This blog is about new ideas which give us new methods and new theorems as the tools to break complex problems in all fields such as Strategic Management, Engineering, Financial Management and so on and finally to solve these problems in the real world in which there is the balance of the cost and the time.
The purpose of this article is to generate new theorems of
probability and to find out some applications of these theorems. In this case,
suppose that we have a covered basket that contains many dices. In many blind
tests, we will reach in and pull out a dice and set it on the table on one row
from left to right. It is clear, each dice has six events (choices) including
1, 2, 3, 4, 5, and 6.
Theorem (1): Rule of fifty plus (50 %+)
I start this theorem by using three dices. At the first, I pull out
one dice from basket then second dice and finally third dice and set them on
the table on one row from left to right.
The question is: What is the
probability for number of third dice less than or equal to average of numbers
first and second dices?
Let us have the dices as follows:
D1 = first dice
D2 = second dice
D3 = third dice
D1 D2 D3
I am willing to know:
P (D3 ≤ ((D2 + D1) / 2)) =?
Definitely, we have 6^3 = 216 permutations with repetition.
We can calculate the probability equal to 54. 1667%.
P (D3 ≤ ((D2 + D1) / 2)) =
Or, what is the probability for number of third dice more than or
equal to average of numbers first and second dices?
The answer is the same:
P (D3 ≥ ((D2 + D1) / 2)) = 54.1667%
Now, another question is: What is the probability number of third
dice less than or equal to twice number second dice minus first dice?
P (D3 ≤ ((2*D2) - D1)) =?
The calculation shows us that the probability is the same equal to
P (D3 ≤ ((2*D2) - D1)) = 54.1667%
Therefore, we can say:
P (D3 ≤ ((D2 + D1) / 2)) = P (D3 ≥ ((D2 + D1) / 2)) = P (D3 ≤
((2*D2) - D1))
Let me expand this idea as follows:
If we assume all dices contains infinity numbers which are members
of Real Number as follows:
D1 and D2 and D3 are subsets of Real Number
In this case, each dice is included a set of Real Number in which
the numbers of sets will be different, then we will have:
P (D3 ≤ ((2*D2) - D1)) > 50%
I name this theorem: The rule of fifty plus (50 %+)
What are the
Here I have brought an example of financial management.
We can apply this theorem to predict assumptions when we have the
final reports of previous years by probability of more than fifty percent (50
%+) for instance, I utilize this theorem for growth rate of sales as follows:
If we replace the dices by years of sales (Income statement) and
use from Monte Carlo method to iterate calculations, we will reach to the rule
of 50 %+. (Please see below pic)
On above spreadsheet, we have sales for year 3, 4 and 5. Then I use
different probability distribution and different growth rate for CUT OFF. By
one way data table, you can see an average of P (x) is equal to 53.9% by
standard deviation of 0.000499.
P (x) = P (year5 ≤
((2*year4) – year3)) = 53.9%
It means that there is a probability more than 50% in which sales
in year 5 will be less than or equal to twice sales in year 4 minus sales in
You can check this theorem (50 %+) for all income statements,
annual reports and so on.
Even though we have found out this theorem, there is still more
than 40 % risk to use this theorem. But, how can we decrease the risk of
projection for assumptions or increase the probability prediction of our
For deducting this risks, we have to generate other theorems by
increasing the number of dices or changing the functions.