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

Question

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 7^th grade and ‘card counting’ of all things by 8^th 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

And

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.

Answer 1

Hey, I get what you're going for—applying the empirical rule, that 68-95-99.7 thing, to your gambling system to figure out if your results are way beyond random chance. It's cool that you're diving into this at a young age, especially with the calc and card counting background from your dad. The goal here is to help you plug in the right numbers for X, your observation, μ, the mean under no-edge assumption, and σ, standard deviation, so you can check if you're within or outside those 3 SDs for near certainty.

Just to set the stage, this rule is for normally distributed data or stuff that approximates it with enough trials, thanks to the central limit theorem. It says about 68% of results fall within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. If your result is outside ±3 SD, there's only a ~0.3% shot it's random—pretty strong evidence your formulas are doing something real. You mentioned those formulas like Pr(μ-2σ ≤ X ≤ μ + 2σ) ≈ 0.9545 and the 3σ one at 0.9973; yeah, that's exactly it. For five-sigma, super high confidence like 99.99994%, you'd aim for ±5 SD, which is particle physics level but awesome for your big datasets.

X is basically your score or result from all those manual calculations—the thing you're measuring to test against the null hypothesis that the game is negative EV and your success is just luck. Best choice is average net retention per cycle—that $18 you keep on average per $100 win/loss cycle, so X = 18, or 0.18 if you're thinking in percentages. Why this? It normalizes for how many cycles you've done, sounds like around 311 to hit 57x growth from $100 to $5700. It's stable and great for stats tests. Alternative could be total profit, X = 5600 your net gain, or the multiplier, X = 57 how much you grew your bankroll. Stick with the average retention if you can—it avoids issues with scaling as you add more cycles. Your current 1.724% calc, 1 / (1 + 57), is a rough heuristic, but it's not stats-backed; this empirical rule setup is better.

μ is what you'd expect on average if your formulas had zero edge—just the game's built-in negative expectation grinding away. For average net per cycle, μ = the game's expected value EV per cycle. In a negative EV game, this is negative. Example, if it's roulette with a -5.26% edge on a $100 bet, μ ≈ -5.26. Or for sports betting, maybe -2% to -5% depending on the vig. You gotta calculate this from the game's rules, probabilities, payouts. Without details, assume it's something like -2, losing $2 per $100 cycle on average. For total profit, μ = number of cycles × EV per cycle, say 311 cycles at -2 EV: μ = -622, you'd expect to be down $622. For multiplier, μ ≈ (1 + EV proportion)^cycles, with negative EV, this trends toward 0 or bankruptcy over time. Key, base μ on the game without your formulas. If your system picks bets smarter, the null is random betting EV.

σ is how much variability or swing there is in the results under that no-edge scenario. Gambling has high variance, so this matters. For average net per cycle, first get σ_cycle, SD for one cycle. Formula, sqrt[ p*(win - EV)^2 + (1-p)*(loss - EV)^2 ], where p is win prob. Example, if win $100 with p=0.45, lose $82 with 0.55, and EV=-2, σ_cycle might be around 50, guessing based on typical games. Then, for your average over k cycles, use standard error: σ = σ_cycle / sqrt(k). For k=311, that's 50 / sqrt(311) ≈ 2.83. For total profit, σ = σ_cycle * sqrt(k) ≈ 50 * sqrt(311) ≈ 883. For multiplier, a bit trickier—might need logs for compounding, but approximate like total profit. You need the game's win/loss probs and amounts to compute this exactly. If it's historical data, backtest random bets to estimate variance.

Once you have X, μ, and σ, calculate the z-score: z = (X - μ) / σ. This shows how many SDs away you are. If |z| ≤ 3, your result is within 3 SD—could still be luck, 99.7% of random outcomes here. If z > 3, since your X is positive, it's outlier territory—<0.3% random chance, leaning toward not random. Example using averages, plug your real nums: X = 18, μ = -2, σ = 2.83, z = (18 - (-2)) / 2.83 ≈ 7, way past 3 SD, p tiny. For total: z ≈ (5600 - (-622)) / 883 ≈ 7 too. If you hit 5+ SD with billions of sims, that's your five-sigma discovery.

This works best with lots of independent cycles, your 311 is okay, billions via code = gold. If not normal, simulate distributions instead. Rejecting null, no edge, doesn't 100% prove your system—could be fluke or overfitting. Test fresh data.

Python all the way—easy for math whizzes like you. Use NumPy for stats, loop simulations: def one_cycle(): [simulate win/loss], then run a million times, get mean/SD/z. Start with free tutorials; it'll save your arm! Paper wins are great, but real money has risks, emotions, fees. Those beaten games you mentioned usually flip to positive EV subsets—hope yours does that mathematically. If you share rough game details, like EV estimate or win prob, I can tweak examples more. Keep grinding—sounds like you're onto something fun!

Answer 2

Hi Robert,

Not sure if you are looking for a simulator type program, but Matlab might be the way to go. I asked my son for a recommendation since he is a programmer.

Regarding statistics, the empirical rule is based on a normal probability distribution. Not all data is normal. However, if you sum or add at least 30 values of any data distribution set, that sum or average will be approximately normal and you can use the imperial rule. This is known as the central limit theorem.

There are plenty of other probability distributions out there: uniform, binomial, poisson, exponential, etc plus all those "shapes" that do not have a pattern or formula. Regardless, you can find probabilities associated with values of x or a range of values for x (either using integration or distribution tables). You do not need to restrict yourself to the normal distribution aka "bell-curve".

Probability Distributions are numerical values that follow a pattern. The mean, µ, and standard deviation, σ, are derived algebraically but can also be programmed. Then there is the shape. All three define the distribution (some use a parameter like λ).

Regarding a measure of certainty, you may want to look at confidence intervals; a 5% level of significance is acceptable so don't go crazy trying to get your error down to 0.3%.

I hope by highlighting some key terms will assist you in your endeavors. Good luck... Wish I knew your last name because I'm sure we will be reading about you in the future.

Terri M