Robert J.

asked • 02/17/20

How to apply the empirical rule (68–95–99.7 rule)

How to apply the empirical rule (68–95–99.7 rule) to the following scenario:

I have spent a significant amount of time trying to do the impossible (i.e., mathematically beat a game with a negative expectation). Yes, I know, mathematicians for hundreds of years have failed at this (but that is only partially true). I can give examples, if need be, where games conventionally thought to be unbeatable have been beaten. My formulas do not involve the simplistic use of input/variable change the way let’s say card counting or other such formulas have been used to win at certain ‘games’ of chance. My dad taught me calculus by 7th grade and ‘card counting’ of all things by 8th grade, so my love of math and ‘beating games of chance’ has been an ongoing love affair over my short life : - )

I have not as yet been able to program/code the formulas to test large data sets, as coding my formulas is proving to be challenging, so I have been performing longhand calculations (very time consuming and tedious). My starting points are: I may only start with a small sum of money and when wagering, I am never increasing the spread between bets (i.e., I am winning by flat betting, but I also created changes to the formulas that allow me to 1.5x, 2x, 4x, etc my flat bet if I wish and still consistently have a positive outcome (i.e., a profit/net gain).

I have now multiplied my ‘random’ hypothetical starting sum 57x which means the odds of this occurring randomly are low: X/(X + (57X)) = 1 / (1 + 57) = 0.01724 (a 1.724%) chance my success is random (assuming I am using that correct algebra to calculate this. I have been told that when I reach 1% or lower (which will take a fair amount of time testing data manually) this is considered by most mathematicians to be ‘near certainty’ that the result I am achieving is ‘not random’. I am current in some areas of math and not current in other areas (i.e., I still have yet to take the courses or teach myself) but I am wondering what variables to plug into the empirical rule to see if I am within 3 standard deviations.

I typically retain 18% of every dollar I win on paper… meaning I will win $100, then lose $82, then win $100 and lose $82, with the retained amount of $18 (as an average) accumulating over time as I get through larger and larger data sets. In other words, the $18 that I retain per win/loss cycle adds to the sum of $ retained (won), and I now have accumulated 57x my random starting sum. E.g. I started with $100 and now have $5700.

What numbers can I plug into 68–95–99.7 rule?

Using this formula:

Pr(μ-2σ ≤ X ≤ μ + 2σ) ≈ 0.9545


Pr(μ-3σ ≤ X ≤ μ + 3σ) ≈ 0.9973

I know where Χ is “an observation from a normally distributed random variable”, but in my case/scenario what is “X”? The number of ‘Sums” [multiples of my starting sum] won? The amount ($) retained (i.e., the net $ retained per win/loss cycle) per $100 of gross winnings? I know it is likely none of these, but what value should I use for X?

μ is the mean of the distribution, and σ is its standard deviation, but in my scenario what numbers do I plug into μ and σ?

My goal in asking this question is to have some method for determining when I am within three standard deviations and therefore my result will be considered "significant and able to qualify as a discovery.

Please do not ask me to share my formulas, as they have significant ‘real world’ value that I plan to apply ‘for public good’. I may be young, but as a kid growing up these days, I am not gullible just because I am young : -) [no insult intended]. The internet has made us aware and savvy in terms of gaining real world skills at a young age. Also, I have nearly lost the use of my writing arm because I have spent so many thousands of hours after school not moving and just writing, that I feel like my left arm is atrophying, so for that reason alone I would not share my formulas. I actually want to use what I have discovered to earn a PhD, but not till I have achieved the level of monetization I seek to apply towards philanthropy, before I share the formulas (if that is required to attain my PhD).

I hope there are some math majors/professors out there who understand statistics as applicable to games of negative expectation and who can help me learn what values to plug into X, μ and σ so I can apply the empirical rule/68–95–99.7 rule to learn when I have attained ‘near certainty’ regarding when my formulas/equations are yielding results that are ‘not random’. I do have a feeling that for the results I am achieving to be considered "significant" I may have to apply the convention of a five-sigma effect (99.99994% confidence) before my results/formulas qualify as a discovery. At the very least, they do currently have real world value. Once I learn how to code my formulas to test data sets in the billions, I can then quickly ascertain where I sit in terms of whether I can achieve the five-sigma effect... (not practical testing/calculating longhand). I am still trying to decide what programming language to learn in order to have the most flexibility in achieving the results I seek. I know there are a lot of similarities in coding languages these days, but I feel one has to be better than the next to assist me in achieving my goals (which in all fairness, I of course have not been able to fully elaborate on in an open forum/environment).

Thank you in advance for any assistance you can provide.

1 Expert Answer


Terri M. answered • 02/22/20

4.9 (195)

I taught Statistics I at Suffolk County Community College for 11 years

Robert J.

Hi Terri... thank you for the reply and kind words. It's funny, all the math professionals and others whose work, theories and research I've studied could only be 'read about' 'after the fact' for the obvious reasons. It's lonely having to be stealth, but a requisite of the field, till the goals are achieved and finally thereafter getting to publish the theorems etc... In any event, thank you so much for the reply, and yes it was helpful, thank you!!


Terri M.

Another avenue for you to explore - take a look at Game Theory and Operations Research. I received my Masters in OR but never figured out how to get an entry level job in that field. So I became an Actuary :)


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