Chapter 3

Card-counting is an advanced gambling tool, which could in theory give a player the advantage in almost any card game. Card-counting was developed in the 1950's by Dr Edward Thorp. Professor Thorp is a character so extraordinary that had he not really lived, then as a character of fiction his exploits would have been dismissed as far-fetched. Unquestionably the greatest mind ever to turn his attention to gambling, Thorp has made small fortunes at blackjack, roulette, and sports betting; he also pioneered the incredibly successful "warrant-hedging" technique on the greatest gambling game of all, the stockmarket. Over $65 billion have passed through his hands over the last few decades, and his personal fortune can only be guessed at. He is without doubt the most successful gambler who has ever lived. All of this was due to an extraordinary ability to understand the mathematics of gambling games and devise practical systems to bend chance to his will.

Experimenting with the game of blackjack, Thorp discovered that when you remove certain cards from the deck, this alters the house advantage. He also found that a player could sometimes have the advantage over the house. For example, removing all the 5's from a deck puts the odds in the player's favour. These favourable situations are outnumbered by unfavourable ones, and so on average an unskilled player would lose more than he wins. But, Thorp discovered, if the player knows when he has the advantage, he can bet more than when the house has the advantage, and so win more money than he loses.

Thorp constructed his highly successful ten-count system, with which he won many thousands of dollars from the casinos at blackjack. They retaliated with a series of countermeasures designed to thwart the "Counters", the disciples who read Thorp's bestselling "Beat the Dealer" and tried their own luck with his system. There are still some professional Counters around today, although the casinos are wise to the danger and bar any player they suspect is counting.

How does counting work? Well, the most popular systems are called "point-counts". Cards are given a value according to how good or bad their removal is for the player. Cards whose removal from the deck is bad are given a minus value, good cards are given a plus value. The counter at all times keeps a "running count" in his head which begins at zero. Every time he sees a card he adds or subtracts its value from the running count in his head. When the running count is high (lots of plus cards are dealt) the player has the advantage and should bet more. When it is negative he should bet the minimum or leave the table. The ratio between the players "large" and "small" bets is called his bet spread, and the larger it is, the more he will win.

A typical count looks like this (this is the popular hi/lo count)

Ace Ten nine eight seven six five four three two

-1 -1 0 0 0 +1 +1 +1 +1 +1

It is hard to guess at the amount of money casinos have lost to blackjack card-counters, but the figure is unlikely to be less than tens of millions of dollars. Thorp had discovered that rare thing, a gambling system which actually worked.

So, of what interest is this to baccarat players? Well, it is a little-known fact that card-counting can also be applied to baccarat. Thorp and other gambling mathematicians began searching around for other games to win at. Baccarat seemed the obvious choice. Like blackjack the cards are dealt out until the pack is depleted, before the cards are shuffled. Therefore the odds of the various bets change, as one hand after another is dealt from the shoe.

Thorp and a fellow academic, William Walden, investigated the possibility of applying card-counting techniques to baccarat, their work being recorded at the taxpayer's expense for the benefit of posterity in "A winning bet in Nevada baccarat" (Journal of the American statistical association, vol 73, 1966). The work was an outgrowth of Walden's PhD thesis which Thorp supervised.

Thorp and Walden, with the aid of a computer, determined the precise expectations for the various bets. They then analyzed random subsets of thirteen cards,a typical minimum number of cards remaining in a deck before a shuffle, to see if either player or banker bet was favourable (the tie-bet had not yet been introduced). In only two occasions out of 58 did Thorp and Walden discover any advantage. Once the player had an edge of 3.2%, once the banker had an edge of 0.1%. Clearly, they concluded, no system based on card-counting could yield a practical winning strategy, for the favourable situations were just too infrequent.

There were two main reasons why the application of card-counting techniques would not work at baccarat as they had been proven so successfully at blackjack. Firstly, the game is dealt from eight or six decks, whereas blackjack was originally dealt from only one. The original order of the cards is slower to change with multiple decks. Secondly, the approximately 1% disadvantage the bettor faces on the player and banker bets, while small when compared with most other casino games, is quite large when compared with blackjack, which is typically 0.5% or less (assuming skilled play).

The Effects of removing a single card on an eight-deck Baccarat Shoe

Card Change in % advantage of banker

and player bets

0 -0.002% +0.002%

1 -0.004% +0.004%

2 -0.005% +0.005%

3 -0.007% +0.007%

4 -0.012% +0.012%

5 +0.008% -0.008%

6 +0.011% -0.011%

7 +0.008% -0.008%

8 +0.005% -0.005%

9 +0.003% -0.003%

However, in a footnote to the text, Thorp stated that a strategy might be feasible when technology progressed further, and that in any case he only ruled out strategies based on card-counting, adding intriguingly that strategies based on an analysis of card-shuffling might produce a winning method.

The casinos also at that time offered two side bets in addition to the main part of the game, on whether or not the banker's first two cards would total eight, or total nine, which paid off at nine to one. Under ordinary circumstances these bets had a large house advantage of approximately 5%, but the academics discovered that when there was a large number of eights or nines remaining to be dealt, i.e. less than the average number of nines and eights had been dealt out, these bets became advantageous to the player. They designed a card-counting system to exploit these favourable opportunities. Because these favourable situations were not too frequent, occuring only 10% of the time in every shoe, it was neccessary for a winning system player to raise his bets by a factor of 40 in order to counter the attrition of the many small waiting bets he would have to make.

With a team of trained players the two went to Nevada with a highly successful application of their system. Thorp records in "Beat the Dealer" that the team averaged $100 an hour in the first casino they visited. After five nights Thorp was "rendered spectacularly rubber-legged and goggle-eyedby knockout drops,courtesy of the house" (Life,March 27,1964,pp 80-91). After the seventh night the team were barred. In the next casino they visited they raised their stakes. They averaged $1000 an hour for two hours before they were barred again. Then the side bets disappeared, throughout the state, never to return. Perhaps this was for the best, for in a Las Vegas which still had close connections with The Mob, it is unlikely that the team would have been allowed to go on winning indefinitely.

It seems somewhat surprising that no other card players had discovered for themselves the system which Thorp and his associates used, particulary since the system was so simple, and the importance of card removal in blackjack and the European forms of baccarat was already recognized (though not fully understood). Perhaps they were more adept at concealing their methods.

Shortly afterwards a side bet began to appear on the tables which paid eight or nine to one whenever the result of a hand was tied. This gradually became a standard feature of the game.

Interest in the application of mathematics to baccarat remained largely dormant for the next decade.

David Sklansky, one of the more innovative of gambling writers, discovered that if the last six cards dealt from a baccarat shoe were three twos and three threes, the player bet enjoyed an 80% advantage. The banker bet enjoys a similiar advantage if only sevens and eights remain in the deck. Sklansky stated that card-counting may indeed be possible at baccarat, and suggested devising a count which weighted twos and threes against sevens and eights.

In March, 1982 the "Gambling Times" published a series of six-card subsets which could give the player an advantage at the very end of the deck. This inspired Joel Friedman to investigate all possible six-card subsets. He discovered that a player with computer-perfect knowledge of the last six-cards could gain an average profit of 26% on that hand. By raising his bets fourfold (or more), his profits could outweigh the loss from making the approximately 80 "waiting" bets on the banker the player would have to make in order to earn the right to bet on this last hand. More than 24% of his gain would come from the tie bet. This finding seemed strange considering the tie bet is almost always disregarded by expert opinion as a frivolous wager that only a fool would make, since it is fourteen times less favourable than the bank or player bets. But, as Friedman's study showed, the tie advantage changes much more rapidly than the player or the banker. The tie is like a golden chalice in a snake pit. Read more about baccarat