Mathematics Of Gambling The Kelly Formula

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The math underlying odds and gambling can help determine whether a wager is worth pursuing. The first thing to understand is that there are three distinct types of odds: factional, decimal,. Presenting the most astonishing formula in gambling mathematics, probability theory at large, widely known now as FFG. Indeed, it is the most essential formula of theory of probability. Indeed, it is the most essential formula of theory of probability.


By Ion Saliu,
Founder of Gambling Mathematics, Founder of Probability Theory of Life

I. Theory of Probability Leading to Fundamental Formula of Gambling (FFG)
II. Fundamental Table of Gambling (FTG)
III. Fundamental Formula of Gambling: Games Other Than Coin Tossing
IV. Ion Saliu's Paradox or Problem of N Trials in Gambling Theory
V. Practical Dimension of Fundamental Formula of Gambling
VI. Resources in Theory of Probability, Mathematics, Statistics, Software

The final version published in December 1997; first capture by the WayBack Machine (web.archive.org) April 17, 2000.
  • Presenting the most astonishing formula in gambling mathematics, probability theory at large, widely known now as FFG. Indeed, it is the most essential formula of theory of probability. This formula was directly derived from the most fundamental formula of probability: Number of favorable cases, n, over Total possible cases, N: n / N. Abraham de Moivre, a French/English-refugee mathematician and philosopher discovered the first steps of this formula that explains the Universe the best. I believe Monsieur de Moivre was frightened by the implications of finalizing such formula would have led to: The absurdity of the concept of God. I did finalize the formula, for the risks in my lifetime pale by comparison to the eighteenth century. God, no doubt, represents the limit of mathematical absurdity, therefore of all Absurdity.

    And thusly we discovered here the much-feared mathematical concept of Degree of Certainty, DC. I introduced the DC concept in the year of grace 1997, or 1997+1 years after tribunicia potestas were granted to Octavianus Augustus (the point in time humans started the year count of Common Era, still in use). The Internet search on Degree of Certainty, DC yielded one and only one result in 1998: This very Web page (zero results in 1997, for DC was introduced in December of that glorious year, with some beautiful snowy days… just before the Global Warming debate started…) For we shall always be mindful that nothing comes in absolute certainty; everything comes in degrees of certainty — Never zero, Never absolutely. “Never say never; never say forever!”

    • The degree of certainty DC rises exponentially with the increase in the number of trials N while the probability p is always the same or constant.
    • DC = 1 – (1 – p) ^ N
    • Simultaneously, the opposite event, the losing chance, decreases exponentially with an increase in the number of trials. That's the fundamental reason why the infamous gambler's fallacy is an obvious absurdity.

1. Theory of Probability Leading to the Fundamental Formula of Gambling

It has become common sense the belief that persistence leads to success. It might be true for some life situations, sometimes. It is never true, however, for gambling and games of chance in general. Actually, in gambling persistence leads to inevitable bankruptcy. I can prove this universal truth mathematically. I will not describe the entire scientific process, since it is rather complicated for all readers but a few. The algorithm consists of four phases: win N consecutive draws (trials); lose N consecutive trials; not to lose N consecutive draws; win within N consecutive trials.

• I will simplify the discourse to its essentials. You may want to know the detailed procedure leading to this numerical relation. Read: Mathematics of the Fundamental Formula of Gambling (FFG).
•• Visit the software download site (in the footer of this page) to download SuperFormula; the extraordinary software automatically does all FFG calculations, plus several important statistics and probability functions.

The probability and statistical program allows you to calculate the number of trials N for any degree of certainty DC. Plus, you can also calculate the very important binomial distribution formula (BDF) and binomial standard deviation (BSD), plus dozens of statistics and probability functions.

Let's suppose I play the 3-digit lottery game (pick 3). The game has a total of 1,000 combinations. Thus, any particular pick-3 combination has a probability of 1 in 1,000 (we write it 1/1,000). I also mention that all combinations have an equal probability of appearance. Also important - and contrary to common belief — the past draws do count in any game of chance. Pascal demonstrated that truth hundreds of years ago.

Evidently, the same-lotto-game combinations have an equal probability, p — always the same — but they appear with different statistical frequencies. Standard deviation plays an essential role in random events. The Everything, that is; for everything is random. Most people don't comprehend the concept of all-encompassing randomness because phenomena vary in the particular probability, p, and specific degree of certainty, DC, directly influenced by the number of trials, N. Please read an important article here: Combination 1 2 3 4 5 6: Probability and Reality. A 6-number lotto combination such as 1 2 3 4 5 6 should have appeared by now at least once, considering all the drawings in all lotto-6 games ever played in the world. It hasn't come out and will not appear in my lifetime... I bet on it... even if I live 100 years after 2060, when Isaac Newton calculated that the world would end based on his mathematical interpretation of the Bible! (Newton and Einstein belong to the special class of the most intelligent mystics in human and natural history.) Instead, other lotto combinations, with a more natural standard devi(l)ation (yes, deviation), will repeat in the same frame of time.

As soon as I choose a combination to play (for example 2-1-4) I can't avoid asking myself: 'Self, how many drawings do I have to play so that there is a 99.9% degree of certainty my combination of 1/1,000 probability will come out?'

My question dealt with three elements:
• degree of certainty that an event will appear, symbolized by DC
• probability of the event, symbolized by p
• number of trials (events), symbolized by N

I was able to answer such a question and quantify it in a mathematical expression (logarithmic) I named the Fundamental Formula of Gambling (FFG):

The Fundamental Formula of Gambling is an historic discovery in theory of probability, theory of games, and gambling mathematics. The formula offers an incredibly real and practical correlation with gambling phenomena. As a matter of fact, FFG is applicable to any sort of highly randomized events: lottery, roulette, blackjack, horse racing, sports betting, even stock trading. By contrast, what they call theory of games is a form of vague mathematics: The formulae are barely vaguely correlated with real life.

2. The Fundamental Table of Gambling (FTG)

Substituting DC and p with various values, the formula leads to the following, very meaningful and useful table. You may want to keep it handy and consult it especially when you want to bet big (as in a casino).

Let's try to make sense of those numbers. The easiest to understand are the numbers in the column under the heading p=1/2. It analyzes the coin tossing game of chance. There are 2 events in the game: heads and tails. Thus, the individual probability for either event is p = 1/2. Look at the row 50%: it has the number 1 in it. It means that it takes 1 event (coin toss, that is) in order to have a 50-50 chance (or degree of certainty of 50%) that either heads or tails will come out. More explicitly, suppose I bet on heads. My chance is 50% that heads will appear in the 1st coin toss. The chance or degree of certainty increases to 99.9% that heads will come out within 10 tosses!

Even this easiest of the games of chance can lead to sizable losses. Suppose I bet $2 before the first toss. There is a 50% chance that I will lose. Next, I bet $4 in order to recuperate my previous loss and gain $2. Next, I bet $8 to recuperate my previous loss and gain $2. I might have to go all the way to the 9th toss to have a 99.9% chance that, finally, heads came out! Since I bet $2 and doubling up to the 9th toss, two to the power of 9 is 512. Therefore, I needed $512 to make sure that I am very, very close to certainty (99.9%) that heads will show up and I win . . . $2!

Very encouraging, isn't it? Actually, it could be even worse: It might take 10 or 11 tosses until heads appear! This dangerous form of betting is called a Martingale system. You must know how to do it — study this book thoroughly and grasp the new essential concepts: Number of trials N and especially the Degree of Certainty DC (in addition to the probability p).

Most people still confuse probability for degree of certainty...or vice versa. Probability in itself is an abstract, lifeless concept. Probability comes to life as soon as we conduct at least one trial. The probability and degree of certainty are equal for one and only one trial (just the first one...ever!) After that quasi-impossible event (for coin tossing has never been stopped after one flip by any authority), the degree of certainty, DC, rises with the increase in the number of trials, N, while the probability, p, always stays constant. No one can add faces to the coin or subtract faces from the die, for sure and undeniably. But each and every one of us can increase the chance of getting heads (or tails) by tossing the coin again and again (repeat of the trial).

Normally, though, you will see that heads (or tails) will appear at least once every 3 or 4 tosses (the DC is 90% to 95%). Nevertheless, this game is too easy for any player with a few thousand dollars to spare. Accordingly, no casino in the world would implement such a game. Any casino would be a guaranteed loser in a matter of months! They need what is known as house edge or percentage advantage. This factor translates to longer losing streaks for the player, in addition to more wins for the house! Also, the casinos set limits on maximum bets: the players are not allowed to double up indefinitely.

A few more words on the house advantage (HA). The worst type of gambling for the player is conducted by state lotteries. In the digit lotteries, the state commissions enjoy typically an extraordinary 50% house edge!!! That's almost 10 times worse than the American roulette -- considered by many a suckers' game! (But they don't know there is more to the picture than meets the eye!)

In order to be as fair as the roulette, the state lotteries would have to pay $950 for a $1 bet in the 3-digit game. In reality, they now pay only $500 for a $1 winning bet!!! Remember, the odds are 1,000 to 1 in the 3-digit game...

If private organizations, such as the casinos, would conduct such forms of gambling, they would surely be outlawed on the grounds of extortion! In any event, the state lotteries defy all anti-trust laws: they do not allow the slightest form of competition! Nevertheless, the state lotteries may conduct their business because their hefty profits serve worthy social purposes (helping the seniors, the schools, etc.) Therefore, lotteries are a form of taxation - the governments must tell the truth to their constituents...

3. Fundamental Formula of Gambling: Games Other Than Coin Tossing

Dice rolling is a more difficult game and it is illustrated in the column p=1/6. I bet, for example, on the 3-point face. There is a 50% chance (DC) that the 3-point face will show up within the first 3 rolls. It will take, however, 37 rolls to have a 99.9% certainty that the 3-point face will show up at least once. If I bet the same way as in the previous case, my betting capital should be equal to 2 to the power of 37! It's already astronomical and we are still in easy-gambling territory!

Let's go all the way to the last column: p=1/1,000. The column illustrates the well-known3-digitlottery game. It is extremely popular and supposedly easy to win. Unfortunately, most players know little, if anything, about its mathematics. Let's say I pick the number 2-1-4 and play it every drawing. I only have a 10% chance (DC) that my pick will come out winner within the next 105 drawings!

The degree of certainty DC is 50% that my number will hit within 692 drawings! Which also means that my pick will not come out before I play it for 692 drawings. So, I would spend $692 and maybe I win $500! If the state lotteries want to treat their customers (players like you and me) more fairly, they should pay $690 or $700 for a $1 winning ticket. That's where the 50-50 chance line falls.

In numerous other cases it's even worse. I could play my daily-3 number for 4,602 drawings and, finally, win. Yes, it is almost certain that my number will come out within 4,602 or within 6,904 drawings! Real life case: Pennsylvania State Lottery has conducted over 6,400 drawings in the pick3 game. The number 2,1,4 has not come out yet!...

All lottery cases and data do confirm the theory of probability and the formula of bankruptcy... I mean of gambling! By the way, it is almost certain (99.5% to 99.9%) that the number 2-1-4 will come out within the next 400-500 drawings in Pennsylvania lottery. But nothing is 100% certain, not even... 99.99%!

We don't need to analyze the lotto games. The results are, indeed, catastrophic. If you are curious, simply multiply the numbers in the last column by 10,000 to get a general idea. To have a 99.9% degree of certainty that your lotto (pick-6) ticket (with 6 numbers) will come out a winner, you would have to play it for over 69 million consecutive drawings! At a pace of 100 drawings a year, it would take over 690,000 years!

4. Ion Saliu's Paradox or Problem of N Trials

We can express the probability as p = 1/N; e.g. the probability of getting one point face when rolling a die is 1 in 6 or p = 1/6; the probability of getting one roulette number is 1 in 38 or p = 1/38. It is common sense that if we repeat the event N times we expect one success. That might be true for an extraordinarily large number of trials. If we repeat the event N times, we are NOT guaranteed to win. If we play roulette 38 consecutive spins, the chance to win is significantly less than 1!

A step in the Fundamental Formula of Gambling leads to this relation:

DC = 1 — 1/e
The limit 1 — 1/e is approximately 0.632120558828558...

I tested for N = 100,000,000 … N = 500,000,000 … N = 1,000,000,000 (one billion) trials. The results ever so slightly decrease, approaching the limit … but never surpass the limit!

When N = 100,000,000, then DC = .632120560667764...
When N = 1,000,000,000, then DC = .63212055901829...

(Calculations performed by SuperFormula, option C = Degree of Certainty (DC), then option 1 = Degree of Certainty (DC), then option 2 = The program calculates p.)

If the probability is 1/N and we repeat the event N times, the degree of certainty is 1 — (1/e), when N tends to infinity. I named this relation: Ion Saliu Paradox of N Trials. Read more on my Web pages: Theory of Probability: Best introduction, formulae, algorithms, software and Mathematics of Fundamental Formula of Gambling.

5. Practical Dimension of Fundamental Formula of Gambling

There is more info on this topic on the next page. It reveals the dark side of the Moon, so to speak. The governments hide the truth when it comes to telling it all; and the Internet is incredibly prone to fraudulent gambling. Read revealing facts: Lottery, Lotto, Gambling, Odds, House Edge, Fraud.
• The Fundamental Formula of Gambling does not explicitly or implicitly serve as a gambling system. It represents pure mathematics. Users who apply the numerical relations herein to their own gambling systems do so at their risk entirely. I, the author, do apply the formula to my gambling and lottery systems. I will show you how to use the gambling formula, my application MDIEditor and Lotto and the lotto systems that come with the application. I will put everything in a winning lotto strategy that targets the third prize in lotto games (4 out of 6).
•• At later times, I also released gambling systems, strategies for: Roulette, blackjack, baccarat, horse racing, sports betting. Is it all? Probably you'll find some more around here…
Click here to go to the lottery strategy, systems, software page

Read Ion Saliu's first book in print: Probability Theory, Live!
~ Discover profound philosophical implications of the Fundamental Formula of Gambling (FFG), including mathematics, probability, formula, gambling, lottery, software, degree of certainty, randomness.

6. Resources in Theory of Probability, Mathematics, Statistics, Combinatorics, Software

See a comprehensive directory of the pages and materials on the subject of theory of probability, mathematics, statistics, combinatorics, plus software.
  • Theory of Probability: Best introduction, formulae, algorithms, software.
  • Bayes Theorem, Conditional Probabilities, Simulation; Relation to Ion Saliu's Paradox.
  • Standard Deviation: Theory, Algorithm, Software.
    Standard deviation: Basics, mathematics, statistics, formula, software, algorithm.
  • Standard Deviation, Gauss, Normal, Binomial, Distribution
    Calculate: Median, degree of certainty, standard deviation, binomial, hypergeometric, average, sums, probabilities, odds.
  • Combinatorial Mathematics: Calculate, Generate Exponents, Permutations, Sets, Arrangements, Combinations for Any Numbers and Words.
  • Caveats in Theory of Probability.
  • The Best Strategy for Lottery, Gambling, Sports Betting, Horse Racing, Blackjack, Roulette.
  • Birthday ParadoxProbability Formula, Odds of Duplication, Software.
  • Monty Hall Paradox, 3-Door Problem, Probability Paradoxes.
  • Couple Swapping, Husband Wife Swapping, Probability, Odds.
  • Download Probability, Mathematics, StatisticsSoftware.

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By Ion Saliu,
Founder of Gambling Mathematics, Founder of Probability Theory of Life

I. Logic of Fundamental Formula of Gambling
II. The Mathematical Solution: Divine Logarithm
III. Mathematics of the God Concept: Formula of Absurdity
    • Why God Fears Mathematics, While Einstein Hates Gambling...
IV. Insider Information: It's All in Our Reason
V. Mathematics of Ion Saliu's Paradox or Problem of N Trials
VI. Software: Divine Tool to Further Empower Reason
VII. Links and Resources Regarding Mathematics, Probability

1. Logical Steps, Algorithm to the Fundamental Formula of Gambling

'Let no one enter here who is ignorant of mathematics.'
(The frontispiece of Plato's Academy)

'The most important questions of life are, for the most part, really only problems of probability.'
(Pierre Simon Marquis de Laplace, 'Théorie Analytique des Probabilités')

Here is how I arrived to, by now, famous Fundamental Formula of Gambling (FFG). When laypersons say: 'It is so simple,' it always represents undeniable mathematics; therefore, undeniable Truth. I thought I had worked it out on my own, because the formula starts with the very essence of probability theory: p = n/N, or reduced to a p = 1/N mathematical relation. A truth becomes (almost) self–evident when a number of people think independently of the same thing. But this human law must be the most undeniable of them all: No Truth is self–evident AND no human thinks totally independently of others.

My first step got my feet wet in my pick 3 lottery software pond. 'Probably win,' that's what I had thought.

I rationalized in this manner. The probability of any 3-digit combination is 1/1000. Therefore, I had expected that the repeat (skip) median of a long series of pick-3 drawings would be 500. It would be similar to coin tossing, where the median of p=1/2 series is 1. In other words, the median of a long series of coin tosses is 1. To my surprise, the repeat median of long series of pick-3 drawings was not 500. It was closer to 700.

I checked it for series of 1000 real drawings and also randomly generated drawings. Then I checked the median against series of 10000 (10 thousand) drawings. The median of the skip was always close to 700. Do not confuse it for the median combination in the set. That value of the median is, in fact, either 499 or 500. The correct expression is 4,9,9 or 4 9 9 or 5,0,0 (three separate digits).

What is that median useful for, anyway? Among other properties, the skip median (or median skip) shows that, on the average, a pick-3 combination hits in a number of drawings. Any pick-3 combination hits within 692 drawings in at least 50% of the cases. Equivalently, if you play one pick-3 combination, there is a 50%+ chance it will hit within 692 drawings, or it will repeat no later than 692 drawings. The chance is also (almost) 50% that you will have to wait more than 692 drawings for your number to hit.

I studied theory of probability (gambling mathematics too!) in high school and in college. Some things get imprinted on our minds. Such information becomes part of our axioms. An axiom is a self-evident truth, a truth that does not necessitate demonstration. We operate with axioms in a manner of automatic thinking.

So, I was analyzing mathematically long pick-3 series, where p=1/1000. Next, I wrote the probability formula of a single pick-3 number to hit two consecutive drawings: p = 1/1000 x 1/1000 = 1/1000000 (1 in 1 million). I have never found useful to work with very, very small numbers in probability.

How about the reverse? The probability of a particular pick-3 number NOT to hit is p = (1 – 1/1000) = 999/1000 = 0.999. This is a very large number. It is almost certain that my pick-3 combination will not hit the very first time I play it.

How about not hitting two times in a row? P = (1 – 1/1000) 2 = 0.999 to the power of 2 = 0.998. Still, a very large number! I reversed the approach one more time: What is the opposite of not hitting a number of consecutive drawings? It is winning within a number of consecutive drawings.

The knowledge was inside my head. Unconsciously, I used Socrates' dialectical method of delivering the truth. (His mother delivered babies.) I also followed steps in De Moivre formula. At this point, I had this relation:

1 – (1 – p) N

where N represents the number of consecutive drawings.

I thought that for N = 500 drawings, the expression above should give the median, or a probability of 50%. So, I calculated 1 – (1 – p) N = 1 – (1 – 1/1000) 500 = 1 – (0.999) 500 = 0.3936 = 39.36%. Thus, my relation became:

0.3936 = 1 – (1 – 1/1000) 500

I made N = 692. I obtained the value:
1 – (1 – 1/1000) 692 = 1 - 0.5004 = 49.96% (very close to 50%).

Next step: I made N = 693. I obtained the degree of certainty:
1 – (1 – 1/1000) 693 = 1 - 0.4999 = 50.01% (very, very close to 50%).

Thus, the parameter I call the FFG median is between 692 and 693 for the pick-3 lottery.

2. The Mathematical Solution: Divine Logarithms

I concluded I should not make more assumptions. What if I don't think I know what N should be for the median (50%), or for any other chance, which I simply called the degree of certainty? I realized I had the liberty to select whatever degree of certainty I wanted to, and only had to calculate N. The relationship became:

DC = 1 – (1 – p) N

Then:

(1 – p) N = 1 – DC

The equation can be solved using logarithms:

The only unknown is N: the number of consecutive drawings (or trials) that an event of probability p will appear at least once with the degree of certainty DC.

The rest is history. I called the relation “The Fundamental Formula of Gambling” almost automatically. Unintentionally, it might sound cocky. Just refer to it as FFG.

Nothing comes with absolute certainty, but to a degree of certainty! That's mathematics, and that's the only TRUTH.

3. Mathematics of the God Concept: The Formula of Absurdity

Unfortunately, O glorious sons and daughters of Logos and Axioma, the idea of God is a mathematical absurdity! It saddened me first, but we must come to grip with reality. The humans fictionalize because we feel we can't live without the comfort of absolute certainty.

* Some humans with mathematical skills will stumble upon an error, when the degree of certainty DC is set to 100%. There is no absolute certainty in the Universe (or probability equal to 1). It leads to an absurdity: Calculating the number of events necessary for an event of probability p to appear with a degree of certainty equal to 100%. It is absurd. No other qualifications apply, such as impossible or erroneous formula. Just remember the relation we had before considering the degree of certainty! We dealt with the probability of losing N consecutive times: (1 – p) N. In this relation, no N can lead to zero (1 – 100%). Not even minus Infinity! A computer program should trap the error and ask the user to enter a DC less than 100% (things like 99.99999999…%).*

Some profound thoughts surrounding this mathematical expression and the false error.
The Fundamental Formula of Gambling (FFG) proves that absolute certainty is a mathematical absurdity. If we set the degree of certainty DC=1 (or 100%), FFG leads to a mathematical absurdity. God is Absolute Certainty, therefore absolute absurdity. I can only imagine de Moivre's reaction when this thought might have crossed his mind: 'Certainty is absurd! How can God be True?' It was the 17th century, and the 21st has just a little changed for the better…

Aporia is a special form of absurdity. A Sophist philosopher — Zeno of Elea — constructed a most famous aporia. It is known as the Paradox of Achilles and the tortoise. Read the first philosophical, logical, and mathematical solution: Zeno's Paradox: Achilles Can't Outrun the Tortoise?

It was very easy to apply FFG mathematics to many types of lottery and gambling games. It is at the foundation of several high probability gambling systems I designed: roulette, blackjack, horseracing, and lottery. It works with stocks, too. There is significant randomness in stock evolution. Many stockbrokers came to terms with the reality that all stocks fluctuate in an undeniably random fashion. I am surprised how many brokerage firms have visited my site!

4. Insider Information: It's All in Our Reason

In the year of grace 2001 my memory dug out a real gem. I wrote about it in a post on my message board: Cool stories of the Truth.

“I found another treasure: A little book in Romanian. Don't they say great things come in small packages? It couldn't be truer than in this case. The book was The Certainties of Hazard by French academician Marcel Boll. The book was first published in French in 1941. My 100-page copy was the 1978 Romanian edition. It all came to life, like awakening from a dream. The book presented a table very similar to the table on my Fundamental Gambling Formula page. Then, in small print, the footnote: “The reader who is familiar with logarithms will remark immediately that N is the result of the mathematics formula: N=log(1-pi)/log(1-p).”

That's what I call the Fundamental Formula of Gambling, indeed! Actually, the author, Marcel Boll did not want to take credit for it. Abraham de Moivre largely developed the formula. Then I remembered more clearly about de Moivre and his formula from my school years. Abraham de Moivre himself probably did not want to take credit for the formula. As a matter of fact, the relation only deals with one element: the probability of N consecutive successes (or failures). Everybody knows, that's p N (p raised to the power of N). It's like an axiom, a self-evident truth. Accordingly, nobody can take credit for an axiom. I thought Pascal deserves the most credit for establishing p = n / N. From there, it's easy to establish p N. And give birth to so many more worthy numerical relations.

5. Mathematics of Ion Saliu's Paradox or Problem of N Trials

Another look at one of the steps leading to the Fundamental Formula of Gambling:
1 – DC = (1 – p)N
We can express the probability as p = 1 / N; e.g. the probability of getting one point face when rolling a die is 1 in 6 or p = 1 / 6; the probability of getting one roulette number is 1 in 38 or p = 1 / 38. It is common sense that if we repeat the event N times we expect one success. That might be true for an extraordinarily large number of trials. If we repeat the event N times, we are NOT guaranteed to win. If we play roulette 38 consecutive spins, the chance to win is significantly less than 1!
1 – DC = (1 – 1 / N) N

I noticed a mathematical limit. I saw clearly: lim((1 – (1 / N)) N) is equal to 1 / e (e represents the base of the natural logarithm or approximately 2.71828182845904...). Therefore:

1 – DC = 1 / e
and
DC = 1 — (1 / e)

The limit of 1 — (1 / e) is equal to approximately 0.632120558828558...

I tested for N = 100,000,000... N = 500,000,000 ... N = 1,000,000,000 (one billion) trials. The results ever so slightly decrease, approaching the limit … but never surpassing the limit!

When N = 100,000,000, then DC = .632120560667764...
When N = 1,000,000,000, then DC = .63212055901829...

(Calculations performed by SuperFormula, function C = Degree of Certainty (DC), then option 1 = Degree of Certainty (DC), then option 2 = The program calculates p.)

If the probability is p = 1 / N and we repeat the event N times, the degree of certainty DC is 1 — (1 / e), when N tends to infinity. I named this relation Ion Saliu's Paradox of N Trials.

    Soon after I published my page on Theory of Probability, Ion Saliu Paradox (in 2004), the adverse reactions were instantaneous. I even received multiple hostile emails from the same individual! Basically, they considered my (1 / e) discovery as idiocy! “You are mathematically challenged”, they were cursing! Guess, what? I saw in 2012 an edited page of Wikipedia (e constant) where my (1 / e) discovery is considered correct mathematics. Of course, they do not give me credit for that. Nor do they demonstrate mathematically the (1 / e) relation — because they don't know the demonstration (as of March 21, 2012)!

    • You can see the mathematical proof right here, for the first time. I created a PDF file with nicely formatted equations:

  • Mathematics of Ion Saliu Paradox.

How long is in the long run? Or, how big is the law of BIG numbers? Ion Saliu's paradox of N trials makes it easy and clear. Let's repeat the number of trials in M multiples of N; e.g. play one roulette number in two series of 38 numbers each. The formula becomes:

1 – DC = (1 – 1 / N)NM = {(1 – 1 / N)N}M = (1 / e)M

Therefore, the degree of certainty becomes:

DC = 1 – (1 / e)M

If M tends to infinity, (1 / e)M tends to zero, therefore the degree of certainty tends to 1 (certainty, yes, but not in a philosophical sense!)

Actually, relatively low values of M make the degree of certainty very, very nearly 100%. For example, if M = 20, DC = 99.9999992%. If M = 50, the PCs of the day calculate DC = 100%. Of course, they can't approximate more than 18 decimal positions! Let's say we want to know how long it will take for all pick-3 lottery combinations to be drawn. The computers say that all 1000 pick-3 sets will come out within 50,000 drawings with a degree of certainty virtually equal to 100%.

Mathematics Of Gambling The Kelly Formula 1

Ion Saliu's Paradox of N Trials refers to randomly generating one element at a time from a set of N elements. There is a set of N distinct elements (e.g. lotto numbered-balls from 1 to 49). We randomly generate (or draw) 1 element at a time for a total of N drawings (number of trials). The result will be around 63% unique elements and around 37% duplicates (more precisely named repeats).

Let's look at the probability situation from a different angle. What is the probability to randomly generate N elements at a time and ALL N elements be unique?

Let's say we have 6 dice (since a die has 6 faces or elements); we throw all 6 dice at the same time (a perfectly random draw). What is the probability that all 6 faces will be unique (i.e. from 1 to 6 in any order)? Total possible cases is calculated by the Saliusian sets (or exponents): 66 (6 to the power of 6) or 46656. Total number of favorable cases is represented by permutations. The permutations are calculated by the factorial: 6! = 720. We calculate the probability of 6 unique point-faces on all 6 dice by dividing permutations to exponents: 720 / 46656 = 1 in 64.8.

We can generalize to N elements randomly drawn N at a time. The probability of all N elements be unique is equal to permutations over exponents. A precise formula reads:

Probability of unique N elements = N! / NN

I created another type of probability software that randomly generates unique numbers from N elements. I even offer the source code (totally free), plus the algorithms of random number generation (algorithm #2). For most situations, only the computer software can generate N random elements from a set of N distinct items. Also, only the software can generate Ion Saliu's sets (exponents) when N is larger than even 5. Caveat: today's computers are not capable of handling very large Saliusian sets!

I wrote also software to simulate Ion Saliu's Paradox of N Trials:
OccupancySaliuParadox, mathematical software also calculating the Classical Occupancy Problem.

Read more on my Web page: Theory of Probability: Introduction, Formulae, Software, Algorithms.

6. Software: The Divine Tool to Further Empower Reason

I wrote software to handle the

All Mathematics Formula Pdf

Fundamental Formula of Gambling (FFG) and its reverse: Anti-FFG or the Degree of Certainty. There are situations when we want to calculate the Degree of Certainty that an event of probability p will appear at least once within a number of trials N. As a matter of fact, this method offers a more precise correlation between an integer

Mathematics Formula Chart

number of trials and a degree of certainty DC expressed as a floating-point number. Furthermore, the program can determine the probability from a data series! The number of elements in the data series is known (N). Sorting the data series can determine the median: The degree of certainty DC equal to 50%!

Mathematics Of Gambling The Kelly Formula 2

FORMULA calculates several mathematical, probability, and statistics functions: Binomial distribution; standard deviation; hypergeometric distribution; odds (probability) for lotto, lottery, and gambling; normal probability rule; sums and mean average; randomization (shuffle); etc.
The 16-bit software was superseded by SuperFormula. Super Formula also calculates the Binomial Distribution Formula (BDF), the Binomial Standard Deviation (BSD), Statistical Standard Deviation (BSD) — and then some.

• Download Software: Science, Mathematics, Statistics, Lexicographic, Combinatorial: SuperFormula, FORMULA, OccupancySaliuParadox from the software downloads site - membership is necessary (a most reasonable fee).

I assembled all my mathematics, probability, statistics, combinatorics programs in a special software package named Scientia.

Read Ion Saliu's first book in print: Probability Theory, Live!
~ Discover profound philosophical implications of the Formula of TheEverything, including Mathematics, formulas, gambling, lottery, software, computer programming, logarithm function, the absurdity of God concept.

Resources in Theory of Probability, Mathematics, Statistics, Combinatorics, Software

Complete List Of Mathematics Formulas

See a comprehensive directory of the pages and materials on the subject of theory of probability, mathematics, statistics, combinatorics, plus software.
  • Theory of Probability: Best introduction, formulae, algorithms, software.
  • Bayes' Theorem, Conditional Probabilities, Simulation; Relation to Ion Saliu's Paradox.
  • Birthday Paradox is a particular case of exponential sets (sets with duplicate elements); software to calculate, generate any form of Birthday Paradox cases.
  • Probability: Pairing, Couple Swapping, Hypergeometric Distribution.
  • Mathematics of Monty Hall Paradox, Classical Occupancy Problem; Ion Saliu's Paradox.
  • Probability Player Index N Draws Ticket Number N.
  • Calculate, Generate Exponents, Permutations (Factorial), Arrangements, Combinations.
  • PI Day, Divine Proportion, Golden Proportion, Golden Number, PHI, Fibonacci.
  • Censorship Ideology Religion Money Jealousy Piracy Hatred Ban Internet.
  • Read all about the famous Fundamental Formula of Gambling.
  • Almighty Number, Randomness, Universe, God, Order, History, Repetition.
    The Fundamental Formula of Gambling (FFG) may well be the Ultimate Formula of The Everything.
  • Press Release: Mathematical Proof of the Absurdity of the God Concept.

    The following pages at this website offer more special mathematical solutions, functions and formulas, especially in combinatorics. There are algorithms and special software to calculate and generate permutations, exponents, combinations, both for numbers and words. Also, lexicographical order or index can be easily calculated for a large variety of sets.

    • Lexicographic, Lexicographical Order, Index, Ranking, Sets: Permutations, Exponential Sets, Combinations.
    • Combination Lexicographic Order, Rank, Index: Comprehensive and Fast Algorithm.
    • Combination Sequence Number or Lexicographic Order Debates.
    • Combination Sequence Number or Lexicographic Order: Mathematical Algorithms.
    • Combination1,2,3,4,5,6: Probability, Odds, Statistics.
    • There are no naturally-occurring Geometric Shapes or Perfect Forms on Earth, indeed in the Universe, to justify a super intelligent, divine force in Cosmos.
    • Only humans create perfect shapes (geometric as in geometry) in an attempt to have relative control over randomness.
    • Download the best software.

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