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 has 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
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
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
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.