Edward Thorp’s Baccarat Adventure

The work was an outgrowth of Walden’s PhD thesis which Thorpsupervised. The banker bet enjoys asimilar 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 againstsevens and eights. The tie is like a golden chalice in asnake pit.

It seems somewhatsurprising that no other card players had discovered for themselves thesystem which Thorp and his associates used, particularly since thesystem was so simple, and the importance of card removal in blackjackand the European forms of baccarat was already recognized (though notfully understood). Every time hesees a card he adds or subtracts its value from the running count inhis head. Under ordinary circumstances thesebets had a large house advantage of approximately 5%, but the academicsdiscovered that when there was a large number of eights or ninesremaining to be dealt, i.e. Hediscovered that a player with computer-perfect knowledge of the lastsix-cards could gain an average profit of 26% on that hand. Perhaps this was for the best,for in a Las Vegas which still had close connections with The Mob, itis unlikely that the team would have been allowed to go on winningindefinitely.


2-0.005% +0.005%

In March, 1982 the”Gambling Times” published a series of six-card subsets which couldgive the player an advantage at the very end of the deck. Perhaps they were more adept at concealing theirmethods.


Ace Ten nine eight seven six five four three two

So, of what interest isthis to baccarat players? Well, it is a little-known fact thatcard-counting can also be applied to baccarat. They retaliated with a series ofcountermeasures designed to thwart the “Counters”, the disciples whoread Thorp’s bestselling “Beat the Dealer” and tried their own luckwith his system. He also found that aplayer could sometimes have the advantage over the house. Cards aregiven a value according to how good or bad their removal is for theplayer. Because these favourable situations were nottoo frequent, occurring only 10% of the time in every shoe, it wasnecessary for a winning system player to raise his bets by a factor of40 in order to counter the attrition of the many small waiting bets hewould have to make. The original order of the cards isslower to change with multiple decks. In only two occasions out of 58 did Thorp andWalden discover any advantage. But,as Friedman’s study showed, the tie advantage changes much more rapidlythan the player or the banker. Like blackjack the cards are dealtout until the pack is depleted, before the cards are shuffled.Therefore the odds of the various bets change, as one hand afteranother is dealt from the shoe. But,Thorp discovered, if the player knows when he has the advantage, he canbet more than when the house has the advantage, and so win more moneythan he loses.

4-0.012%+0.012%. This gradually became astandard feature of the game.



Thorp and Walden, withthe aid of a computer, determined the precise expectations for thevarious bets. This findingseemed strange considering the tie bet is almost always disregarded byexpert opinion as a frivolous wager that only a fool would make, sinceit is fourteen times less favourable than the bank or player bets.

However, in a footnote tothe text, Thorp stated that a strategy might be feasible whentechnology progressed further, and that in any case he only ruled outstrategies based on card-counting, adding intriguingly that strategiesbased on an analysis of card-shuffling might produce a winning method.

0-0.002% +0.002%

Experimenting with thegame of blackjack, Thorp discovered that when you remove certain cardsfrom the deck, this alters the house advantage. Over $65 billion have passed through his handsover the last few decades, and his personal fortune can only be guessedat. When it is negativehe should bet the minimum or leave the table. After five nights Thorpwas “rendered spectacularly rubber-legged and goggle-eyedby knockoutdrops,courtesy of the house” (Life,March 27,1964,pp 80-91).After the seventh night the team were barred. By raisinghis bets fourfold (or more), his profits could outweigh the loss frommaking the approximately 80 “waiting” bets on the banker the playerwould 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. They averaged $1000 an hour for twohours before they were barred again. Professor Thorp is acharacter so extraordinary that had he not really lived, then as acharacter of fiction his exploits would have been dismissed asfar-fetched.

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

With a team of trainedplayers the two went to Nevada with a highly successful application oftheir system. less than the average number of nines andeights had been dealt out, these bets became advantageous to theplayer. They then analyzed random subsets of thirteen cards,atypical minimum number of cards remaining in a deck before a shuffle,to see if either player or banker bet was favourable (the tie-bet hadnot yet been introduced). The counter at all timeskeeps a “running count” in his head which begins at zero. Unquestionably the greatest mind ever to turn hisattention to gambling, Thorp has made small fortunes at blackjack,roulette, and sports betting; he also pioneered the incrediblysuccessful “warrant-hedging” technique on the greatest gambling game ofall, the stockmarket. In the next casino theyvisited they raised their stakes. Thorp and other gamblingmathematicians began searching around for other games to win at.Baccarat seemed the obvious choice.


Card-counting is an advanced gambling tool, which could in theory givea player the advantage in almost any card game. Then the side bets disappeared,throughout the state, never to return. Thorp records in “Beat the Dealer” that the team averaged$100 an hour in the first casino they visited.

CardChange in % advantage of banker

Thorp constructed hishighly successful ten-count system, with which he won many thousands ofdollars from the casinos at blackjack. Once the player had an edge of 3.2%,once the banker had an edge of 0.1%. There are still some professional Counters aroundtoday, although the casinos are wise to the danger and bar any playerthey suspect is counting. Card-counting wasdeveloped in the 1950s by Dr Edward Thorp. Cards whose removal from the deck is bad are given a minusvalue, good cards are given a plus value. This inspiredJoel Friedman to investigate all possible six-card subsets. They designed a card-counting system to exploit thesefavourable opportunities. He is without doubt the most successful gambler who has ever lived.All of this was due to an extraordinary ability to understand themathematics of gambling games and devise practical systems to bendchance to his will.

Chapter 3

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


and player bets

(An extract from the book Baccarat For The Clueless)

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

How does counting work?Well, the most popular systems are called “point-counts”. 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, andso on average an unskilled player would lose more than he wins. Clearly, they concluded, no systembased on card-counting could yield a practical winning strategy, forthe favourable situations were just too infrequent.

There were two mainreasons why the application of card-counting techniques would not workat baccarat as they had been proven so successfully at blackjack.Firstly, the game is dealt from eight or six decks, whereas blackjackwas originally dealt from only one.

David Sklansky, one ofthe more innovative of gambling writers, discovered that if the lastsix cards dealt from a baccarat shoe were three twos and three threes,the player bet enjoyed an 80% advantage. The ratio between theplayers “large” and “small” bets is called his bet spread, and thelarger it is, the more he will win.


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

It is hard to guess at theamount of money casinos have lost to blackjack card-counters, but thefigure is unlikely to be less than tens of millions of dollars. Shortly afterwards a sidebet began to appear on the tables which paid eight or nine to onewhenever the result of a hand was tied. Thorphad discovered that rare thing, a gambling system which actuallyworked.

The casinos also at thattime offered two side bets in addition to the main part of the game, onwhether or not the banker’s first two cards would total eight, or totalnine, which paid off at nine to one.

Thorp and a fellowacademic, William Walden, investigated the possibility of applyingcard-counting techniques to baccarat, their work being recorded at thetaxpayer’s expense for the benefit of posterity in “A winning bet inNevada baccarat” (Journal of the American statistical association, vol73, 1966). When the running count is high (lots of plus cards are dealt)the player has the advantage and should bet more. Secondly, the approximately 1%disadvantage the bettor faces on the player and banker bets, whilesmall when compared with most other casino games, is quite large whencompared with blackjack, which is typically 0.5% or less (assumingskilled play)

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