Monday, September 19, 2016

The Generating New Probability Theorems

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)) = 54.1667%

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 54. 1667%

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 applications?

Here I have brought an example of financial management.
Let me again refer you to my article of "A Template for Financial Section of a Business Plan (Con)"
posted on link:
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 year 3.
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 assumptions?
For deducting this risks, we have to generate other theorems by increasing the number of dices or changing the functions.