Mathematics Of Gambling The Kelly Formula

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The Kelly Formula. This would imply an even-money bet. In such an even-money bet, the Kelly Formula can be simplified to f = 2p - 1. To use the Kelly Criterion, then, a player must be able to estimate the odds, the probability of winning and the probability of losing the bet. Aug 12, 2019  Like how most successful gambling stories go, Kelly applied mathematics from information theory to create the Kelly Criterion (KC) 1. From KC, a gambler could make the best bet size to get the most money in the long run, if she knew her odds of winning and losing.

BornAugust 14, 1932 (age 87)
Chicago, Illinois, U.S.
CitizenshipAmerican
Alma materUCLA
Scientific career
FieldsProbability theory, Linear operators
InstitutionsUC Irvine, New Mexico State University, MIT
ThesisCompact Linear Operators in Normed Spaces(1958)
Doctoral advisorAngus E. Taylor
InfluencesClaude Shannon

Edward Oakley Thorp (born August 14, 1932) is an American mathematics professor, author, hedge fund manager, and blackjack researcher. He pioneered the modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.[citation needed]

Thorp is the author of Beat the Dealer, which mathematically proved that the house advantage in blackjack could be overcome by card counting.[1] He also developed and applied effective hedge fund techniques in the financial markets, and collaborated with Claude Shannon in creating the first wearable computer.[2]

Thorp received his Ph.D. in mathematics from the University of California, Los Angeles in 1958, and worked at the Massachusetts Institute of Technology (MIT) from 1959 to 1961. He was a professor of mathematics from 1961 to 1965 at New Mexico State University, and then joined the University of California, Irvine where he was a professor of mathematics from 1965 to 1977 and a professor of mathematics and finance from 1977 to 1982.[3]

  • 1Computer-aided research in blackjack

Computer-aided research in blackjack[edit]

Thorp used the IBM 704 as a research tool in order to investigate the probabilities of winning while developing his blackjack game theory, which was based on the Kelly criterion, which he learned about from the 1956 paper by Kelly.[4][5][6][7] He learned Fortran in order to program the equations needed for his theoretical research model on the probabilities of winning at blackjack. Thorp analyzed the game of blackjack to a great extent this way, while devising card-counting schemes with the aid of the IBM 704 in order to improve his odds,[8] especially near the end of a card deck that is not being reshuffled after every deal.

Applied research in Reno, Lake Tahoe and Las Vegas[edit]

Thorp decided to test his theory in practice in Reno, Lake Tahoe, and Las Vegas.[6][8][9]Thorp started his applied research using $10,000, with Manny Kimmel, a wealthy professional gambler and former bookmaker,[10] providing the venture capital. First they visited Reno and Lake Tahoe establishments where they tested Thorp's theory at the local blackjack tables.[9] The experimental results proved successful and his theory was verified since he won $11,000 in a single weekend.[6] Casinos now shuffle well before the end of the deck as a countermeasure to his methods. During his Las Vegas casino visits Thorp frequently used disguises such as wraparound glasses and false beards.[9] In addition to the blackjack activities, Thorp had assembled a baccarat team which was also winning.[9]

News quickly spread throughout the gambling community, which was eager for new methods of winning, while Thorp became an instant celebrity among blackjack aficionados. Due to the great demand generated about disseminating his research results to a wider gambling audience, he wrote the book Beat the Dealer in 1966, widely considered the original card counting manual,[11]which sold over 700,000 copies, a huge number for a specialty title which earned it a place in the New York Times bestseller list, much to the chagrin of Kimmel whose identity was thinly disguised in the book as Mr. X.[6]

Mathematics Of Gambling The Kelly Formula 2017

Thorp's blackjack research[12] is one of the very few examples where results from such research reached the public directly, completely bypassing the usual academic peer review process cycle. He has also stated that he considered the whole experiment an academic exercise.[6]

In addition, Thorp, while a professor of mathematics at MIT, met Claude Shannon, and took him and his wife Betty Shannon as partners on weekend forays to Las Vegas to play roulette and blackjack, at which Thorp was very successful.[13]His team's roulette play was the first instance of using a wearable computer in a casino — something which is now illegal, as of May 30, 1985, when the Nevada devices law came into effect as an emergency measure targeting blackjack and roulette devices.[2][13] The wearable computer was co-developed with Claude Shannon between 1960–61. Thefinal operating version of the device was tested in Shannon’s home lab at his basement in June 1961.[2] His achievements have led him to become an inaugural member of the Blackjack Hall of Fame.[14]

He also devised the 'Thorp count', a method for calculating the likelihood of winning in certain endgame positions in backgammon.[15]

Stock market[edit]

Since the late 1960s, Thorp has used his knowledge of probability and statistics in the stock market by discovering and exploiting a number of pricing anomalies in the securities markets, and he has made a significant fortune.[5] Thorp's first hedge fund was Princeton/Newport Partners. He is currently the President of Edward O. Thorp & Associates, based in Newport Beach, California. In May 1998, Thorp reported that his personal investments yielded an annualized 20 percent rate of return averaged over 28.5 years.[16]

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Bibliography[edit]

  • (Autobiography) Edward O. Thorp, A Man for All Markets: From Las Vegas to Wall Street, How I Beat the Dealer and the Market, 2017. [1]
  • Edward O. Thorp, Elementary Probability, 1977, ISBN0-88275-389-4
  • Edward Thorp, Beat the Dealer: A Winning Strategy for the Game of Twenty-One, ISBN0-394-70310-3
  • Edward O. Thorp, Sheen T. Kassouf, Beat the Market: A Scientific Stock Market System, 1967, ISBN0-394-42439-5 (online pdf, retrieved 22 Nov 2017)
  • Edward O. Thorp, The Mathematics of Gambling, 1984, ISBN0-89746-019-7 (online version part 1, part 2, part 3, part 4)
  • Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone
  • The Kelly Capital Growth Investment Criterion: Theory and Practice (World Scientific Handbook in Financial Economic Series), ISBN978-9814293495, February 10, 2011 by Leonard C. MacLean (Editor), Edward O. Thorp (Editor), William T. Ziemba (Editor)

See also[edit]

References[edit]

  1. ^Peter A. Griffin (1979) The Theory of Blackjack, Huntington Press, ISBN978-0929712130
  2. ^ abcEdward O. Thorp. 'The Invention of the First Wearable Computer'(PDF). Edward O. Thorp & Associates. Retrieved April 26, 2010.
  3. ^'Founding professor of math donates personal, professional papers to UCI Libraries'. UCI News. UC Irvine. June 12, 2018.
  4. ^Understanding Fortune’s Formula by Edward O. Thorp Copyright 2007 Quote: 'My 1962 book Beat the Dealer explained the detailed theory and practice. The “optimal” way to bet in favorable situations was an important feature.In Beat the Dealer I called this, naturally enough, “The Kelly gambling system,” since I learned about it from the 1956 paper by John L. Kelly.'
  5. ^ abTHE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET by Edward O. Thorp Paper presented at: The 10th International Conference on Gambling and Risk Taking Montreal, June 1997
  6. ^ abcdeDiscovery channel documentary series: Breaking Vegas, Episode: 'Professor Blackjack' with interviews by Ed and Vivian Thorp
  7. ^The Tech (MIT) 'Thorpe, 704 Beat Blackjack' Vol. 81 No. I Cambridge, Mass., Friday, February 10, 1961
  8. ^ ab'American Scientist online: Bettor Math, article and book review by Elwyn Berlekamp'. Archived from the original on April 23, 2007. Retrieved March 18, 2006.CS1 maint: BOT: original-url status unknown (link)
  9. ^ abcdIt's Bye! Bye! Blackjack Edward Thorp, the pensive professor above, is shaking the gambling world with a system for beating a great card game. He published it a year ago, and now the proof is in: it works David E. Scherman January 13, 1964 pp. 1–3 from SI Vault (beta)(CNN) Quotes: 'The unlikely trio was soon on its way to Reno and Lake Tahoe, where Thorp's horn-rimmed glasses, dark hair and fresh, scrubbed face hardly struck terror into the pit bosses. (p. 1)', 'But Edward Thorp and his computer are not done with Nevada yet. The classiest gambling game of all—just ask James Bond—is that enticing thing called baccarat, or chemin de fer. Its rules prevent a fast shuffle, and there is very little opportunity for hanky-panky. Thorp has now come up with a system to beat it, and the system seems to work. He has a baccarat team, and it is over $5,000 ahead. It has also been spotted and barred from play in two casinos. Could it be bye-bye to baccarat, too? (p. 1)' and 'But disguises frequently work. Thorp himself now uses a combination of wraparound glasses and a beard to change his appearance on successive Las Vegas visits. (p. 3)'
  10. ^Breaking Vegas “Professor Blackjack.”Archived December 21, 2008, at the Wayback Machine Biography channel Rated: TVPG Running Time: 60 Minutes Quote: 'In 1961, lifelong gambler Manny Kimmel, a 'connected' New York businessman, read an article by MIT math professor Ed Thorp claiming that anyone could make a fortune at blackjack by using math theory to count cards. The mob-connected sharpie offered the young professor a deal: he would put up the money, if Thorp would put his theory to action and card-count their way to millions. From Thorp's initial research to the partnership's explosive effect on the blackjack landscape, this episode boasts fascinating facts about the game's history, colorful interviews (including with Thorp), and archival footage that evokes the timeless allure and excitement of the thriving casinos in the early `60s. '
  11. ^'Blackjack Hero profile'. Blackjackhero.com. Retrieved April 26, 2010.
  12. ^A favorable strategy for twenty-one. Proceedings of the National Academy of Sciences 47 (1961), 110-112
  13. ^ ab'Poundstone, William: 'Fortune's Formula : The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street''. Amazon.com. Retrieved April 26, 2010.
  14. ^Anthony Curtis. 'Las Vegas Advisor on Ed Thorp'. Lasvegasadvisor.com. Retrieved April 26, 2010.
  15. ^Chuck Bower (January 23, 1997). 'Cube Handling in Races: Thorpe count'. bkgm.com. Backgammon Galore. Retrieved May 8, 2013.
  16. ^'Thorp's market activities'. Webhome.idirect.com. Archived from the original on October 31, 2005. Retrieved April 26, 2010.

Sources[edit]

  • Patterson, Scott D., The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It, Crown Business, 352 pages, 2010. ISBN0-307-45337-5 via Patterson and Thorp interview on Fresh Air, February 1, 2010, including excerpt 'Chapter 2: The Godfather: Ed Thorp'

External links[edit]

  • Edward O. Thorp at the Mathematics Genealogy Project
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Edward_O._Thorp&oldid=929403226'

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 NFormula 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.

Mathematics formula sheet

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). Ainsworth free slot games.

How to make a blackjack drink. 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!..

Two basic function of an expansion slot. Nov 13, 2018  In this picture, there are three different types of expansion slots: PCI Express, PCI, and AGP. How many expansion slots does my computer have? Every computer motherboard is different, to determine how many expansion slots are on your computer motherboard identify the manufacturer and model of the motherboard. PCI: The PCI slot is the most common form of internal expansion for a PC. Some PCs have a mixture of PCI and PCI Express slots. If so, go with PCI Express when you have that option. AGP: This type of expansion slot was specifically designed to deal with graphics adapters. In fact, AGP stands for Accelerated Graphics Port. Aug 07, 2017  M.2 is a slot that can interface with SATA 3.0 (the cable that’s probably connected to your desktop PC’s storage drive right now), PCI Express 3.0 (the default interface for graphics cards and other major expansion devices), and even USB 3.0. Sep 13, 2011  The function of expansion slots are exactly like their name. They are meant to hold extra devices. They 'expand' your systems ability to have hardware.

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

Mathematics Of Gambling The Kelly FormulaThere 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.
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  • Monty Hall Paradox, 3-Door Problem, Probability Paradoxes.
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  • Download Probability, Mathematics, StatisticsSoftware.

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