lifetime of a person, maybe 60 years, is 10 million
victims.
If the cost of one error can be \$1 or \$1 million, what is the cost of this law? And the second question is: is it
possible to compute the costs of social values, and personal damage. Will you accept \$1 million as a price for
your life and willingly let someone kill you? Will the cost of loss of part of your freedom be calculable?
Therefore, while we can compute the number of victims from the error in a law, it is extremely if not impossible
to do it for the cost of this error.
Since a 1% error means 99% accuracy in a law, it becomes an error magnified one hundred million or billion times
in a large country, where this error can become 10^(7), or 10 million percent. The flip side of this is that accuracy
is decreased 10 million times from 99% accuracy to 0.0000001% accuracy. This is 10 million times less accuracy.
This means that if congress adopts laws with 99% accuracy, then to achive this accuracy level for the entire
population, the
Required Political Accuracy = Accuracy rate per person × population size
needs to become 99% × 100,000,000 persons, to become at the level of 0.9999999999% accuracy.
This level of accuracy is the reserve of fields such as detailed engineering or advanced physics. While the cost of a
failed advanced space machine can be calculated or limited, the cost of a single law can be much greater.
If the requirements of accuracy from a scientific field are determined by the cost of error, the field of political
science comes first among all.
From this we can see the need for highest accuracy standards and the needed mathematical education for any
student contemplating the study of political science or law.
In the early days of formulating probability theory, with Bernoulli, De Moire, Newton, La Place and others, the
field was called by some “ The Study Of Errors”. It seems like a fitting title for the study of probability theory and
now for the subject of politics and law.
A single voter is equivalent to a coin

When a regular person from the public is voting in a poll, we can observe that: the voters disagree,
regardless of the voters different attributes, the candidate or the issues that is the focus of the poll.
Since two opposing political groups vote in opposition to each other, we have to assume equal selection
ability by each voter, and that two opposing members of the political parties have equal chance of
being correct in their choice. This forces YES for each of them to be 0.5.
Similarly, for law makers voting in a poll on a proposed law.
Similarly for a jury of judges in a court or supreme court.
No matter your education or expertise, when voting in a group, all the group members are assumed to
be equally likely to make the correct selection.
This is one of the most important principles in voting.
Principal: members of a voting group are assumed to be equally likely to make the
correct selection. This means the probability of a member voting In Favor or Yes is 0.5
universally, for any single-event unrepeatable experiment. This is useful in the
calculation of Group Size Accuracy when there is only a single poll taken. If the
experiment is repeatable, YES can be adjusted to reflect this based on the
experiments sample size.
Therefore:
Principal: The voter probability Yes is determined after the poll is complete, and is the same as a coin
toss experiment.
If instead of saying “the voter is asked to select Yes, or No” we say for the purpose of making this easy to
remember:
Tossing a fair coin with Yes, No sides is the same as tossing a voter with a Yes, No
sides
Principal: tossing a voter is the same experiment as tossing a coin, and the sample mean is equal to the
group population mean (since the sample is the entire population of the group) and is assumed to be
the true probability.
Example: if tossing 100 voters gives results of 80% Yes, this is the poll percentage of Yes and the
probability YES.
Therefore, all the mathematics of a coin tossing is transferable to our constraint
condition voting analysis

Since it is established that a coin toss experiment is a Bernoulli Trial, satisfying its conditions, then we
can conclude that our voting experiment is also a Bernoulli Trial.
I can further conclude:
An experiment of tossing a single coin 1000 times, is equivalent to an experiment of
tossing 100 identical coins ten times, or tossing 10 identical coins 100 times
The percentage of Yes of any of these different experiments is expected to be the same.
Similarly for voters.
A poll is defined as a group voting on a single issue, with only Yes, No choices. And, Yes% + No% =
100%
I chose to use word poll, because using the word vote, as is common now, is not a well-defined word
since it is used for singular and plural. Example : saying “ there is a vote” means a poll, and “my vote was
Yes” could mean a personal vote being Yes.
Poll or Group Guess requirements

The difference between polling voters and polling scientists.
We established that a voter in any poll is presumed to have a Yes voting probability of 50%.
A poll is:
1. A purely statistical voter survey method used to answer a question. (The resulting answer is not derived
analytically using the established standards of mathematical.)
2. The resulting answer is never 100% certain
3. A poll question cannot be used to establish the correctness of a specific scientific equation. The correctness
of an equation is established by using scientific methods, and not by polls.
If a poll is done with voters all being scientists, testing their knowledge of a common equation, then we
should expect the poll results to be nearly 100% Yes or 0% Yes, (meaning they know the equation or
how to use it). Because, the correctness or incorrectness of an equation, is already established prior to
the poll, and there should be no deviation in voters choice.
From this we can conclude the principal that: When a poll results for a question are 100% Yes, it means
this is the best guess this group can produce. This poll answer may still be 100% wrong if put to a
scientific testing standard, since a poll is not a scientific standard for measuring correctness.
4. A poll result is a guess, composed from many guesses.
Warning
To remind educated and uneducated voters that a poll is a guess, it is helpful to give a poll a common
use name that reminds of this fact. For example, governments, groups, etc. can use the word “group
guess” or something similar in their official language .Instead of people saying “we have a poll today on
this person being the correct choice for office”, people would say “we have a group guess today on this
person being the correct choice for office”. Or instead of people saying “we have a poll on a proposed
advisory , people would say “we have a group guess on a proposed advisory”.
Worth noting that personal or group interests may create strong opposition to use of such accurate
language because inevitably it reminds people that their field of work or source of authority is inherently
based on guessing. In fields such as politics and law, this can be very damaging to such interests. However,
proper use of language is critically important for proper thinking.
There is also need for terms that clearly specify if the answer to a question is “scientifically correct”, that
is, derived by scientific method, or the answer is a guess. Since saying the poll results are 100% Yes, the
100% word may immediately conjure in the mind of uneducated people, that the chosen answer to the
question in the poll is “correct”, even if a poll cannot establish correctness. Therefore, a language should
have two words, a single word to denote “established to be correct using scientific analytical methods”,
such as the existing English word “correct” and the other to denote “seems correct using empirical
guessing methods” similar to “guessed correct” or “circumstantially established” or “group guess” .
Common language should reflect the proper underlying scientific language.

Example: the group guess results were 100% Yes.

Definition: A coin toss, or in short a toss, is tossing a coin with two sides labeled Yes and No and noting
the landed side . With the words Yes and No capitalized to denote a choice, and YES, NO, all letters
capitalized to denote the percentage of Yes or No in a series of tosses or a single toss of many coins.
Definition: A vote toss or opinion toss, or in short a toss, is the equivalent of a coin toss.
Definition: A poll is a group toss, or simply tosses.
These language terms are meant to serve in connecting mathematical probability language to common
language. The other possible words that also come to mind are True and False, correct and incorrect or
error, or right and wrong.
Definition: A fair coin has the probability of Yes = No = ½, and the ½ is the target number or expectation.
Definition: A fair person has the probability of Yes = 1, and the 1 is the target number or expectation.
Definition: Intelligence is what is left after removing randomness and error?.
Perfect intelligence has YES equal to one.

Perfect randomness has YES equal to 1/2.
Definition: The intelligence interval starts at perfect randomness and ends at perfect intelligence and is
[0.5 , 1] . At YES = 0.5, at perfect randomness, intelligence is 0, and at YES = 1, intelligence = 1.
The words randomness and bias are used for non-thinking objects. The words intelligence, no
intelligence or zero intelligence, or randomness are used for thinking objects. Therefore:
Intelligence = 1 – randomness

Mathematical note: I define the symbol ~ placed on top of a variable, to imply a distribution (of
numbers as a random variable). Then:
intelligence~ = 1~ – perfect randomness~

A Mental Experiment In Guessing: Humans
Versus Coin Machines

Intelligence Versus Randomness or Dumbness : Unrepeatable Experiment

If we have a closed room with two groups. One group is made from 100 coin tossing machines. The coins
are fair, with YES = 0.5 and therefore they represent Perfect Randomness.
We make a statement and ask if the statement is true, with two reply option: Yes, No. We request only
one of the two groups return the answer, but we do not know which group will be answering.
The answer is written on a small glass window as percentage of Yes of the group.
How do we know which group answered?
If the answer returned had a YES near 50%, then it is likely the coin flipping machines answered, or the
question was not solvable by the human group (maybe too difficult or its subject is unknown to them).
If YES of the answer was near 100%, then we can say it is much more likely that the answer came from
the human group.
A coin machine has zero knowledge and zero intelligence, but it randomly produces 50%
correct answers to a question or an exam. So a human being getting 50% correct answers on an
exam, or answering questions, or a political candidate receiving 50% support in a poll, or
congress getting 50% support on a proposed law is a demonstration of zero knowledge and zero
intelligence. We already established that the voting scale begins at 50% and is 0% at that point,
now we see that the same applies for correctness or intelligence.
The other related question is: how is getting 51% correctness or support different from 50%. We
have learned that there is practically mathematically none.
If we increase the groups size to 10,000 adult humans versus 10000 coin tossing machines, then when
YES is near 50%, we can conclude that the answer came from the coin tossing machine or we can
conclude that the voters have zero intelligence about the subject of the question. If YES was near 100%,
we can conclude that it is likely the voters have perfect intelligence about the issue in the poll.

The implication of these results are critical to understand, because it represents the
essence of the voting problem in politics and its mathematics
In politics or law etc. when a poll, a group guess, is talking about a political candidate, a proposed law, a
court verdict, often, the poll is done only a single time on the issue involved. Therefore, no matter the
answer, the only conclusion we can make is that the answer is likely random or likely intelligent, based
on how near it is to 50% or 100%.

Intelligence Versus Randomness or Dumbness: Repeatable But Not Identical
Experiment

Next, we look at an experiment where we can ask many questions
The coin machine versus the student
If in the experiment above we put in the room one student and one coin tossing machine.
As an exam to test the student, we ask one hundred exam questions. Only the machine or the student
are allowed to answer all of them.
When the answers YES results are delivered, how can you tell if the answers came from the coin tossing
machine or the student?
If Yes was near 50%, we can say it likely the coin machine answered or if it was the student, then the
student likely does not know his subject.
The implication of these results are critical to understand, because it represents the
essence of the problem in grading a test exam and its mathematics.

Next, if we have a chance, to ask more questions, then as the number of questions increase, then we can
become more and more sure of the knowledge or ability level of the student. After 1000 or 10000
questions, we can become certain of the results YES and what it implies. If after 10000 questions, the
YES of answers is near 50%, then we say the answer came from the coin machine or a student without
knowledge or ability. Which puts the student at the same level of knowledge or ability as a coin tossing
machine. In short we can say his intelligence in the subject being tested is 0.
If the YES results were near 100%, we can say the answer came from the student and the student has
perfect intelligence as it relates to the subject of the exam.
Note: we can see how changing the number of questions (experiment) and the number of people can
effect confidence in the results, as per the Group Size Accuracy

Combining the two experiments to establish with confidence the intelligence level of a group
If we modify the first experiment to have 100, 1000 or 10000 questions then we would be able to make
with greater confidence how intelligent this group is. For example: looking at 1000 previous votes of
congress, we can make a statement about their level of intelligence. Same for looking at the voting