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Gambling Science = Limited-Step Martingale Progressions Applied to STREAKS (Win WLoss L)

By Ion Saliu,
Founder of Gambling Science, Founder of Blackjack Mathematics

Probability Theory: Formula, formulae, Software.

I. Gambling and Dreaming
II. My Mental Blackjack System, Fundamental Formula of Gambling, Martingale
III. Other Gambling Systems, Foundation, and Honesty
IV. The Fundamental Prayer of the Occult Science of Gambling
V. Resources in Gambling, Mathematics, Systems, Software, Books

Read about gambling science and dreaming formulating occult gambling.

I. Gambling and Dreaming

  • Written on August 14, 2004 (4 WE); later updates (the later, the better!)
  • First capture by the WayBack Machine ( on August 13, 2004.

    I finally decided to go back to the Atlantic City casinos. My bankroll was smaller than last year. I had that clear dream with Winston Churchill in the Irish Pub. I went directly to the inn of the Irish Pub and rented a room for a week. I got a good deal. Pay for six nights, get the seventh for free.

    The first day was not meant for real gambling. I wanted to check a new casino: Borgata. It looked very stylish from a distance. I was surprised how good the table games were compared to the Boardwalk casinos. My games of interest were blackjack, roulette, and baccarat. The minimum and the maximum bets were better than in any of the old casinos. Blackjack tables with $10 minimum limit are available even late at night. Besides, the casino is absolutely the most stylish in Atlantic City.

    I just wanted to pass the time while testing the blackjack game. I didn't want to use my notebook on that occasion. My intention was to be much disciplined. Use my notebook, start with at least $1000 bankroll, and avoid alcohol at the table. This first night would be the only exception. I started with only $200 bankroll. I had drinks during the game. I wouldn't open a notebook. Still, I did pretty well. I played my style, keeping track (mentally) of two types of streaks: W+ and L-.

    I also played against the (old) blackjack basic strategy (OBS) when I thought the casino dealer was too hot. For example, I did split two tens against dealer's 6 on two occasions. The BJ dealer busted both times. Other players were intrigued. I'm sure some cursed me silently. One Asian American, a good player, said to me: “You are amazing! But you know how to play your streaks!” I suspected him to be a casino guy. One of the blackjack dealers, another Asian American, a young lady, asked me what other casino games I play. The Asian Americans are the best-represented ethnic group in the Atlantic City casinos. At baccarat/mini-baccarat a clear majority of the dealers and players are Asian Americans. They put trust in their own minds.

    The first night was good. I had two more winning days and nights. I was comfortably ahead. I only played blackjack, without using my notebook. I started with no more than $300 in bankroll. I always had drinks at the table. One half of me was angry at my new style of playing. That half didn't agree with my new play. I was angry sometimes. I wanted to go home immediately. I was not disciplined. I was just a tourist on a vacation in a gambling environment.

    My intent was to become a professional gambler. In fact, my second day in AC I started to look for places to rent. I got discouraged by the difficulty to find a place in Atlantic City. People at my inn told me that Atlantic City was very much a closed place. On the other hand, I saw a lot of abandoned buildings around!

    Then I had two bad nights. Long streaks of 8 or 9 consecutive hands started to creep in. I didn't cry foul play, 'cuz I don't think there was dealer cheating. One night I had 7 consecutive losses in one shoe. The next shoe gave me 8 consecutive losing hands. The following shoe gave me 11 losing hands in a row! To add insult to the injury, a casino boss came over to congratulate the dealer! That was the thing that I hated the most during my trip! Bastard behavior on that casino man's part! I had started to hesitate in my play. I had already abandoned martingaling my losing streaks. I'll present my system in the next section.

    I was mad at myself and meself only. The realization was I had no chance of becoming a professional gambler. Two key elements were missing: a place to live in Atlantic City, and number two, a much larger bankroll. Tourism and professional gambling are a contradiction in terms. I realize I must treat it as a job. Sleep and eat at home. Go to work, preferably from 9 to 5. Drive my car directly to the casino of the day and play only in that place. When I go to work, I don't drink alcohol. I am well rested also. During my gambling adventure I was very, very tired. My room was close to the boardwalk. It was noisy at least until midnight. Then… noise again early in the morning… The seagulls make their own contribution starting to scream early in the morning. Then, I wondered around for places to eat... I had no legs under me!

    The size of the bankroll is the other essential element of professional gambling. The minimum minimorum is $10,000 for the current conditions in Atlantic City. The bankroll has no other purpose but casino gambling. It's not money to eat or to pay the rent. It's money for gambling and nothing else.

    Then, there is the starting bankroll. If I go to a table, I start with double the amount of the maximum bet at the table. Rarely, and very rarely, the starting bankroll should be equal to the maximum limit at the table. Most players I saw started the blackjack operation with just $100. Most of them were quickly cleaned up. Some chipped in with another $100. They should have started with the $200. At least, I always started with $200-$300. It is necessary to have enough money to weather bad moments, as I had. If I ever gamble as a tourist (no more than three-day trips) it won't happen if my bankroll is under $5,000 (five thousand).

    I recovered a little after those two bad days. All in all, my trip cost me about $1,000. I considered it a $1,000 vacation — the most expensive in my life. I discovered also a new blackjack game. It only has one table in Atlantic City — at Caesar's. It's called Double Attack Blackjack. It is not Spanish 21.

    The game has no Tens, but has A, J, Q, K. Blackjack pays even money (too bad!) It has two betting boxes. One is the betting box like at the regular blackjack. The second box is the interesting one. The first card in the round is dealer's face card. Seeing the first card, the player has the option to place an equal bet in the second betting box. It's like a double up your bet. Usually, I double attacked dealer's 2, 3, 4, 5, 6 (and rarely 7) face card.

    There is another intriguing bet: Bust It! At the beginning of the game, before the dealer shows the first card, a blackjack player places the initial bet. At that time, the player can place from $1 to $50 in the Bust It! box. If the dealer busts in three cards, the player receives additional winnings depending on the bust card. The bonus wager pays out according to what the dealer's third card is, as shown in the table below:

    Dealer's Third Card            Payout
    Ten                            3 to 1
    Nine                           6 to 1
    Eight                          8 to 1
    Seven                         10 to 1
    Six                           15 to 1

    If the dealer has an eight in her hand, and draws an eight of the same color or same suit as the eight already in her hand, there is a bigger bonus payout as follows:

    Dealer's Third Card           Payout
    Eight of the same color 	   50 to 1
    Eight of the same suit	  200 to 1

    I bet once $5 that the dealer would bust in three cards. It happened. The third card that busted dealer's hand was an 8. The payout was 8 to 1. There are two bonuses. If dealer busts with 8, 8, 8 of the same color, the payout is 50 to 1. If dealer busts with 8, 8, 8 of the same suit, the payout is 200 to 1. The first time I played the game, the dealer had 8, 8 of the same color. He called the pit boss to witness the third card. It wasn't a bonus 8.

    That first time also, they had a dealer who was learning the game. In my last hours in AC, at the same game table, I taught three casino people how to play the game! I'm not kidding! They didn't know well the double down and split situations. I played the new blackjack twice and I was a winner twice. Not big winnings, though. I treated it as a learning experience. I don't know how long that game would last. Another player and I won a few times in a row, including Bust It! bets. A floor man said that the blackjack professional counters win that game! Baloney… card counting is!

    • Read the official rules of Double Attack Blackjack (DABJ) as approved by New Jersey Casino Control Commission:

    We must count for all 3-card Double-Attack Blackjack hands in order to calculate the dealer bust in three cards. Thus, the total-total possible 3-card hands at DABJ is Arrangements (48, 3) = 48 * 47 * 46 = 103,776. And that leads to Bust It! odds of 15864 / 103776= 15.3% (or 1 in 6.5). Surely not a good deal if you bet on Bust It! every hand.

    The expectation is two BJ dealer busts in 13 hands (2 or 3 cards per hand only). The degree of certainty that the dealer busts at least 2 times in 13 hands is 61.3%. You can count the rounds in groups of 13 at a time. The dealer busts about two of three such groups.

    I wrote a unique piece of software that attempts, for the first time ever, to generate all possible blackjack hands. I uploaded the output file for 2-card and 3-card hands for Double-Attack Blackjack:


    You'll have a better understanding of the Bust It! wager.

    Total DABJ Hands: 47400
    Total Non-Bust Hands: 31536
    Total Dealer BUSTS: 15864 (33.47%)
    Total BlackJacks (10+A): 96 (4.26%)

    Bust It! Analysis
    Total possibilities (arrangements): 48 * 47 * 46 = 103,776
    Bust It! Odds: 15864 / 103776= 15.3% (1 in 6.5)

    Study blackjack betting systems, formula of gambling, Martingale, Fibonacci progressions.

    II. My Blackjack System, Fundamental Formula of Gambling, and Martingale

    I founded my blackjack system on the Fundamental Formula of Gambling (FFG). My full system, as used with a notebook, keeps track of several streaks: Win (W), Loss (L), Dealer Bust (DB), Player Bust (PB), Dealer Blackjack (DJ), Player BJ (PJ). In my trimmed down version, without the notebook, I only keep track of two streaks: Win (W+), Loss (L-). This is a mental gambling system — the records are kept in my memory. As such, the mental gambling system requires discipline. Unfortunately, I didn't apply my gambling system with the required discipline in 2004.

    My count always starts at: 0 - 0. I say mentally zero – zero.
    ~ The first number represents my consecutive losses;
    ~ The second number represents my consecutive skips without two or more consecutive wins. This parameter is trickier to grasp. I still have difficulties keeping accurate track of it mentally. It is much easier on paper.

    If I win my first hand, the count becomes 0 / 0-and-a-half. I say mentally zero – zero-and-a-half. The half simply indicates that the parameter is still running. If I lose the following hand, the count becomes 1 - 1. I say mentally one – one. That means I lost one consecutive hand; I also have a streak of one without winning two hands in a row.

    • If I win my first hand, the count becomes 0 - 0 and a half. I say mentally zero – zero-half. If I win also the following hand, the count becomes 0 - 0. I say mentally zero – zero. That means I have no consecutive losses; I also won two hands in a row. Every time I win 2 consecutive blackjack hands, the second count is automatically set to 0. The mental count becomes something – zero.

    If I win my first hand, the count becomes 0 / 0 and a half. I say mentally zero – zero-half. If I lose the following hand, the count becomes 1 - 1. I say mentally one – one. That means I lost one consecutive hand; I also have a streak of one without winning two hands in a row. I lose also the following hand. The count becomes 2 & 1. I say mentally two – one.

    • I win the following hand. The count becomes 0 - 1 and a half. I say mentally zero – one and a half. I lose the following hand. The count becomes 1 & 2. I say mentally one – two. It means I lost one consecutive hand; I also have a streak of two without winning two hands in a row. That is, in two consecutive situations, my win was not followed immediately by another win.

    Let's say I won one blackjack hand. The count is 0 - 0 and a half. I lose the following three blackjack hands. The count is 3 - 1. I say mentally three – one. I win the following hand. The count becomes 0 - 1-half. I say mentally zero – one and a half.

    I lose the next two blackjack hands. The count is 2 - 2. I say mentally two – two. If I lose the very next hand, the left count increases by one, while the right count remains unchanged. Mentally: three – two.

    I win the following hand. The count becomes 0 - 2 and a half. I say mentally zero – two-half.I lose the next four blackjack hands. The count is 4 - 3. I say mentally four – three.

    • Every time I lose a blackjack hand, the left count increases by one. Every time I win a blackjack hand, followed immediately by another win, the right count is set to 0. Every time I win a blackjack hand, followed immediately by a loss, the right count increases by a half. And so on… and so forth…

    With record-keeping on paper (notebook), I track the entire session. I can see the streaks of all lengths. I know I will lose 4, 5, 6, 7 … more blackjack hands in a row. I know also I will win 4, 5, 6, 7… more blackjack hands in a row. I look at the past 30-50 blackjack hands and decide that I should expect soon a 4-5 winning streak. Or, I shouldn't expect a losing streak longer than 4-5, because I already had a longer losing streak. There is no gambler's fallacy, really; or reversed gambler's fallacy. Read the pages in the Resources section for undeniable mathematical facts.

    I noticed that the second count is less streaky than the first one. That is, I win more regularly two (or more) hands in a row. I keep that in mind when I martingale my streaks. The Fundamental Formula of Gambling tells me that I win at least once in 7 hands with a degree of certainty of 99%.

    I martingale more aggressively at the beginning of my blackjack game. I martingale the first count when it reaches 4. Since the second parameter is more consistent, I martingale it after it reaches 3. I go up to 7 in both cases (2-4-8 and 2-4-8-18). I stop at 7. My probability software Streaks proves the more aggressive betting at the beginning of a blackjack (or gambling) session. The degree of certainty DC for long streaks (losing or winning streaks) is lower.

    I become more cautious after 100 blackjack hands or so. I martingale the first count after it reaches 6 (or 5 earlier in the game) and the second count after it reaches 5 (or 4 earlier in the game). I go as much as I can or allowed. If I sense that the streak is real bad, especially inside the same shoe, I stop the Martingale after 7, and start another one with 3 units after the previous streak ends.

    Mental blackjack strategy system wins consistently thousands of dollars a day in heads-up play.

    I released in April 2005 blackjack software to analyze streaks: Blackjack, BJAQK. BJAQK stands for: Best on Jacks Aces Queens Kings – and all other cards. See image above. The blackjack streaks software is available from the Software Download Site, software category 5.5. The program is a probability and statistical analyzer of thousands of Black Jack hands from the perspective of a strict blackjack old basic strategy (OBS) player. This program is a great complement to the free blackjack strategy presented on this page.

    The software generates truly random blackjack hands based on the probability of winning/losing/pushing from the standpoint of applying strict basic strategy rules. The software user can choose to generate and automatically analyze any number of hands, from dozens to millions (even billions, if the PC is out of this world!) The analysis is saved to a text file that can be viewed in any text editor. The report is so logically organized that even a modest blackjack player will comprehend it.

    • • Alternatively, you may consider the Fibonacci progressions (sequences) as well. In the traditional Fibonacci series, the Nth term is the sum of the two preceding terms. 1, 1, 2, 3, 5, 8, 13, 21, etc. I also devised series where:
    ~ the Nth term is the sum of the three preceding terms;
    ~ the Nth term is the sum of the four preceding terms;
    ~ the Nth term is the sum of the five preceding terms.

    I also created software to generate Fibonacci sequences. The programs also calculate the ratio between consecutive Fibonacci numbers and compare the result to the Golden Number (Divine Proportion). Go to this page: Fibonacci Progressions, Mathematics, Gambling.

    Gambling and betting systems from the perspective of mathematics and honesty.

    III. Other Gambling Systems, Foundation, Honesty

    I have founded my systems on mathematics, especially the Fundamental Formula of Gambling. Furthermore, I apply probability theory to the gambling mathematics of streaks.

    Probability software proves probability theory is gambling mathematics of winning - losing streaks.

    There are tons and tons of gambling systems out there. Some are outrageously priced. To me, only absolute idiots or absolutely insane players could pay such amounts of money. Also, it seems impossible to find a system where its author presents some facts. Is it founded on mathematics? “Yes,” every author out there says so. But they don't offer a single illustrative element. Rarely, some vendors offer a money-back guarantee, which is very, very rarely fulfilled.

    The most famous gambling systems are for the game of blackjack, namely card counting systems. I have written quite a bit on the fallacy of card-counting systems. Mathematically, card counting is worthless for all intents and purposes. The permutations are staggering. Besides, there is more than one player at the table. What makes you believe you are the chosen one to benefit from a so-called positive count? Also, the counts come in streaks. At the beginning of the round, the count might be very positive. When the second cards are dealt, however, the count goes very negative!

    Counting cards at blackjack makes money only for those few who sell gambling systems, books about card counting, or newsletters about counting cards. I know of several card-counting gurus who don't even play blackjack anymore! They only write about card counting at their websites. They also try to sell their gambling systems, or books, or newsletters, or videos, or software — all for naught!

    Edward Thorp, the founder of card counting, is NOT a blackjack player — never was and never will! Ken Uston was a blackjack player. But he made his money from selling blackjack books and especially from a successful lawsuit against a casino (scripted lawsuit). He also advertised for casinos and worked for a Kuwaiti business, instead of playing blackjack. If I had just $100,000 in bankroll, I guarantee 99.99% I would win an average of $10,000 a day.

    There are also tons of roulette systems. The roulette systems seem to me even more outrageous than card counting. Track the wheel speed, or the moon movement, or the dealer gender, etc. The authors never play roulette. They only try to find suckers to sell their systems or books or newsletters or videos.

    The authors of gambling systems target a special demographic group. It is a group of people with a special addiction. They are dreaming players or gamblers. They only dream of becoming gamblers. Luckily, they suffer from a safe form of addiction. They convinced themselves that they would start gambling after they discovered and learned one or two good systems. That's why they spend a lot of time in all kinds of public gambling and lottery places.

    They cruise every day gambling and lottery forums, and message boards, and newsgroups, and chat rooms, etc. They are waiting for Godot... James Godot. But that's safe addiction, you know. They avoid gambling; therefore they avoid serious financial losses. They must also realize that they should not play with real money for as long as they feel hesitation. They better grow mossy in those gambling forums dedicated to the APs. AP stands for Arithmetical Psychotic… him card counter who can count from -1 to +1 and believes with blind faith that a plus count is an automatic win… Alas!

    Presenting a gambling system based on win-lose streaks WL and smart martingale.

    4. The Fundamental Prayer of the Occult Science of Gambling

    I had another clear dream during my winning streak in Atlantic City.

    Learn blackjack systems based on fundamental formula of gambling and martingales.

    Addendum, October 2004 — The Fear_Survival System Never Sleeps!

    Things are clearer when they have the perspective of time. I reread this material a few times. I put very hard on myself. As I said above: “I was mad at myself and meself only”. In fact, the truth is very different. I did what my Fear-Survival decided, not what my mind wanted.

    There were hints before my adventure. I didn't want to pay attention. One thing was I didn't prepare my formidable ally — the notebook — for battle. The pages need to be formatted like spreadsheets, with rows and columns. In fact, I never used a notebook in the casinos during my 2004 gambling action. I did in 2003 and saw how efficient my action was. By the way, I picked the wrong casino for lawsuit: Taj Mahal is under bankruptcy protection! No court would accept a case against Taj Mahal.

    One of the clues was kind of weird. It took place before my gambling trip, and repeated thereafter. I was on one of the hiking trails adjacent to the Appalachian Trail. I passed a couple and reached a bifurcation. After a few minutes I met the couple again. They said “Oh, you turned back…” I answered something like “I think you took the long branch of the bifurcation…” It was end of July. I did it again in October. I passed an elderly lady. I reached the same bifurcation, without paying any attention to it. After a few minutes I met the same lady again. I was hiking the opposite direction I thought I was heading to! That was the “Aha!” moment. I had the picture clear and decided to write this addendum.

    The most revealing clue is the dream I had in Atlantic City. My Fear_Survival system was aware of an important discovery related to the Fundamental Formula of Gambling. At that time I thought I had discovered a new gambling system. In truth, I had discovered what now is known as the Ion Saliu Paradox of N Trials.

    The FFG dream was the loudest signal. My Fear-Survival did not want me in the casinos before solving the Ion Saliu's Paradox of N Trials. I just did that, and, boy, am I proud of it, or what?! Now even the new gambling system based on the paradox is far better defined. I also feel now that I can go back to the casinos and be the best I can be. Certainly, I would not make mistakes again. My Fear-Survival system shows me the green light…

    Find here resources in gambling, mathematics, blackjack, roulette, craps, systems, software.

    5. Resources in Gambling, Mathematics, Systems, Software, Books

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

    Resources in Theory of Probability, Mathematics, Gambling, Fraud

    Visit the outstanding resources in theory of probability, mathematics, statistics, combinatorics. The most original theories are based on truth exclusively. The convincing formulas must be backed by data. The best data validation is offered by software.

    Fear not Martingale betting, even if playing Double Attack Blackjack - gamblers fallacy is idiocy.

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