
Please be advised that this is a replica of an original web page serving the purpose of backup — in case the real page becomes unavailable. This page is NOT updated. For the latest, always go to saliu.com/Saliu2.htm:
Fundamental Formula of Gambling: Theory of Probability, Mathematics, Chance, Degree of Certainty.
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.
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.
• Click the following if you want to know the detailed procedure leading to this numerical relation (worth reading!): Mathematics of Fundamental Formula of Gambling.
• • Click here if you want to download scientific software that automatically does all the calculations (plus a whole lot more): SuperFormula.exe (32bit super software) and FORMULA.EXE (16bit for DOS or command prompt). The program allows you to calculate the number of trials N for any degree of certainty DC. Plus, you can calculate the very important 'binomial distribution' formula (BDF) and 'binomial standard deviation' (BSD).
Most software titles require a onetime payment to download – all software for one very reasonable price. The software is then free to run forever. Pay safely online via PayPal —
Let's suppose I play the 3digit lottery game (pick 3). The game has a total of 1,000 combinations. Thus, any particular pick3 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 and Pascal demonstrated that hundreds of years ago. Evidently, the combinations have an equal probability, but they appear with different frequencies. Please read an important message in my forum: Combination '1,2,3,4,5,6': Probability and Reality. The standard deviation directly influences the degree of certainty DC, in addition to the influence played by the probability p and number of trials N.
As soon as I choose a combination to play (for example 214) 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 (FFG) 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.
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).
DC ¯ 
p= .90 
p= .80 
p= .75 
p= .66 
p= 1/2 
p= 1/3 
p= 1/4 
p= 1/6 
p= 1/8 
p= 1/10 
p= 1/16 
p= 1/32 
p= 1/64 
p= 1/100 
p= 1/1,000 
10%                    1  1  3  6  10  105 
25%              1  1  2  3  4  9  18  28  287 
50%  1  1  1  1  1  1  2  3  6  7  10  21  44  68  692 
75%  1  1  1  2  2  3  4  7  11  13  21  43  88  137  1,385 
90%  1  2  2  2  3  5  8  12  17  22  35  72  146  229  2,301 
95%  1  2  2  3  4  7  10  16  22  29  46  94  190  298  2,994 
99%  2  3  3  4  7  11  16  25  34  44  71  145  292  458  4,602 
99.9%  3  4  5  6  10  17  24  37  52  66  107  217  438  687  6,904 
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 5050 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 even 10 or 11 tosses until heads appear! This dangerous form of betting is called a Martingale system. Beware of it! 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 edge . 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 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 3digit game. In reality, they now pay only $500 for a $1 winning bet!!! Remember, the odds are 1,000 to 1 in the 3digit 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 antitrust 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...
Dice rolling is a more difficult game and it is illustrated in the column p=1/6. I bet, for example, on the 3point face. There is a 50% chance (DC) that the 3point face will show up within the first 3 rolls. It will take, however, 37 rolls to have a 99.9% certainty that the 3point 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 easygambling territory!
Let's go all the way to the last column: p=1/1,000. The column illustrates the wellknown 3digit lottery 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 214 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 5050 chance line falls. In numerous cases it's even worse. I could play my daily3 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 214 will come out within the next 400500 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 (pick6) 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!...
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. Click here for important facts: Lottery, Lotto, Gambling: Math, Odds, House Edge, Fraud, Integrity.
• 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 do not sell for any price such gambling systems, however. Please do not send any inquiries regarding my gambling or lottery systems or software. I only offer publicly a limited version of my lottery software. It is unarguably the best on the market, but it is not the best software I use for myself. Probably everybody is aware of the rule that really winning systems (90%+) are always secret. It is logical and human. Nevertheless, I will make an exception. 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'). Click this to go to the system page: Lottery Software, Lotto Software: Create Winning Strategy, Systems, Lotto Wheels. 
Most software titles require a onetime payment to download – all software for one very reasonable price. The software is then free to run forever. Pay safely online via PayPal —