II. Fundamental Table of Gambling (

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

IV.

V. Practical Dimension of Fundamental Formula of Gambling

VI. Resources in Theory of Probability, Mathematics, Statistics, Software

**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__.

• 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

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

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,

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

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:

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

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

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...

Let's go all the way to the last column: * p=1/1,000*. The column illustrates the well-known

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

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

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

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

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:

• The does not explicitly or implicitly serve as a Fundamental Formula of Gambling. 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 systemgambling 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. Is it all? Probably you'll find some more around here…sports bettingClick here to go to the page
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~ Discover profound philosophical implications of the ** Fundamental Formula of Gambling (FFG)**, including mathematics, probability, formula, gambling, lottery, software, degree of certainty, randomness.

: Best introduction, formulae, algorithms, software.**Theory of Probability**.*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**

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