We have fair balls and one edge.

 

Which side the first ball will fall down? To the left (0) or to the right (I)?
Only if the center of every ball is exactly placed on the edge.
We only know a probability. What is the probability?
It is the opposite of the certainty. We only know some possibilities.
It is our certainty! Certainty is equal to
a sum of allowable possibilities.
Possibility is the state which could be realized. The ball will not stop
on the edge - for example. The ball ha
s one certainty
- to fall down to the left or to the right. That´s all.

The probability P is the ratio of one possibility (n) to all possibilities (A).
The possibilities are independent to each other - in that case.

The equation for the probability:

In our case with balls we have two possibilities – the left side (0)
and the right side (I). The sum of possibilities
A is equal to number 2.
But only one side have to fall down. Then n is equal to number 1 .
With our equation we obtain:

In our case the probability P is equal to the number ½ (one over two).

We have 50% chance to two possibilities - the ball will fall to the left (0)
or to the right (I). In the beggining whether the ball is placed on the edge
we only know the chance of two possibilities. After the falling of the ball
we know the certainty which possibility had realized. We could write down
on the paper the progression of numbers 0 or I. Theese numbers are
random numbers. We don´t know which number follow.

Summary: Before the experiment (the fall down) we know the certainty
of the ball falling down. But we don´t know exactly the side. We only know
the two possibilities (two ways). The certainty is divided to two possibilities.
After the experiment we know which possibility become real.

If we take N balls we could write down on the paper the progression
of numbers 0 or I.

For example we can have such progression

II0I0III0I00I0II00III0I

We have an interesting question. Is it possible to have such progression
as IIIIIIIIIIIII, or IIIII0000 or IOIOIOIOIOIOIOIOI or
other else.
And our answer? Yes,
all progressions are possible with fair ball.
But with different probability.

Imagine the progression IIIII – fifth times ball had to fall to the
right side. Is it fair? How many possibilities are there then?
See the table below

00000 00I00 0I000 0II00 I0000 I0I00 II000 III00
0000I 00I0I 0I00I 0II0I I000I I0I0I II00I III0I
000I0 00II0 0I0I0 0III0 I00I0 I0II0 II0I0 IIII0
000II 00III 0I0II 0IIII I00II I0III II0II IIIII

The sum of possibilities is equal to number 32. The certainty is a set
that has 32 subsets. Every subset has 4 members. Every subset is different,
unique. In the other hand, every ball has its own path arround the edge.
Let´s say the turning or the motion of every ball is unique.
In the result all different paths were over in two sides.

We see a set with subsets below

C is the set of certainty and P1 to P32 are subsets of possibilities.


The probability of our progression (IIIII) is 1 over 32. We have to let fall
down the ball 128 times. 32 times 4 is 128. In that serie we could find
out our progression. Not only to find out. This progression (IIIII or 00000)
must be in such progression of 128 balls fallen down. If we expect fair
conditions of our experiment.
We see from the table

00000 00I00 0I000 0II00 I0000 I0I00 II000 III00
0000I 00I0I 0I00I 0II0I I000I I0I0I II00I III0I
000I0 00II0 0I0I0 0III0 I00I0 I0II0 II0I0 IIII0
000II 00III 0I0II 0IIII I00II I0III II0II IIIII

there are some progression (e.g. with two or three I) often repeated
- green colour. The progresion with two and three I or 0 is more frequently
then the progression only with one I or one 0 - yellow colour.
The progression without I or 0 has only two members - red colour.

If we get a progression IIIIIIIIIIIIIIIIIIIIIIII.....IIIIIIII. What it means.
The condition are "unfair" or tip the edge.

If the the position of the edge to every ball is alike :

a) then the ball always will fall down to the left (0)
b) then the ball always will fall down to the right (I)
c) then the ball mostly will fall down to the left (O)
d) theb the ball mostly will fall down to the right (I)

Only when the edge is placed exactly in the center of every ball
we only know the 50% chance to each side .
But we put a candle to the right side of the edge and suddenly
the balls mostly will down to the right - as d). The reason is the
thermal expansivity - the edge is deformed to the left from
the center of every ball.

How we recognize fair conditions from "unfair" then? Or how we recognize
a change of conditions during our experiment? We can use a progression
0,I,0,I,0,I, ... . There are not random numbers. We know what number will
follow. In the other hand a distribution of such progression would be the same
with pure random numbers. That progression would be a "basic unit" for us.
We can measure the distribution of random progression or other progression:
E.g. the progress of number π or other numbers as e or √2.

to be continued


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