Study gambling mathematics of blackjack, professional card counting, counting cards.

Blackjack Card Counting Systems, Gambling Mathematics, Cult, Fraud

By Ion Saliu, Founder of Blackjack Mathematics

Apply mathematics, gambling probability formula, card sequences to blackjack card counting.

All gambling systems to count cards in blackjack are fraudulent and lead to big financial losses.

I. Prolegomenon: Mathematics Invalidates Blackjack Card Counting Strategy
II. In-Depth Mathematical Analysis of BJ Counting Card Methodology III. Card-Counting Blackjack Strategy Amounts to Deception, Fraud
IV. Shuffling Software Simulates Sequences of Blackjack Cards
V. Edward Thorp Fathered BJ Card Counting; Ken Uston Made it a Cult
VI. Aura and Mystique of Card Counting at Blackjack
VII. Blackjack: Software, Content, Resources, Systems, Basic Strategy Charts

Casinos falsely fight blackjack card counters as a form of advertising and attracting more gamblers.

1. Prolegomenon: Mathematics Invalidates Blackjack Card Counting Strategy

Published on August 19, 2000; later updates.

Mathematically, the card counting system has a minor influence today, if any at all. I showed in a previous article here that the only impact counting can have in a single deck blackjack game, only one Player against Dealer. A +2 count, generously translated into a 2% Player's advantage results in a 0.01% better chance to get a blackjack (a natural 21). Such a chance increases, at best, from 4.77% to 4.78%. Much ado about nothing!

The main mathematical aspect is the sequence of the remaining cards. Card counting does not even attempt to “predict” the card sequence. In reality, it is impossible to detect the sequence of the remaining cards. The number of blackjack sequences is staggering. They are calculated by the factorial operator (N! = 1 * 2 * 3 * ... * N).

Let's exemplify briefly in a most ideal situation. Before Edward E. Thorp's “Beat the Dealer” book (an important work, indeed) the Dealer dealt all the cards in a single deck. Therefore, the penetration was 100%. A card counter would wait up until the last quarter of the deck or so. Let's say the counter estimated that around 13 cards were remaining in the 52-card deck. The legend has it that many struck it big at the blackjack tables simply by betting huge amounts towards the end of the deck.

Mathematics, however, does not confirm such a claim — in fact, mathematics invalidates the wishful thinking. To calculate the number of possible sequences of 13 cards in blackjack, we use 13! = 1 x 2 x 3 x … x 12 x 13. The result is: 6,227,020,800 (6 billion, 227 million, 20 thousand, 800 hundred sequences). Let's be fair. There would be only 10 unique cards among 13, since 10=J=Q=K. In this case, 10! = 3,628,800 (over 6 million possible sequences)! Imagine an even better situation for Player, that the count is +3 and that there are no neutral cards remaining (i.e. 7, 8, or 9; we count Ace with Tens as high cards).

2. In-Depth Mathematical Analysis of BJ Counting Card Methodology

  • Axiomatic ones, this is the most in-depth analysis of how blackjack card-counting works. We analyze a most ideal game case: One quarter of a deck, no neutral cards, a +3 count. That count figure is equivalent to a +12 count in one full deck of cards – as frequent an occurrence as once in 600 years of play (the lifespan of Biblical Noah who “played” with his daughters to save the human species!!!)

    It is obvious now that we work with sequences of 5 cards at a time: 2 for Player, 3 for Dealer. We simplify a great deal if we apply the numbers sets known as combinations instead of the arrangements type of sets. Combinations of (13 numbers taken 5 at a time) = 1287. We have only 8 high cards but only 5 low cards.

    So, among the 13 cards left in the deck, we can only have 2 5-card sequences of HH for Player and HLH for Dealer (BUST). Each of the two cases lead directly to a Player's win. By the same token, Dealer will have a mathematically equal chance to get 2 sequences in the format HH for her, HLH for Player (BUST). Evidently, the advantage and disadvantage cancel out. No harm, no charm. But there is that big problem. The majority of the situations did not fulfill Player's Dream. The player expected wins in all those situations and bet bigger than in non-positive counts.

    In fact, in this Neverland case scenario, Mrs. Dealer busts less by 8%. Meanwhile, another assumption of card counting theory is that Dealer will bust more often in positive counts. Go figure!

    Mathematical Conclusion

          Before the "favor" of counting cards comes to fruition, the card counter wrongly assumes winnings and increases his bets, drastically at times, in every positive-count situation. Mathematically, there are more losing than winning sequences for the blackjack player even in positive counts. (As you'll see below, the negative counts might be better for the player, after all!) No wonder the blackjack counters lose so much in shorter periods of time compared to straight basic strategy players. Virtually all of them counters encountered heavy losses and abandon the trade.

    Inquisitively axiomatic one, I discovered one paradox regarding counting cards. I ran my special software that calculates the blackjack odds precisely mathematically. Clearly, Player would have an advantage in negative-count situations! There is a high percentage of double down sequences: 4+6, 5+5, 5+6, even 3+6, 4+5. All these situations, however, are squandered because they must occur in a low count (negative). Of course, the dealer and the player have an equal chance to get the double-down hands. You know the difference: the blackjack dealer may not double down.

    A new blackjack system called reversed card counting seeks negative counts that lead to double down.

    Blackjack Reversed Card Counting System

    The reversed card counting blackjack strategy can be highly superior to the positive count system.

  • In the best-case scenario of One Player versus Dealer, winning in the plus count situations is equitable to random play, which clearly favors the house (the blackjack dealer). In multiple-player cases (by far the most common), the so-called true count (TC) diminishes with the number of players. Should we apply a new concept? I coined it as the per capita true count (PCTC). The total true count is divided equally amongst all players plus the dealer. In the example above, TC = +12; PCTC = 12 / 8 = +1.5 if there are 7 players at the table (don't forget the dealer!)

    3. Card-Counting Blackjack Strategy Equates Deception, Fraud

    The fraud of the card-counting blackjack system is two-pronged. It is defended by two pretexts, two absurd justifications: long run and variance.

    Some claim that blackjack simulation software always shows an advantage for card counting play. Baloney, I say. I coined this saying: Card counting authors, developers, vendors COOK THE BOOKS. The crooks do that with some simulation software applications. I experienced an interesting case in BJ Bishop Arnold Snyder's blackjack forum. One card counting system author was strikingly candid in a scrimmage the two of us had. The counting-cards developer candidly admitted that his system had failed 3 consecutive times in his attempt to create an illustrative YouTube video. He succeeded to show winning for his card-counting system in his 4th try. So, he uploaded to YouTube that fourth video that showed a success for his system!

  • Evidently, the long run never reaches an end, according to the crooks of blackjack card counting desperados (read: vendors)! Theory of probability, however, applies to any situation: Any event of any probability p, any number of trials N, any degree of certainty DC.

    Ion Saliu's Paradox precisely calculates the long run. Just 50 repeats of the N trials constitutes a sufficient mathematical long run. If card counting expectation is a little better than 50%, then probability p is just a little better than 1/2. Just 50 repeats of 2 gives 100 blackjack hands. 1000 BJ hands is that much better — a larger than sufficient long run. By the same standard, a sufficient long run in roulette would be 2000 spins. Every analysis of real-life casino data proves that every roulette number came out very, very close percentage-wise to formula calculations.

    And, again, the reverse imposes itself upon the card-counting crooks. How about a real long run when the counters do not lose: Play billions of blackjack hands before encounter a loss!

    No wonder the casinos are so bullish in touting and promoting counting cards at blackjack! Ever seen that failed casino movie entitled 21 dedicated to BJ card counting?

    I've written this article in harsh language, indeed. It's the only language the krookoos understand. They'll jump at one's throat if one dares to critique card counting. They attacked me virulently every time and everywhere I attempted to express my opinion on counting cards at blackjack. Those with authority banished me quickly in their “public” forums. As soon as I jotted down words such as mathematics or formulas, them crooks would go beserk!

    4. Shuffling Software Simulates Sequences of Blackjack Cards

    My latest blackjack software calculates the odds of dealer busting in the most accurate mathematical fashion. We can generate all possible BJ hands mathematically — option A = Arrangements. That is, the order counts. We will notice undeniably that 10 followed by 6 occurs equally to 6 followed by 10. Even the so-called basic strategy stands on thin ice mathematically.

    Generate blackjack hands when the order counts: card counting at the table is worthless.

    Let's play the first sequence now, between one Player and the Dealer.

  • Player gets L, Dealer gets H. Player gets H, Dealer gets H. Player stands, Dealer stands. Dealer wins one hand. Player gets H, Dealer gets H. Player gets L and stands, Dealer gets H and wins the 2nd hand. Player gets L, Dealer gets H, Player gets H, Dealer gets L and draws L; more probably, Dealer wins the hand.
  • In the 2nd sequence, Player gets H, Dealer gets L, Player gets H and stands, Dealer draws L, and must draw to LH, gets an H and busts. Player wins the 1st hand. Player gets L, Dealer gets L, Player gets H and stands, Dealer gets H and draws another H (bust). Player wins the 2nd hand.

    Now, let's change the perspective from mathematics to logic. That is, instead of numerical relations, let's use words.

    It becomes very, very complicated in a hurry. We are dealing with over three and a half million sequences. The matters are a lot more complicated when we consider the neutral cards and multiple players. The impact of card counting appears to be near negligible. I count cards sometimes up to the latest card. I call it the “real running count”. At the end of a round the count is positive, for example. When it is my turn to make a decision after my first two cards, the count turns out to be very negative. A blackjack card counter would have bet bigger at the beginning of the round. When it was his/her turn to make a decision, the count would be clearly negative. The player might expect a low card. Quite often, Player gets the opposite, a busting high card. I noticed frequently such sequences. They seem to be totally random. If you are a faithful card counter, you should expect to lose serious money for a significant number of rounds (hands).

    Sometimes, but rarely, card counting might help in hit / stand or double-down decisions. Once, I shocked the audience when I doubled-down on 12 against dealer's 6. The blackjack dealer announced his pit boss about my play. Curiously, I won! The count was very negative — just before my third card. I did it more for showmanship… my bet was at the minimum…

    5. Edward Thorp Fathered BJ Card Counting; Ken Uston Made it a Cult

    The personal side of my previous posts on Thorp and Uston relied on blackjack and gambling books I have read. I have never claimed I experienced firsthand Thorp's or Uston's casino playing. First of all, I praise the two computer programmers who took on a very difficult challenge. Trying to discover rules in random events such as gambling represents an enormous task. Yes, the casinos did change the rules after Thorp published his book. The simple fact that the casinos added penetration to the deck diminished the effect of counting down to fractions of a percentage point. The multiple decks made it impossible to have a true count. It is impossible to divide accurately cards remaining in the deck by an approximate number of decks remaining in play — and get a true count! Contradiction in terms!

    One problem I have with the two blackjack authors and researchers is the fact that they reached a point of worshipping card counting. I am sure they did a large number of computer simulations. I am sure they noticed the negligible effect of the system. Yet they continued to influence a large number of potential casino players that card counting was the road to the riches. I would like to present a few more excerpts on Edward Thorp and Ken Uston. Carl Sifakis writes in his 1990 Encyclopedia of Gambling (pg. 36-37).

    The late gambling expert John Scarne derided the counters and challenged Edward Thorp to a $100,000 match at blackjack. He later extended the challenge to other leading counting advocates. "There were some nos, then yeses, then considerably more backing and filling; the blackjack contest never came off." After long hesitation, Thorp accepted Scarne's challenge. Edward Thorp lost badly and he admitted he was a lousy blackjack player. His card-counting system didn't play any miracle!

    Edward O. Thorp is associated with another gambling “system”: the Big Toe Method of beating the Wheel of Fortune (Big Six). Says Carl Sifakis in the same book I quoted above: “Is the Thorp Big Toe Method still in use? The bottom line is that the money wheels of all kinds are still in place in the casinos, something that would not be the case if there was a way they were being beaten considerably.” (page 31).

    6. Aura and Mystique of Card Counting at Blackjack

    Why this aura of legend surrounding card counting at blackjack? Even more mystique is added when considering that Las Vegas is still barring card counters from playing blackjack! Says Carl Sifakis: “And what of the casinos today? Blackjack is a much bigger game today than it was before "Beat the Dealer" appeared. More people than ever patronize the tables and casinos today make more money from the game than ever before. That's a significant bottom line.”

    Roger Gros, senior editor of Casino Player Magazine writes in his 1996 “Casino Gambling the Ultimate Play-To-Win Guide”: “After all, casinos make most of the money they make at the table games via blackjack. It is great advertising when someone reports a big win at the blackjack tables. Players have been encouraged for many years to believe that blackjack can be beaten, and the casinos don't want to do anything to disrupt that message.” (page 30).

    Methinks the legend of blackjack card counting plays as the most successful means of advertising for the casinos. It certainly attracts a large number of players who believe counting at blackjack is a road to riches. There are also other ways that the card counting legend favors the casinos. Read any blackjack book on card counting and virtually all of them contain the same cliché. “If you are a card counter, make some bonehead plays so that the pit bull won't ban you!” I think many card counters take the advice seriously. They do make bonehead, stupid plays from time to time just to hide their card-counting skills. What a stupid strategy for the blackjack player! What a profitable play for the casino!

    It is admitted that counting cards offers no more than a 2% advantage for the blackjack player. That's a slim margin by all standards. Making bonehead plays can easily wipe out the slim, potential 2% advantage. The casinos owe big time to all authors of card-counting books. Then, in places where it is legal to ban skilled players from the blackjack action, the casinos commit downright robbery.

    From what I have read, the casinos show a strong bias towards barring a blackjack player when he/she is at a serious LOSS! I read once that a known card counter was losing some $14,000. Exactly at that point, the pit boss approached the counter and asked him to leave the blackjack game! Get it? The $14,000 went immediately to the casino bottom line.

    Meanwhile, the player was deprived of a reasonable chance to recuperate his loss. After all, they trumpet, blackjack is almost an even-odds game for a player using the basic strategy! It is fair to expect swings in the winning and losing columns of the player. Of course, the banned player is allowed to play blackjack again. The pit bosses pretend to have forgotten him/her, until another significant loss for the player. The player is thrown out again! If the known counter is winning, the rationale is that a swing in the fortune will follow. Sooner or later, the player will encounter a severe loss. That's when you ask a counter to get out! What casino would be happy if a winner does do take the money and run?!

    You can read some shocking details (2008). It is not only about mathematics: It is about flat-out crime. If you win big, they slam you to the ground and take all your money under the "legal" pretext of BJ card-counting:

  • Legal Cases Against Casino Abuse of Laws.

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    You learned here about blackjack gambling mathematics and the thin foundation of card counting.

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    Ken Uston, Edward Thorp and the legend of blackjack of card counting will not help the gamblers.