A Prof Beats the Gamblers
The advantage in two-handed blackjack, long supposed to lie with Ihe dealer or the house, was converted recently to the profit of the player by EDWARD O. THORP,a young assistant professor in the mathematics department of New Mexico State University. A detailed exposition of his theory for winning at blackjack will be published in book farm next fall by Blaisdell.
GAMBLERS have learned through experience that games of chance can be run in such a Way that a certain percentage favors one side at the expense of the other side. That is, if the game is played a sufficient number of times, the winnings of the favored side are generally near a certain fixed percentage of the total amount of all bets placed by the opponent. The modern gambling casino takes the side of the gambling games which has proved in practice to be favorable. If necessary, the casino alters the rules of the game so that the casino advantage is sufficient to cover expenses and yield a desirable rate of profit on the capital which the owners have invested.
There have been many attempts to overcome the casino advantage. But all of them have the same flaw. The casino always sets a limit to the amount that may be bet. With this limit on bets, the casino wins the same percentage of the gross bets which it normally wins, even though a player uses a complicated betting scheme. It was no surprise, then, when it was proved, by using the mathematical theory of probability, that for most of the standard gambling games no betting scheme can ever be devised that will have the slightest effect upon the casino’s long-run advantage. In view of this mathematical proof and the painful experience of millions of gamblers, informed people and uninformed people alike firmly believe that it is impossible to beat any of the modern casino gambling games.
I was well acquainted with these facts, and therefore I did not harbor the belief that gambling in the casinos was a likely way to make money. I was, however, a frequent visitor in Nevada. One Christmastime during school vacation, just before my wife and I left U.C.L.A. to spend a few days in Las Vegas, a professor called my attention to an article in one of the mathematics journals. The article described a strategy for playing blackjack which assertedly limited the house to the tiny edge of .62 percent. This allows the player an almost even break, so I made up a little card with the strategy on it and brought it along on our trip.
As played in the casinos, blackjack, or twenty-one, involves a dealer employed by the casino and one to six players. After players make their bets, hands of two cards each (hole cards) are dealt to each of the players and to the dealer. The players in turn, and then the dealer, are allowed to draw additional cards. The goal is a total as close to twenty-one as possible without exceeding it. The dealer’s strategy is fixed: he must draw to (hit) totals of sixteen or less and may not draw to totals of seventeen or more. Players can draw or not, as they please. They also have the option of doubling down with their hole cards — that is, they can double their bet and draw exactly one more card.
The numerical value of cards is ten for all tens and face cards; it is as labeled for cards two through nine, and aces may be counted as either one or eleven, as desired. A pair of hole cards with the same numerical value may be split, to form two hands. An additional bet equal to the original one must be placed on the new hand.
Bets are usually paid off at even money. If the player’s total exceeds the dealer’s, he wins. If the totals are equal, it is a tie. If the player has a lower total, he loses. If the player’s total exceeds twenty-one, he “busts” and immediately loses. If the two hole cards of either the player or the dealer total twenty-one, they are termed a natural, or blackjack. Naturals are immediately turned face up and win against all non-naturals. A player is paid at the rate of three to two for a natural. There are many details which have been omitted from this sketch of rules. Also, casinos often have minor variations in these rules.
WHEN I arrived at the blackjack tables I purchased ten silver dollars. This was more than the total amount of all bets I had made in my life up to that time. I was resigned to losing, but I wanted to see how long my stack would last, as well as to try out the strategy under fire.
In a few moments the slowness of my play and the little card in my palm amused and attracted bystanders. The dealer could not conceal his scorn for one more system player. These sentiments were laced with pity when they saw me playing. Who ever heard of splitting a pair of lowly eights and doubling the amount of money being risked when the dealer’s up card was the powerful ace? Who doubled down on ace, two against a five? Why stand on a piteous twelve against a four?
To add to this beginner’s misery, the dealer was having a very strong run of luck. Every player at the table was losing heavily. Surely my ten crumbs would soon be swept away. Or would they? Somehow these weird plays kept turning out right. As the other players lost heaps of chips, my little stack held. It even inched up once.
And then a strange thing happened. I was dealt ace, two. I drew a two, then a three. I now had ace, two, two, three. The dealer had a nine up. The dealer might not have nineteen. Only a fool would draw again and risk the destruction of such a good hand. I consulted my card and drew. With no little satisfaction and several tsk-tsks, the amused onlookers saw me draw a six for fourteen. Serves me right. I drew an ace for fifteen. Tough luck; I drew again. A six! I now held ace, two, two, three, six, ace, six — a sevencard twenty-one. This is an event so rare that it happens only once per several thousand hands.
After a moment of shock, some of the bystanders said I had a twenty-five-dollar bonus coming. The dealer said no, it was only paid at a few places in Reno. I was unaware of such a bonus. But I thought it might be amusing to create the impression that I’d sacrificed my soft eighteen because I foresaw the seven-card twenty-one. The amusement and patronization of some bystanders changed to respect, attentiveness, and even to goose pimples.
After another fifteen minutes and the obliteration by the dealer of all my fellow players, I was behind a total of eight and one half silver dollars and decided to stop. The atmosphere of ignorance and superstition that pervaded my little experience securely planted in my mind the suggestion that even good players didn’t know the fundamentals of this game.
When I returned home, I carefully studied the mathematics article by Baldwin, Cantey, Maisel, and McDermott on blackjack. In a flash of mathematical intuition I realized that it must be possible for the player to beat the game. The idea was this: the basic strategy by Baldwin was a complete set of instructions telling the player the best possible way to play. To simplify the calculations, it was assumed that the deck always had its average composition — that is, that all hands were dealt from a complete shuffled deck. However, when the game is actually played, used hands are placed face up on the bottom of the deck, and subsequent rounds of play come from a progressively more depleted deck. In general, the proportions of the various cards in the depleted deck wall not be the same as in a complete deck. Thus, the casino advantage should fluctuate. Mathematical considerations suggested that the fluctuations should often be much larger than .62 percent, and further, the player should have the advantage frequently. If the player were to bet very heavily when he had an advantage and very lightly when he had a disadvantage, he would not need to have the advantage very often in order to make a handsome profit.
The basic problem, then, was to determine when the player has the advantage and how large this advantage is. Baldwin’s calculations had taken four capable young men a total of twelve manyears of off-duty army time, working with the aid of desk calculators. My first step was to master every detail of these calculations.
The next step was to analyze the effect on the casino advantage when changes were made in the proportions of various cards in the deck. For example, suppose all the aces are removed from the deck. No naturals are then possible, and there are a number of other changes in the game. The result is that the player is seriously hurt.
It would be necessary to make a number of calculations similar to the Baldwin calculation, but because greater precision was desirable, they would each be many times as lengthy. Since ten thousand man-years at desk calculators were required, the problem could be done only by an electronic computer. I had access to the IBM 704 in the M.I.T. computation center. Writing up instructions for the computer turned out for this problem to be a long and tricky business. However, when I finished, it took the computer a mere seven hours to type out the answers — enough numbers to fill an average-size book.
The answers amazed me. The typical casino advantage over a player using the best complete deck strategy, hereafter called the basic strategy, was less than .21 percent. I have since learned that, in fact, the player using the basic strategy has a small edge of about .16 percent over the house, without keeping track of the cards. If various groups of cards were used up in play, the advantage surged wildly back and forth between casino and player. (The player, if he only knew it, has the advantage about half the time.) The four fives cause a bigger swing than any other four cards. When they are gone, the player has an edge of more than 3.3 percent. The next largest effect occurs when the four aces are gone. Then the casino has an edge of about 2.7 percent. When I examined the effect of variations in the proportion of the sixteen ten-value cards present, I learned that player or house advantages of more than 10 percent were frequent. Occasionally the advantage approached 100 percent! I developed a detailed system of play which involved keeping count of the number of ten-value and non-ten-value cards remaining to be played. Experiments with several people showed that twenty hours of practice was generally adequate to train them as ten-count players for casino play.
I THOUGHT that the simpler but less efficient strategy based on counting fives might make an interesting talk at an upcoming annual meeting of the American Mathematical Society in Washington, D. C. A few days before the meeting, the society published abstracts of the two hundred or so talks that were to be given. Included was my abstract describing the fives strategy, “Fortune’s Formula: A Winning Strategy for Blackjack.”
Following my talk, I was asked to give a press conference. Then I was televised by a major network and interviewed on a number of radio programs. Over the next few weeks I received hundreds of letters and long-distance phone calls. The bulk were requests for information. Interspersed among these were several offers to back me in a casino test of my system. The amounts proffered ranged from a few thousand dollars to as much as $ 100,000. Together they totaled a quarter of a million dollars.
The decision was being thrust upon me of whether or not to go to Nevada and test my ideas in play. I finally decided to go. Part of the reason was, I suppose, to silence that irritating question, “Well, if you’re so smart, why aren’t you rich?” Another was that I decided to write a book expounding my ideas, and I felt that the reader should know that in this case theory really works.
What may have finally clinched matters was scoffs and boasts from casinos that my claims were ridiculous. Their arrogance was succinctly summarized by a casino operator being interviewed on a nationwide television program. When he was asked whether the customers ever walked away winners, he said, “When a lamb goes to the slaughter, the lamb might kill the butcher. But we always bet on the butcher.“
The most attractive offer was one of SI00,000 made by two New York multimillionaires whom I will refer to as Mr. X and Mr. Y. They are both large-scale gamblers. Mr. Y has lost $100,000 in one of the casino games without being seriously hurt financially. Mr. X’s gambling activities involve hundreds of thousands, and even millions in profit. At the invitation of Mr. X and Mr. Y, I flew from Boston to New York to discuss the system and to plan a trip to Nevada.
There were two main approaches we could adopt for betting. One, which I’ll term “wild,” involved betting the casino limit whenever the advantage to the player exceeded some small figure, say one percent. This method produces, on the average, the greatest gain in the shortest time. However, in a short run of a few days, the fluctuations in the player’s total capital generally are violent, now favoring the player, now favoring the casino, so a large bankroll is required. Mr. X and Mr. Y said that they would back this to the extent of $100,000, and that they would go further if necessary.
I was not in favor of this approach, since there were too many things I did not know about the gambling world. I also had no idea how I would be affected if I were to get behind, say $50,000, and were betting each minute more than my monthly salary. Further, the purpose of the trip, from my point of view, was to test my system rather than to make big money for Mr. X and Mr. Y, so I preferred being certain of a moderate win, rather than a probable, but somewhat uncertain, big win. I therefore favored conservative play, betting twice the minimum bet I was making when the advantage was one percent, four times my minimum when it was 2 percent, and finally leveling off at ten times my minimum when it was 5 percent or more in my favor. If my bets ranged from $50 to $500 (the highest casino maximum generally available), $6000 or $7000 would probably be adequate capital. To be safe, we took along $10,000.
When the M.I.T. spring recess came, Mr. X and I flew to Reno, where Mr. Y was to join us. We checked into one of the large Reno hotels about 2 A.M. and immediately went to sleep. Early the next morning we investigated casinos.
Our plan was to start small betting, $1 to $10, and gradually increase the amount of the bets as I gained experience. Eventually we planned to bet $50 to $500, always using the ten-count system.
FIRST we drove to a casino outside of town. In an hour or so of play I won a few dollars, and the establishment closed for three hours because of Good Friday, so we returned to Reno. We next investigated a number of casinos in Reno to determine which rules were most favorable. As a nice spot for practicing, we selected a casino that dealt down to the last card and allowed the player to double down on any hand, split any pair, and insure. This is a more favorable set of rules than is ordinarily found.
After a lavish dinner and a rest, I returned about 10 P.M. I began by playing for fifteen or twenty minutes at a time and then resting for a few minutes. When I sat down again, I always chose the table with the fewest players. I also paused for thought and stared at all the cards played. My behavior pattern made it apparent that I was using some system. But system players are frequent, if not common, in the casinos. They are welcome as long as they are losing. Playing $1 to $10, I gradually fell further behind, until at 5 A.M. I was down $100.
Then business fell off sharply, and I was able to get a table completely to myself. My new dealer was particularly unfriendly. When I asked to be dealt two hands, she refused, saying that it was house policy that I must bet $2 per hand to play two hands. This change in the scale of betting would have confused my records of the evening’s play, so I refused. Besides, I was getting tired and irritable. After a sharp exchange of words, she dealt as rapidly as she could.
Just then a one percent advantage arose. I decided to go to the $2 to $20 scale and bet $4. I won, and the advantage coincidentally advanced to 2 percent. I let my $8 ride and won. My advantage rose to 4 percent. I let my $16 ride and won again. I left $20 on the table with the remark that it was time to take a small profit. I continued with $20 bets as the deck remained favorable. At the end of the deck, I had recouped my $100 loss and had a few dollars profit besides.
The doubling-up betting pattern I used in the last few minutes makes no sense in games where the house has the advantage, but in blackjack, with counting methods, it is as profitable as any other way of putting down money at favorable times. Further, since it is so widely and so unsuccessfully practiced, it makes excellent camouflage.
Sandy-eyed and stiff, I woke up early Saturday afternoon and had breakfast. Mr. X and I again visited the casino outside of town. Within minutes, playing $10 to $100, I won two or three hundred. Then Mr. X jumped in. After two hours, we had accumulated $650, and the house began to “shuffle up” — that is, they would shuffle the deck several cards before the end. Favorable situations arise with greatest frequency toward the end of the deck, so shuffling can sharply reduce the rate of profit. Since we were practicing, it seemed discreet to leave and hope we could come back later for a few full-scale hours.
Mr. Y arrived Saturday evening, and we set out to seek our fortune. First we visited one of the most famous clubs in the center of downtown Reno. We began play at the $ 500-maximum tables. (The maximum generally ranges from $100 to $500 in Nevada, varying from casino to casino, and frequently from table to table within a given casino. With our capital, we preferred the highest maximum possible.) In fifteen minutes we won $500 while warming up at $25 to $250. Then our dealer pressed a concealed button under the table with her foot. In a few minutes the owner and his son arrived. There were many pleasantries and politenesses exchanged, but they made their point: the deck would be shuffled as often as necessary to prevent us from winning.
The owners had learned, over the last decade, that some players wait for very special combinations of cards to arise at the end of the deck. Then they sharply up their bet, sometimes going from $1 to $500. These players are stopped by shuffling five or ten cards from the end of the deck.
To be safe, the owners instructed our dealer to shuffle twelve to fifteen cards from the end. Fortunately for them, they waited to see the results. The tens strategy locates favorable situations after the first hand has been played, even as early as after only four cards have been dealt.
A few minor situations appeared and were exploited by us. Then the deck was shuffled twenty-five cards from the end. Still, occasional minor favorable situations arose. Finally the dealer began shuffling forty-two cards from the end, after only two hands had been played. During this twenty minutes of fencing, bad luck in addition to this particular club’s unfavorable rules and the shuffling allowed us to squeeze out only an additional $80, so we stopped.
We next visited the casino in one of the large hotels, where we had been told that a cheat dealer was used on big-money players. After being cheated on the very first hand, we moved on.
At our next stop, the maximum was $300, but this was compensated for by excellent rules: the player could insure, split any pair, and double down on any set of cards. We purchased $200 in chips from the cashier and selected an empty table. I lost steadily, and at the end of four hours of play I was $1700 behind and quite discouraged. However, I decided to wait for the deck to become favorable just once more so that I could recoup some of my losses.
In a few minutes the deck obliged, suddenly producing a 5 percent advantage calling for the maximum bet of $300. Curiously, my remaining chips amounted to precisely $300. As I tried to decide whether to quit if I lost this one, I picked up my hand and found a pair of eights. They must be split. I flung three $100 bills from my wallet onto the second eight. On one of the eights I was dealt a three. I had to double down. So I flung three more $100 bills onto this hand. Nine hundred dollars was now lying on the table, the largest bet I’d yet made.
The dealer was showing a six up and had a ten under and promptly busted. Now I was only eight hundred dollars down. This deck continued favorable, and the next deck went favorable almost at once. In a few minutes I wiped out all my losses and went ahead $255. Then we quit for the evening.
For the second time, the tens system had shown a feature that would appear again and again — moderately heavy losing streaks, mixed with “lucky” streaks of the most dazzling brillance.
The next afternoon the three of us visited the casino outside of town again. Before sitting down to play, I made a phone call. When I came back my friends told me the casino had barred us from play, but that they would be only too happy to pick up our meal tab. I called the floor manager and asked him what this was all about. He explained, in a very friendly and courteous manner, that they had seen me playing the day before and were very puzzled at my steady winning at a rate that was large for my bet sizes. He said that they decided that a system was involved.
We returned to our hotel, and while my friends took care of business, I spent a couple of hours betting $5 to $50 at our hotel’s tables. Despite the annoying presence of a shill, I won about $550. At this point, the pit boss asked me to stop playing at the hotel. He said that this also went for Mr. X and Mr. Y and any other friends.
IT WAS almost suppertime Sunday when the three of us revisited the casino at which I had made the $900 bet. I was warmly remembered as the rich playboy of the night before who had been down $1700 before wriggling off the hook by some quirk of fate. We were invited to dine, courtesy of the house, as a prelude to the evening’s gaming festivities. After two $4 entrees of assorted baked oysters on the half shell and various supporting dishes, capped with wine, I set out somewhat unsteadily for the gaming tables. Within a few minutes, however, I was at peak alertness. After four hours of betting $25 to $300,I was ahead about $2000. I was beginning to tire, so with the utmost reluctance I returned to my hotel.
I remember that casino fondly: the fine cuisine, the spacious attractive modern dining room, and the casino with its little clusters of blackjack tables, the favorable rules, the courtesy and hospitality, and, last but not least, the free money.
We were again ready for action early Monday afternoon. We decided to drive to the south end of Lake Tahoe. About 6 P.M. we arrived at a large brightly lighted gambling factory. It was jammed. I was barely able to get a seat at the blackjack tables.
A few minutes after I placed on the table $2000 worth of chips I had purchased from the cashier, a drooling pit boss rushed over to invite me to dinner and the show. I requested that my two friends be included. In a few minutes, I won $1300, and Mr. X, betting wildly, won $2000. Then we took time out for our free dinner — filet mignon and champagne. Within hours, destiny would present us with a bill for our free dinner. The charge? Eleven thousand dollars.
After dinner we strolled over to another casino. There was a $500 limit and acceptable rules. As usual, I purchased $2000 in chips from the cashier and selected the least busy table. From the beginning I was plagued by $1 bettors who came and went, generally slowing down the game and concealing cards so that they were hard to count.
Whenever one arrived, I pointedly reduced my minimum bet from $50 to $1. After a few minutes the pit boss got the message and asked me if I would like a private table. When I said it would transport me with ecstasy, he explained that the club didn’t like the psychological effect of a private table on the other customers. With a trace of a smile, he said that a $25-minimum game could be arranged, and wondered if that would be satisfactory. I promptly agreed, and a sign to that effect was installed, clearing the table of all customers but me. A small crowd gathered. They were quiet, perhaps anticipating the imminent slaughter of their somewhat plumpish fellow lamb.
After I had won a few hundred, my friend Mr. X jumped in. I then took the responsibility for keeping the count and calling the signals for both of us. Within thirty minutes we had emptied the table’s money tray — the blackjack version of breaking the bank. The smiles of the pit boss were replaced by signs of fear.
The employees began to panic. One of our dealers bleated to her boyfriend higher-up, “Oh, help me. Please, help ine.” The pit boss was trying to explain away our win to a nervous knot of subordinates. While the money tray was being restocked, the crowd swelled. They began to cheer their David on against the casino Goliath.
In two hours, we broke the bank again. The great heaps of chips in front of us included more than $17,000 in profits. I won $6000, and Mr. X, betting wildly, had won $11,000. I was tiring rapidly. The aftereffects of our huge dinner, the increased effort in managing two hands, and the strain of the last few days were telling. I began to find it very difficult to count properly and observed that Mr. X was equally far gone. I insisted that we quit, and I cashed in my $6000. As I did so, I was startled to find three or four pretty girls wandering back and forth across my path smiling affectionately.
After wending my Ulyssean way back to the tables, I watched horror-stricken. Mr. X. refusing to stop play, poured back thousands. In the forty-five minutes or so that it took to persuade him to leave, it cost the two of us about $11,000 of our $17,000. Even so, when we returned to our hotel that evening, we were ahead $13,000 so far on the trip.
Tuesday we paid a series of visits to a downtown club which had bad rules and shuffled five to ten cards from the end. We also had poor luck, and playing $50 to $500, we lost almost $2000.
We remembered that the club where I first practiced so lengthily had excellent rules and made a practice of dealing down to the last card in the deck. We decided to return there. I purchased $1000 in chips and began to win. Within minutes the owner was on the scene.
In a panic, he gave the dealer and the pit boss instructions. Whenever I changed my bet size, the dealer shuffled. Whenever I varied the number of hands I took (I could now play from one to eight hands at a time, and faster than the best dealers could deal), the dealer shuffled. The dealer whom I had played against last in my practice session was standing in the background saying over and over, in reverent tones, how much I had advanced in skill since the other night. Finally I happened to scratch my nose, and the dealer shuffled! Incredulous, I asked her whether she would shuffle each time I scratched my nose. She said she would. A few more scratches convinced me that she meant what she said. I then asked whether any change in my behavior pattern, no matter how minute, would cause her to shuffle. She said it would.
I was now playing merely even with the house, as the shuffling destroyed nearly all my advantage (except that gained from seeing the burned card). I then asked for some larger denomination chips — $50 or $100 — as all I had were twenties. The owner stepped forward and said that the house would not sell them to us. He then had a new deck brought in and carefully spread, first face down, then face up. Curious, I asked why they spread them face down. Although the practice is a common one in the casinos, seldom do they examine the backs of the cards for a couple of minutes, as these people were doing. The dealer explained that it was believed that I had unusually acute vision (I wear glasses), could distinguish tiny blemishes on the backs of the cards, and that this was what enabled me to foretell what cards were going to be dealt. I scoffed, but the owner, still panicky, brought in four new decks in five minutes.
My behavior was unchanged, in spite of the new decks, so they gave up on that. In whispers, they formulated a new theory. I asked them, then, what they thought my secret was. The dealer claimed that I could count every card as it was played, so that I knew exactly which cards had not yet been played at each and every instant. It is a well-known fact to students of mnemotechny (the science of memory training) that one can readily learn to memorize in proper order a deck of cards as it is dealt. However, I am familiar enough with the method involved to know that the information, when so memorized, cannot be used quickly enough for play in blackjack. So I challenged the dealer by rashly claiming that no one in the world could watch thirty-eight cards dealt quickly off a pack and then tell me quickly how many of each type of card remained.
She answered by claiming that the pit boss, next to her, could do just that. I told them I would pay $5 on the spot for a demonstration. They both looked down sheepishly and wouldn’t answer. I made my offer $50. They remained silent and ashamed. Mr. Y increased the offer to $500. There was no response. We left in disgust.
I had proved the system, and the millionaires had business elsewhere, so we agreed to terminate our little gambling experiment. In thirty manhours of mediumand large-scale play, we built $10,000 into $21,000. At no point did we have to go into our original capital further than $1300 (plus expenses). Our experiment was a success, and my system performed in practice just as the theory predicted.
The day of the lamb had come.