Author: Arnold Snyder

Edward O. Thorp in Las Vegas, 1962

By Russell T. Barnhart

(From Blackjack Forum Volume XX #1, Spring 2000)
© 2000 Blackjack Forum

In the early 1960s much publicity occurred concerning a 28-year-old professor of mathematics, Edward O. Thorp, first of the Massachusetts Institute of Technology (MIT) at Cambridge, then of New Mexico State University at Las Cruces, finally of the University of California at Irvine, who was making blackjack history.

I read articles about him in the New York Herald Tribune (January 29, 1961), the Paris edition of the New York Herald Tribune (by Tom Wolfe, December 9, 1962), the Atlantic Monthly (January, 1963), Time (January 25, 1963), Sports Illustrated (January 13, 1964), Life (March 27, 1964), The New York Times (April 3, 1964), and so on.

What was the wherefore of all this publicity? Here’s how the Time article explained matters:

The omnipresent computer, whose attention often seems to be concentrated on the welfare of moon travelers and submariners, may have at last produced a palpable boon for the common run of mankind: a system for winning money in a gambling house.

A 30-year-old mathematics professor named Edward O. Thorp claims to have made this historic breakthrough by feeding the equivalent of 10,000 man-years of desk-calculator computations into an IBM 704 computer and arriving at a set of discoveries about the way the odds fluctuate in the game of blackjack, or twenty-one.

This system enables the initiate to bet heavily when the odds are with him, lightly when they are against him. What’s more, the cost of the system — including a set of palm-sized, sweat-resistant charts to take to the casino — is only $4.95, which happens to be the cost of Thorp’s book, Beat the Dealer (Blaisdell [Random House]). (By this year, 2000, more than 700,000 copies have been sold.)

Hard Hands & Soft. Thorp’s system is based on the fact that blackjack is not what mathematicians call an “independent trials process,” in which, as in craps or roulette, each play is uninfluenced by the preceding plays. As each card is played in blackjack, it changes the possibilities for both player and dealer by diminishing the number and the variety of cards that may be dealt.

Hence the basic blackjack strategy, according to Thorp’s computer, is that the fewer cards valued at two to eight that are left in the pack, the greater advantage to the player. On the other hand, a shortage of nines, tens, and aces gives the dealer an advantage. [In other words, high- or ten-value cards favor the player.]

A scarcity of fives, Thorp’s figures indicate, is more advantageous to the player than a shortage of any other card; when all four fives have been played, the player has an edge of 3.29% or, expressed roughly in odds, 52-48 in the player’s favor. Thorp has devised a series of charts to show when to split a pair (‘always split aces and eights, never split fives and tens’), when to double and when to stand.

Knowing when to stand and when to ask for another card is, of course, the very heart of the game….

This is Thorp’s “basic” strategy; his “full-dress” system [the Ultimate Strategy] involves a much more complex technique of betting in terms of the number of tens, aces, and fives remaining in the deck in relation to the number of cards left in the pack before the next shuffle.

The Plan that Made Blackjack History

As an individual who has always had trouble finding even the On button both on a computer and in life itself, naturally all this intrigued me. Hence before Beat the Dealer was published in late 1962, I approached a life-long friend of mine, William F. Rickenbacker (1928-1995), a businessman, writer, and journalist, to whom I referred at the time as Mr. X, and suggested the following gambling plan, which was to make blackjack history.

If Rickenbacker and one or two others would together put up a bank of $10,000, why could not Edward O. Thorp, with his remarkable blackjack system favoring the player, and Mickey MacDougall (1906-1996), with his expertise in protecting gamblers against cheating casinos, along with Russell Barnhart, his humble servant and stake holder, travel to Las Vegas to earn for us all an honest, capitalistic dollar?

Next I telephoned Mickey MacDougall, who lived in Forest Hills, Queens, and if Henri Daniel, founder of the first magazine for gamblers in Monte Carlo, made his living by repairing typewriters, Mickey MacDougall made a far more lucrative one by being a wholesale and retail numismatic dealer. It turned out that, as MacDougall would be in Boston the following weekend to attend the coin show of the American Numismatic Association, it would be convenient for him to meet Professor Thorp at the latter’s home in nearby Cambridge.

Finally I telephoned Professor Thorp, introduced myself politely, and suggested that he might like to meet Mickey MacDougall and me to hear of our tentative plan to go to Las Vegas and win money using Thorp’s new blackjack card-counting system. As Thorp was also amenable to the meeting, the following Sunday we met for the first time at his home in Cambridge.

The upshot of our meeting was that the three of us agreed to work cordially together to beat the blackjack tables on a trip to Las Vegas. That was our optimistic and rational plan, and I so informed Mr. X (not to be confused with the Mr. X of the first trip to Las Vegas, described in Chapter 6 in Thorp’s Beat the Dealer, a gentleman named Manny Kimmel). What follows is what actually happened.

Mickey MacDougall’s Role

Now let’s turn once more to Mickey MacDougall. For 20 years, MacDougall wrote for the McClure Syndicate on gambling and other subjects, a national newspaper column, “The Inside Straight,” and what follows is part of a column he wrote after our trip to Las Vegas. Titled, “Even ‘Honest’ Vegas House Cheats,” it appeared on December 2, 1962, in 193 American newspapers, including The Sunday Star-Ledger, Newark, NJ:

. . . Professor Edward O. Thorp . . . wanted me to accompany him on a trip to Nevada . . . . certain he had evolved a system whereby the player would beat the house at the blackjack tables, wanted me along to make certain he didn’t get cheated.

I told the Professor he was mistaken on two counts. (1) No one could beat the dealers at their own game, and (2) the better casinos never cheated, so even if his system did work, he wouldn’t need me to protect him from chicanery.

I had previously played at every temple of luck in Las Vegas as a special investigator for the Nevada Gaming Control Board. Like every other agent, my bets were limited, seldom exceeding $20.

However, Dr. Thorp’s system necessitated wagers of $500. And I learned a bitter lesson. Some casinos which wouldn’t think of cheating a small bettor have no qualms about taking an unfair advantage of a big bettor.

I have in mind one particular gambling house [the Riviera] . . . . which gets world-wide favorable publicity . . . . I recommended it highly to Dr. Thorp .… The very next day the cheating began…. I noticed the pit boss signal an idle dealer and whisper some instructions . . . . I edged nearer, overheard the dealer say: “Okay, I’ll give it to ’em . . . .”

A stout man, whom I had seen working as a shill at the dice games, came hurrying down the aisle [and] sat down directly to the right of Dr. Thorp.

On the very next hand Dr. Thorp was dealt eleven. Automatically he turned his cards face up, shoved out $100 to signify he was going down for double.

The newcomer played first . . . . He drew a card, a queen. He drew a second card, a ten. Then he tossed his hand in, face down, so I couldn’t see the cards, and said, ‘I’m busted . . . .’ I thought then that the shill was an “anchor man,” put into the game to draw cards when it would injure Dr. Thorp’s play, or to stand pat when that would be more advantageous for the house.

On the next hand . . . . I watched the dealer’s eyes [This is the key to MacDougall’s method of catching cheaters.] Sure enough, he constantly glanced at the pack in his hand, at the same time bending the corner of the top card so he could see its index . . .

I gave Dr. Thorp the signal that I was suspicious . . . . Soon the anchor man got careless. I could glimpse his cards. Once he stood on eleven, another time he drew to nineteen . . . .

Dr. Thorp, the modern Wizard of Odds, has published a book detailing his winning system, Beat the Dealer. Now, to balance the scales, a Nevada gambler [casino owner] should write a book detailing crooked gambling methods and entitle it, Beat the Player.

Of all Mickey MacDougall’s articles on crooked gambling casinos, perhaps one of the best, explaining both the founding and operation of the State of Nevada Gaming Control Board, the police arm of the State Gaming Commission, as well as sleight-of-hand methods using false dice and cards, is “Why Nevada’s Gamblers Toe the Mark” (Look, October 28, 1958), an article I recommend.

As MacDougall’s article on one Strip casino’s use of an anchor man against Thorp, just quoted in part, wasn’t published , of course, until a few months after our blackjack trip to Nevada, when we three arrived in Las Vegas on Tuesday, January 23, 1962, none of us knew what fate held for us at even the best casinos on the Strip, so our optimism, though tempered, ran fairly high. Our trip lasted until February 1st, when we departed Reno for our several homes.

That First Historic Blackjack Trip

Our first setback occurred when all three of us came down with colds brought on by our ignorance, in the forty-degree cold, of how to operate the heating control buttons in our individual rooms in the City Center Motel on Fremont Street. MacDougall began taking anti-histamine tablets, I visited the local hospital emergency center to obtain sulfa tablets, and Edward O. Thorp tried both. A bad head cold hardly helps to count cards under casinos conditions.

I hired a self- drive car, a light blue Rambler, to drive us from one casino to another. “You’re our wheel man,” declared MacDougall whimsically, who favored the rear seat, the initiative Thorp always sitting on my right so he could see, figuratively and literally, where we were going.

In the Tables on pages 14 and 15, I list the amounts that we won or lost at blackjack at each gambling casino, reserving my text for a brief discussion of what happened to us at each of these so-called temples of chance in both Las Vegas and Reno. At all casinos Thorp played his Ultimate Strategy of counting aces-tens- and-others, which he explains at length in Beat the Dealer.

As all of us were quite tired from traveling to Las Vegas, on our first evening we ate down the street at the El Cortez Casino and then decided, for our financial backers, Mr. X and his partners, to test the waters with only small stakes, so at the El Cortez, Thorp sat down and played his count system for an hour. In wagers his spread was from $1 to $5 as we note in the first entry on page 14. Already suffering from a worsening head cold, MacDougall kept on his overcoat and usually sat on Thorp’s right to watch for any cheating by a dealer.

In contrast, when MacDougall was working as an undercover agent for the Gaming Control Board, he would naturally never sit, even when alone, and stare with even casual suspicion at a dealer’s hands as the latter dealt the cards. Instead, as he explained to me, to catch any cheating he employed only casual, peripheral vision. “There are fifteen different ways they can cheat you at blackjack,” he explained. “I sit there and casually bet and, in my mind, just go down my list, eliminating one by one the methods the dealer isn’t using until I come to the one I see him or her using — if any.”

This brings to mind an amusing experience MacDougall had on another trip at the Desert Inn.

“It was late in the evening, and I was playing blackjack for minimum stakes,” the Gambling Detective recounted to me. “I wasn’t winning or losing much, and the dealer, though odd in mannerisms, hadn’t been cheating so far, so possibly to draw him out, I said…

“‘I’m glad this is an honest house. I didn’t do so well at the Sands though I suppose it was just bad luck.’

“‘We’re an honest house because the Gaming Control Board sends in undercover agents to watch us to see if we cheat.’

“‘They do?’

“‘Yes. One of their main agents,’ he whispered confidentially, ‘is a man called Mickey MacDougall. So every night I keep a look-out for him on my shift.’

“‘What does he look like?’ I whispered back.

“‘I’m not sure. They say kind of tall…’ [Note: MacDougall was actually a rather short man whose appearance was mundane. He had the air of an ordinary businessman — as that’s what he was.]

“‘There are lots of guys that look like that. How’ll you know which one is he?’

“‘By the way he bets. I can always tell an agent.’

“‘How?’

“I told you — by the way they bet. Agents always bet like shills — only one dollar at a time. They just buck us to death.’

“Naturally I reported this conversation to the Gaming Control Board,” concluded MacDougall, “and from then on,every agent has been given enough money so he can spread at least a little bit — no more ‘bucking them to death.’”

Be that as it may, while Thorp and MacDougall sat playing blackjack cooperatively at the El Cortez, what was I doing? I just wandered around this small casino room as unobtrusively as possible, betting small amounts on red and black at roulette, or on two dozens with my Creep Along system, while keeping a weather eye on the pitbosses and various dealers, none of whom paid me any heed.

But why should they have? I did nothing unconventional, being, after all, the most unimportant individual of our gambling trio. Indeed, inconspicuous was my appropriate manner in all subsequent casinos though I often joined MacDougall to peer over Thorp’s shoulder to watch the master play and see if I could detect any cheating on the dealer’s part.

After an hour of warm-up play at the El Cortez we’d lost $33. So for a second hour of play we went down to the Fremont Casino, where Thorp spread from $5 to $50 and lost $100. Then we went for a quarter of an hour to Benny Binion’s Horseshoe Casino, where the Professor of Mathematics spread again from $5 to $50 and won $40.

At this point, as all our colds had worsened, we decided to call it a night and returned to the City Center Motel for a restless sleep in the deep freeze. All told, by the end of our warm-up night, we had lost $93.

The foregoing figures comprise, for our first day of gambling, the results I’ve listed in the Tables on pages 14-15, to which the reader may refer as I discuss matters concerning our remaining eight gambling days in Las Vegas and Reno.

Methods of Casino Cheating at Blackjack in 1962

As Mickey MacDougall demonstrated them to Thorp and me, here are the main ways that the casinos used to cheat us frequently on our trip to Nevada to play blackjack. The reader will find more extensive elucidation comprising Chapter 7, “How to Spot Cheating,” in Thorp’s cited book on blackjack, Beat the Dealer, whence my own name was omitted by Thorp at my personal request, for to become known hobbles further learning and winning.

(1) The anchor man. This method was explained in MacDougall’s article quoted earlier in part. See also Chapter 7 in Beat the Dealer, in which reference to Mickey MacDougall and our blackjack trip begins on p. 133.

(2) The peek and deal. MacDougall elucidated this method as he saw it during his three subsequent trips to Hot Springs, Arkansas. The six illegal casinos were run by the New Orleans mafia according to MacDougall; I myself visited Hot Springs twice.

(3) The turn over. When the dealer holds a deck in his left hand, he openly places discarded cards cumulatively face up under the face-down ones on top. As he picks up these discarded cards, however, he secretly stacks them in an order later favorable to himself. Between hands the dealer slowly inches the pack down in his left hand and out over his left fingers preparatory to the rapid sleight-of-hand move. Then as he reaches with his free right hand to pick up the discarded cards of some last hand, he snaps the deck leftward extremely quickly, thereby invisibly turning it over.

According to MacDougall, this sleight occurs often about half way through a whole deck deal, and the warning tip-off is the inching down of the pack, which one may perceive. Thus the stacked cards, formerly uselessly face up, are now face down on top, ready to be dealt out to the dealer’s advantage. I saw this method of cheating used subsequently at the illegal Automobile (gambling) Club in Nashville, Tennessee.

(4) The high-low pick up. As the dealer overtly picks up the cards discarded from each player’s hand, he covertly and unobtrusively stacks them in the following order favorable to himself: high-low, high-low, high-low. After a false shuffle, if there be an odd number of players, everyone, including the dealer, will get a bad hand consisting of one high and one low card, for a mediocre total like, say, 15. Naturally everyone will ask for another card, and the next high card will put him over 21.

True, the dealer must bust too, but as all the players have lost before him, he’ll thereby win. If there be an even number of players, knowing the stacked high-low order, the dealer, through sleight-of-hand, merely deals the second card from the top, thereby righting the order, rendering it favorable to himself. See Beat the Dealer, p. 132, in which Thorp states that, in his opinion, during our nine-day trip to Nevada the high-low pick up was “the most widespread single method of dealer cheating.”

(5) Marked Cards. MacDougall elucidated on this method as used during his three trips to Hot Springs, Arkansas. To both Thorp and me he confessed that, as an undercover agent in Nevada, he often saw casinos using the Bee brand, sort-edge deck, both along the Strip, in downtown Las Vegas, as well as in Reno and elsewhere in the state, like Winnemucca, Elko, etc.

(6) The Kentucky or seven-card step up (and variations). Ahead of time the dealer stacks the following eight cards thus: 9, 10, 10, J, and 10, 10, Q, A. After the usual false shuffle, the dealer crimps or bridges lengthwise the deck. If the player cuts to the crimp or bend, then on the first hand, he gets 19, while the dealer gets 20. On the second hand the player gets 20, while the dealer gets 21.

While picking up the discards on both hands, the dealer rights their order and crimps again the last card, thereby readying the stack again for the next deal. On the other hand, if the player doesn’t cut to the crimp, no matter, for when the dealer perceives during the deal the crimped or subtly bent card, he may adjust the order of the stack if need be by skillfully dealing the second card one or more times. Remember that, to himself, the dealer often deals a good hand or, to the player, a bad hand, letting chance deal a favorable second hand to himself. In other words, he doesn’t employ the second deal compulsively.

To Thorp and me MacDougall explained that the Kentucky step up was invented by a crooked dealer name Charley in an illegal gambling house in Newport, Kentucky (near Covington). “He was very proud of his invention,” declared MacDougall.

I asked MacDougall if he’d ever seen the Kentucky step up in the casinos in Hot Springs, Arkansas. “No, never,” he replied, “just the peek and deal.”

(7) The Nevada step up. Ahead of time the dealer stacks thirteen cards of various suits in the ordinary serial order thus: 2, 3, 4, 5… 10, J, Q, K, A. Although this step up may be obviously substituted for the Kentucky step up, its great drawback is that, owing to its obvious regularity of order, the dealer may not use it a second time in a row.

(8) The short shoe. This method of cheating wasn’t used against us three gamblers in our January, 1962, trip to Nevada, because at that time almost all blackjack games were dealt from a single deck held in the dealer’s hand. When a casino wants to use several decks — commonly 4, 6, or 8 — it employs a regular dealing shoe, and these became common in Nevada in the 1970s. According to MacDougall, the Gaming Control Board advocated the use of shoes to preclude the sleight-of-hand cheating methods which had theretofore become rampant.

After Mickey MacDougall explained the short shoe to me around 1975, with his permission I explained it in turn to Edward O. Thorp, because its elucidation would require mathematics way beyond my ability and training. So besides writing a magazine article on the short shoe, Thorp included his computational results in Chapter 2 of his book, The Mathematics of Gambling (Lyle Stuart, 1984), which I recommend.

According to MacDougall, the short shoe remained the favorite way of cheating by a casino whenever it chooses to deal from several decks.

Blackjack History, Day 2 (Or How Professor Thorp Missed MacDougall’s Exit Signal)

Let’s return to what happened on my trip with Professor Thorp and Mickey MacDougall in January, 1962. For the extent of Thorp’s spread and the sums of our wins and losses, I again refer the reader to the Win/Loss Tables, pages 14 and 15.

As I mentioned, on our first evening, in our warm up at the El Cortez Casino, Thorp lost $33, and then at the late Benny Binion’s Horseshoe Casino, he won $40.

On our second day, January 24th, as MacDougall suspected that at the Dunes the dealer was using the high-low pickup, we soon left. “Better safe than sorry” was his admirable motto in these matters, for although it was indeed our goal to seek out an honest dealer in a game in order to gain Thorp’s average plus PC of about 3% from his blackjack count system, by the same token it obviously behooved us to scuttle a dealer and his casino when MacDougall reasonably suspected that dealer of cheating. In other words, we always tried to strike an intelligent balance between, at one extreme, baseless suspicion that carks the soul, and, at the other, an equally absurd and inevitably disastrous faith in a casino’s honesty.

And after MacDougall had demonstrated, on more than one occasion, to Thorp and me the sleight-of-hand and other cheating methods that I’ve already enumerated, we two other gamblers would have had to be blind not to detect them occasionally ourselves when a crooked dealer employed them.

Once, for example, when we three had just come out of a rear door to the parking lot of the Desert Inn, MacDougall turned to Thorp and said, “Go back and look at the dealer shuffling at that last table and then tell me what you think.” In the meantime, MacDougall and I loitered by our light-blue Rambler. After a short interval, a cynically smiling Thorp rejoined us: “I think sometimes he’s doing the high-low pick up, and sometimes he’s not. Then he uses the push-through shuffle.” MacDougall smiled paternally: “That’s exactly what I thought myself.”

So we got into the Rambler and drove down the Strip to the Hacienda Casino. Prior to Thorp’s play the evening before at Binion’s Horseshoe Casino, MacDougall had advised Thorp that if he, MacDougall, perceived any cheating, he would poke the mathematical gambler between the shoulder blades and announce: “Let’s get a bite to eat.” That became our signal to leave a casino at once. So when MacDougall saw the blackjack dealer at the Hacienda peeking at his top cards, fearing the next move would be a second deal and another loss to our beneficent Mr. X and his partners, he accordingly poked Thorp between the shoulder blades.

Unfortunately in the heat of his concentrated play Thorp had forgotten that “Let’s get a bite to eat” was a dire signal indeed, so he replied indifferently, “No, I’m not hungry yet.” Again, MacDougall poked the mathematician between the shoulder blades. “I already told you,” replied Thorp irritably, “I’m not hungry yet!” Given this misunderstanding, when we left the Hacienda, we’d lost $60, which sum the reader will duly note as the fourth entry under Day 2 in the Tables.

I mention this trifling misadventure, because it’s most important for us all to realize that at one time or another all of us become fallible under fire, and anon I’ll give much more important illustrations of my own vanity, ignorance, and ineptitude.

Blackjack History, Day 3

On our third gambling day, January 25th, while my two partners were jousting in the arena of aces-tens-and-others, I drove over to meet in his boarding house the elderly host at the Sands, Frederick O. Brown, who’d written an autobiographical pamphlet about his own gambling experiences. Once a year, in the annual Las Vegas Helldorado parade, Frederick O. Brown would stand up in a convertible and, waving to the spectators, lead the parade.

Although I liked the effusive Brown, when I asked him courteously about his claim in his pamphlet that he’d become the confidant of the late King Farouk of Egypt and that both of them, seated side by side at the baccara and roulette tables at the Monte Carlo Casino had been royally cheated there, he admitted having made up the whole episode. He’d never even met Farouk. I mention this trifling incident, because sometimes knowledge of the honorability of a casino arises from some chance remark or fugitive pamphlet like Brown’s and requires verification.

Indeed, I’d like to go on record as saying that I myself have never heard of anyone who could adduce evidence of any cheating whatsoever by the Monte Carlo Casino. In my own opinion, it is now, and always has been, a perfectly honest gambling house — one of the reasons I chose to gamble there.

Meanwhile, back at el rancho Vegas, as it were, Thorp and MacDougall had gone on by themselves to win $180 from the Silver Palace, only to lose back $176 of it to the Stardust, as per the Win/Loss Tables. They told me later that at the Silver Palace, although the dealer had dealt honestly, the pitboss had been visibly upset by their winning even $180.

In Las Vegas and Reno each dealer’s shift lasted for twenty minutes, so my two partners had become suspicious when the dealer hadn’t gone off his shift after the usual twenty minutes, especially after they saw the pitboss go and telephone someone. After thirty minutes with the first dealer they saw a second man burst through the front door of the Silver Palace, and without stopping even to tie on the regulation green apron he came over and tapped the first dealer on the shoulder to relieve him.

Feeling that the second dealer was probably a crooked dealer on call — in the cheating trade, called a mechanic — my two partners left immediately, following Mac- Dougall’s prudent motto, “Better safe than sorry.”

[For confirmation of how corrupt this era was, see Easy Money: Inside the Gambler’s Mind by David Spanier (London, 1987), and especially his All Right, Okay, You Win: Inside Las Vegas (London, 1992 and 1993); see also the book, Casino: Love and Honor in Las Vegas, by Nicholas Pileggi (New York, 1995) and the movie by the same name about Lefty Rosenthal (at the Stardust Casino).]

At this point, I rejoined them, and MacDougall related to Thorp and me that, as on a trip to Las Vegas six months earlier, when he’d detected the Silver Slipper using the Kentucky step up, he wanted to show it to the two of us as “the epitome of what I’ve been trying to teach you two guys.” So we got into the Rambler and drove back to the Silver Slipper. After my two partners got out of the automobile and went into the Silver Slipper, I had some trouble finding a parking place, taking a bit longer than usual. Then I too got out of the Rambler and began walking toward the casino’s front door.

Lo and behold, out popped MacDougall and Thorp, grinning.

“Hey — where are you two guys going?” I cried. “Mickey — you promised to show Ed and me somebody in there using the Kentucky step up.”

“And we saw it,” exclaimed Thorp.

“But it took me only two minutes to park the car.”

“That’s when we saw it,” explained MacDougall smiling.

Sure enough, the two adventurers had gone into the Silver Slipper and walked directly to a blackjack dealer standing behind an empty table. They didn’t bother even to sit down. Thorp made his two bets standing up. On his first hand he bet $1: he got 19 and the dealer got 20. On his second hand he bet another $1: he got 20, and the dealer got 21.

At that point they thanked the chuckling dealer for his cooperation and left the casino, bumping into me as I was heading toward its front door. So all three of us went over and got back into the Rambler. “It cost Mr. X two dollars,” commented MacDougall, comfortably adjusting himself in the back seat, “but wasn’t the lesson worth it? ‘Seeing is believing.’”

Blackjack History: Professor Walker Joins Team Thorp

On our fourth gambling day, January 26th, as we’d won $738 on the day before, Thorp suggested to me the option of our inviting Professor Elbert Walker, one of his mathematical colleagues at New Mexico State University, to join us, thereby doubling our win rate. Thorp described Walker as being as good a counter as he was.

My new role was to stand behind Walker and spot any cheating. In other words, with Walker I was to take the role that Mickey MacDougall took with Thorp.

I mentioned earlier that I would cite an illustration of my own vanity, ignorance, and ineptitude, and herewith is an outstanding example. For as the official and responsible stake holder representing our beneficent Mr. X and his partners I should have had the mature judgment to hold fast to the single partnership approach which had so far produced tangible financial rewards for us all. And in this regard it must be most importantly borne in mind, that had we not been so frequently and unquestionably cheated by practically every major casino on the Strip, in the Win/Loss Tables the plus figures in the Net column would have been far larger, and by the same token, the minus figures far smaller.

Be that as it may, like a perfect fool, instead of conservatively protecting Mr. X, I acceded, largely through vanity, to the option of Professor Walker’s joining us and flying up to Las Vegas at Mr. X’s expense. Thus from our fifth gambling day, January 27th, through our seventh, January 29th, a period of three days, our so-called syndicate was composed, on the one hand, of Thorp and MacDougall, and on the other, of Walker and Barnhart. [In Beat the Dealer, neither Professor Walker’s name nor mine is mentioned.]

As it turned out, under fire, spreading from only $1 to $10, Walker wasn’t so proficient as Thorp himself, for in the middle of a deal, Walker, alas, would often freeze and forget his count. Yet Walker’s lack of experience under fire wasn’t at all the prime cause, in my opinion, of our debacle — it was my own absurd vanity of considering myself, a mere tyro in the harsh world of card and dice cheating, sufficiently knowledgeable and competent under casino conditions to detect cheating and protect Walker from cheating dealers.

That was the chief cause of Walker’s and Barnhart’s losing money at the tables. Try as I did, I was simply never competent enough to perceive when a dealer may or may not have been cheating Professor Walker. To be as proficient as Mickey MacDougall as a gambling detective requires a lifetime of experience under fire. In considering Thorp’s proffered option concerning Professor Walker I should have immediately admitted to myself my own lack of experience. I did not do so.

The wins and losses of Professor Walker I haven’t included in the Win/Loss Tables, probably because I don’t care to look at them again myself. Although Walker was a fine mathematician, a congenial, soft-spoken Texan, friendly and cooperative, for the sake of Mr. X’s dwindling resources, after our three losing days as a second count team I had to see Walker off on a plane back to New Mexico State University in Las Cruces.

And what of MacDougall and Thorp, Mr. X’s premier count team? As these two men had won relatively large sums at the Riviera Casino, they decided to try it again, preferring its tranquility and lack of crowds to the Thunderbird, the deafening clamor of whose lounge acts disturbed them. So to the Riviera they went, but it was at this casino that they were cheated by Fatso, the anchor man, who even followed them invidiously from table to table. For details of this casino’s predatory response I refer the reader to MacDougall’s article, “Even ‘Honest’ Vegas House Cheats,” of December 2, 1962, which I quoted in part earlier.

Disgusted with the anchor man at the Riviera Casino, MacDougall and Thorp moved to the Desert Inn, where MacDougall again spotted the Kentucky step up. So although Thorp’s Ultimate Strategy of aces-tens-and-others rarely calls for the player to take insurance when the dealer has an ace showing, when MacDougall spotted the step up appearing on Thorp’s second hand, he warned Thorp against following the correct mathematical strategy, declaring, “You’d better insure — I think he’s got the ten of hearts in the hole.” Sure enough, when the dealer turned over his hole card, it was the ten of hearts.

Disgusted again, MacDougall and Thorp left the Desert Inn, where on their way out, much to their consternation, they spotted, leaving the dealers’ staff room, the next dealer coming on duty, who had the effrontery to be arranging in his hands the very deck of cards he was to use on the next table of gullible blackjack players.

Meanwhile, at the Sands I’d been standing behind Professor Walker, who’d also been playing the aces-tens-and-others count, spreading only very minimum bets of from $1 to $10. Soon another player sat down on Walker’s immediate right. He was a middle-aged man, who kept grinning and chatting loquaciously with the dealer. Overly friendly, effusively affable, the man proceeded to draw and stand in such a way to ruin Professor Walker even though the latter was making the lowest possible bets.

When Walker and I moved to a second empty blackjack table, the garrulous man had the effrontery to follow us there immediately, plopping himself down again on Walker’s right, declaring, “I’m glad you guys moved — my luck at the first table was lousy too.” The man then proceeded to stand on 9 and draw on 18, a strategy so absurd even a beginner wouldn’t follow it. Most anchor men try to be quiet and inconspicuous. At the Sands their anchor man strove to be a garrulous pest. He certainly succeeded. Walker and I left. And remember: Walker was a very low roller.

On our seventh gambling day, Monday, January 29th, while MacDougall went to talk to Butch, the Sheriff of Clark County (containing Las Vegas), Thorp and I repaired on our own to the Stardust, whose 25 blackjack tables in the morning were largely deserted. The two of us seated ourselves at the first table and watched the dealer shuffle the cards suspiciously. When he dealt Thorp the first hand, he used the second deal, which both of us immediately perceived. When the dealer began doing the high-low pick up, Thorp and I got up and moved down to the fifth empty table. (MacDougall later remarked that, in the cheating trade, the high-low pick up is called hen pecking.)

At the fifth table the new dealer shuffled in such a way that the large bottom stock obviously remained unmixed. Annoyed, Thorp asked, “Would you mind shuffling the cards once more — just for luck?” Blushing, the dealer refused to reshuffle. So Thorp called over the pit boss, who made the dealer reshuffle. Just the same, once more the cards had been so inadequately mixed that, again, too many of them came out in the stacked order of high-low, high-low, high-low. In disgust Thorp and I got up and forsook the Stardust.

That night MacDougall rejoined us, and I drove Thorp and him once more to the Flamingo. As the two partners started off by winning, another anchor man soon sat down on Thorp’s right and proceeded to stand and draw to injure Thorp. But in this process the anchor man was so obvious that, incensed, for the first time, MacDougall stood up and declared angrily to the pit boss: “I’m surprised that in a place like this you’d use an anchor man!” The pit boss reddened and just stared in silence at the floor.

When he so wishes, MacDougall may be outspoken, hard-hitting, and to the point, and although Thorp and I obviously agreed with him, just the same, now at least the Flamingo management realized that that so-called businessman customarily sitting next to the blackjack-playing Professor of Mathematics — that businessman, his overcoat folded across his lap, his fedora perched jauntily on the back of his head — obviously knew a great deal more about the gambling world than they have ever suspected; so on MacDougall’s part, his chastisement of the Flamingo pit boss, however justifiable, was perhaps also a moment of imprudent self-indulgence. As the boys would say over at the poker table, “Never tip your mitt.”

That night, at the City Center Motel, we three held a council of war. Although Thorp’s Ultimate Strategy had won for Mr. X almost $2,000 (more precisely, $1,849), we felt that, owing to the rampant cheating we’d received at the hands of the Strip casinos, this sum was ludicrously small. So we decided that on the morrow we’d fly north to Reno, for although Reno blackjack rules were and are comparably less favorable to count systems, perhaps Washoe County might contain, in its untutored, provincial way, fewer cunning rogues.

As the three of us rose to our feet, I remarked, “Well on the Strip there remain at least two honest casinos — The Tropicana and the Thunderbird.”

“No,” responded MacDougall cynically, “because though the Thunderbird didn’t cheat us, while Thorp and I were playing at our table, I noticed the dealer at the next table doing the high-low pick up. So that leaves just one honest casino — The Tropicana — and how long can they hold out?”

When I said goodnight to MacDougall at his motel door, he stared gloomily down the corridor and, avoiding my gaze, declared: “Do you know why casinos cheat?” “No,” I replied. “Because if they cheat, they make more money than if they’re honest. Goodnight.” With that, he closed his door, and if that’s what the premier agent of the Gaming Control Board said, that’s how I’m quoting him.

Blackjack History: On to Reno with Team Thorp

So on our eighth gambling day, Tuesday, January 30th, we three resolutely boarded an early afternoon plane bound for Reno, Nevada. Our carrier was Bonanza Airlines, owned by the late eccentric billionaire, Howard Hughes, and our equipment was a relatively small turboprop plane, the number of its propellers being just two. Up to that day Bonanza Airlines had never suffered a single fatal crash.

To fly from Las Vegas to Reno, Nevada, is like flying over the moon. Looking down through the passenger cabin ports, the three of us viewed the bleakness of a lunarscape. Toward all horizons we saw nothing but dun-colored mountains, treeless and usually grassless too, interspersed with barren plains: no houses, no people, no animals, nothing. On rare occasions we might sight a long pickup truck slowly jouncing its way over a rutted road leading in the desert from nowhere to nowhere.

Twenty minutes after takeoff, in our cabin I occupied an aisle seat next to Thorp, on my right. Farther forward, on the left side, MacDougall occupied another aisle seat, deep in his old mystery novel, Death in a White Tie. Besides us three, the cabin contained, scattered in seats about its rear, four other travelers, including a middle-aged lady, who kept coughing slightly.

As Thorp and I chatted about blackjack, our attention became gradually focused on the slowly swinging door separating our passenger cabin from the cockpit of the two Bonanza airline pilots. When the turboprop plane tilted gently to port, the door would creak slowly open. When the plane tilted gently back to starboard, the door would creak slowly closed. So half the time, Thorp and I could glimpse the blue-uniformed shoulders of the pilot and copilot, and half the time we could not. The constant swinging and creaking were mesmerizing.

Suddenly I turned to the Wizard of Odds and said, “Just suppose that, the next time the door creaks open, instead of the two pilots facing forward and flying the airplane we see them facing each other, casually playing blackjack, and the pilot remarks to the copilot, ‘Hit me.’”

“Even better,” responded Thorp immediately, “suppose that the next time the door creaks open, instead of the two pilots actually working in their cockpit we see nothing but open sky — nothing but air!”

While guffawing at our own absurdities, we hadn’t noticed that the airplane had become beset with mechanical difficulties. After completely feathering our right engine, the pilot had turned the aircraft around, and we were headed back to Las Vegas. But would we make it? Our problem wasn’t one of insufficient fuel. Would our overheated port engine alone be able to last for the twenty-minute return flight to Las Vegas? Or would it fail too? While dismally contemplating his end, everyone in the cabin watched as the surrounding hills and desert hove closer and closer.

Finally getting up, I walked a few paces forward and tapped MacDougall on the shoulder.

He didn’t even glance up from his mystery novel. “I know,” he murmured, “we’re going to crash. We’ve been losing altitude form some time.” He continued to read calmly.

I went back and took my seat again next to Thorp. Like everyone else in the cabin we too had become silent and grim. I no longer chose to glance out the ports to assess conditions. Why gaze at the dun-colored desert coming ever nearer and bringing only death?

Just as we’d lost so much altitude that everyone aboard the plane assumed his life was near its end, however, the ever-approaching desert suddenly turned into the concrete runway of McCarran International Airport in Las Vegas. We’d been saved, but it had been very close.

As we’ve all heard, after an unnerving experience on an airplane, to regain one’s positive attitude toward life, a person should immediately board the next flight — to anywhere. So that’s what we three adventurers immediately did — but not to regain our emotional stability but to reach as quickly as possible the blackjack tables of Reno, Nevada. So in the late afternoon of Tuesday, January 30th, we belatedly touched down in a second Bonanza airplane at the Reno airport.

In Las Vegas, with our benevolent Mr. X’s $10,000 bank behind us, we’d managed to win roughly $2,000. During our two days in Reno, how did we do? If the reader glances again at Win/Loss Tables, he’ll note, under the final Reno section, at least three dismal minus figures. In other words, in Reno we managed to lose back to the casinos most of the $2,000 we’d won from them in Las Vegas.

What was the cause of our financial loss in Reno? Did the casinos there cheat us even more viciously than had those in Las Vegas? Not at all. In Reno we found the casinos neither more nor less honest than those in Las Vegas — indeed, MacDougall gave both Harrah’s and Harold’s Club an admirably clean bill of health.

Blackjack History: Assessing the Play

Here are a few of my own comments.

As the responsible stake holder for Mr. X, as I’ve already mentioned, in Las Vegas my own primary mistake was, for me — a mere tyro — to substitute for the experienced Mickey MacDougall, a professional in detecting the cheating methods of gambling casinos. In Reno I continued my purblindness. Thus, on the evening of Tuesday, January 30th, at the Holiday Inn there I foolishly substituted for MacDougall again, and in only fifteen minutes Thorp lost the largest sum of our entire nine-day trip — $600. Although two detectives for the Gaming Control Board later experienced the same cheating methods at the hands of the same crooked female blackjack dealer, whom the Holiday Inn management then fired, her just deserts hardly served to compensate Mr. X, who was out $600!

As for our premier gambler, Edward O. Thorp, according to MacDougall, Thorp had a stubborn tendency to ignore prior warnings, so that instead of making small initial bets to see if a dealer was honest, he would imprudently make large initial wagers. As too often, alas, the dealer turned out to be a crook, Mr. X lost therefrom.

Secondly, like many fighters, Thorp never wanted to admit defeat and opt for a strategic withdrawal, conserving his resources for the morrow. According to MacDougall, “ten o’clock at night was Thorp’s witching hour,” after which fatigue would naturally tend to cloud his judgment, and he would begin losing too much even to honest dealers. Sometimes we had trouble getting him away from the tables. [Retired from teaching at the business school at the University of California at Irvine, Prof. Thorp now runs a hedge fund.]

As for Mickey MacDougall himself, of the three of us I believe he had the most difficult job of all, for as both Thorp and I readily concede, from a technical standpoint it’s excessively difficult to know for sure when a dealer is or is not cheating a player. Just the same, according to us, we’re sure that we both perceived cheating going on when MacDougall insisted that nothing of the sort was occurring.

In the general matter of who ran Las Vegas in the 1960’s and the kind of people who run it today, a matter in which Mickey MacDougall, according to him, had little changed his mind, I think it only fair to give MacDougall the final say. So I quote in part from his second column on our blackjack trip, “Cards Stacked Against Big Vegas Winners,” from the November 29, 1964, issue of The Sunday Star-Ledger of Newark, NJ:

Every so often I read of a “lucky” gambler who wins a fortune in Las Vegas. Sometimes it’s $50,000. Sometimes it’s $100,000. Occasionally even more.

Don’t you believe it. No one, and I mean no one, can ever hit the gangster-controlled game in Las Vegas for any sizable sum….

As steady readers of this column know, I accompanied Edward O. Thorp, Professor of Mathematics at New Mexico State University, on a Las Vegas venture…. Professor Thorp thought he was being cheated and asked me along on his second trip to protect him from the tricksters.

All suspicions were soon verified. At casino after casino, once Thorp’s wagers hit the $500 maximum, skullduggery started. It was soon apparent that the hoodlums who control Las Vegas weren’t going to give the sucker an even break. After all, who ever heard of the lamb killing the butcher?

The story of Russian Louie will serve as a perfect example. Louis Strauss wasn’t a tourist passing through Las Vegas. He lived in the gambling capital, had many friends among the gangsters who owned the gambling houses…. So what had Russian Louie to fear if he won a fortune? His friends would protect him.

Russian Louie did win a fortune — $92,000 to be exact. He started playing blackjack at the Desert Inn and got about $6,000 ahead. In an effort to recoup, the gamblers asked him to join a high-stake poker game in one of the private rooms. Russian Louie, who felt he could hold his own in any company, agreed. His luck held. Came dawn, and he had won another $86,000.

Like all big winners he developed a headache and said he wanted to quit…. Then his very good friend, Jack Dragna, suggested a trip to Palm Springs for a weekend of rest and relaxation…. Soon a shiny black Cadillac pulled away from the Desert Inn. Marshall Caifano, like Dragna, a capo mafioso and one of the most feared gunmen in the gambling underworld, was driving. Jack Dragna sat in the back seat, Russian Louie in front with Caifano.

When the big car arrived in Palm Springs, Caifano was still driving, Jack Dragna was sitting next to him. Russian Louie, the $92,000, and a valuable diamond ring had vanished.

“We had a little argument,” Dragna explained, “and Louie decided to hitchhike.”

When I say Russian Louie vanished, I mean just that. He has never been seen since. Rumor has it that he is buried in the desert, minus the $92,000 which couldn’t be identified, but still wearing the expensive solitaire, which could be identified.

“What do the hoodlums themselves think happened to the unlucky winner? Well nowadays, when a Las Vegan wants to postpone something indefinitely, he says, “I’ll do it when Russian Louie gets back to town.”

So the next timer you read of someone winning a fortune in Las Vegas, remember the tale of Russian Louie and decide not to buck the tables there until Russian Louie gets back in town.

On Thursday, February 1, 1962, at the Reno airport Mickey MacDougall and I bid Edward O. Thorp a most cordial farewell. Although both of us believed he verily deserved one, we didn’t pin onto Thorp’s chest a gold medal for blackjack bravery — perhaps because we feared that, if only to get rid of it, he’d give it right back to us.

So far as the money itself goes, from our nine-day trip how much did we three finally win? For the final results, I refer the reader to the two columns of the Summary Table on page 15, which indicate that from the whole incredible venture, we cleared only $317.

When I got back to New York City, on the following Saturday afternoon, as was my occasional custom, at an eatery off Times Square called the 42nd Street Cafeteria, I joined my fellow amateur magician friends for refreshments and a discussion of the best way to saw a woman in half.

One of the younger magicians, a card manipulator with a side interest in gambling, named Persi Diaconis, now a Professor of Mathematics at Cornell University, approached me and whispered confidentially, “Did you hear the rumor that three mathematicians hit the Vegas blackjack tables?”

“No, I hadn’t heard that. How much did they win?”

“A hundred thousand dollars.”

I wanted to say something about that, but I decided to wait until Russian Louie gets back to town.

Thorp’s Card Counting Wins and Losses

The wins and losses at each casino visited by Edward O. Thorp, Michael MacDougall, and Russell Barnhart during their nine-day blackjack trip to Las Vegas and Reno, Nevada, from Tuesday, January 23, through Wednesday January 31, 1962.

In Las Vegas

Day	Casino	Hours Spent	$ Bet Spread	Net
1	El Cortez	1	1-5	-33
	Fremont	1	5-50	-100
	Horseshoe	1/4	5-50	+40
			Day 1	-93
2	Fremont	3/4	5-50	+135
	Stardust	1	5-50	+145
	Dunes	1/2	10-100	+170
	Hacienda	1	5-50	-60
	Tropicana	1/2	5-50	+60
	Sands	3/4	5-50	+175
	Sands	1 3/4	10-100	-540
	Sands	1 1/2	7-75	+130
	Thunderbird	1	10-100	+20
			Day 2	+235
3	Fremont	1	10-100	+105
	Silver Palace	1/2	10-100	+180
	Stardust	1 3/4	7-75	-176
	Desert Inn	3/4	10-100	+260
	Riviera	1/2	7-75	+250
	Riviera	1/2	7-75	+153
	Sahara	1/2	5-20	+30
	Flamingo	1/4	5-50	-64
			Day 3	+738
4	Fremont	1/2	7-75	-165
	Horseshoe	1/4	5-20	-13
	Las Vegas Club	1/4	5-50	-23
	Desert Inn	1	10-100	+200
	Riviera	1/4	10-100	+160
	Sands	5 min.	5-50	+70
	Desert Inn	3/4	5-50	+370
	Sahara	1/2	10-100	+40
	Tropicana	1/2	10-100	+190
	Flamingo	1/2	5-50	-100
	Riviera	3/4	5-10	-65
	Sands	1	5-50	-165
			Day 4	+499
5	Stardust	1	5-50	-80
	Riviera	1/2	20-100	+285
	Desert Inn	1/4	20-75	-75
	Desert Inn	3/4	25-100	+380
			Day 5	+510
6	Tropicana	3/4	20-200	+325
	Desert Inn	1	20-200	-192
	Desert Inn	3/4	25-200	-575
	Riviera	5 min.	25-200	+160
	Stardust	3/4	25-200	-100
	Sahara	1	15-75	+245
	Sands	1 1/4	20-200	-5
			Day 6	-142
7	Sands	1 1/4	5-50	+40
	Stardust	1/2	5-50	-170
	Flamingo	1/2	10-100	+40
	Dunes	1/2	1-20	-20
			Day 7	-110
8	Tropicana	1 1/4	5-50	-25
			Day 8	-25

In Reno

8	Holiday Inn	1/4	25-75	-600
	Golden Hotel	1	5-40	-215
	Harrah’s	1/2	25-200	+70
	Harrah’s	2 1/2	25-200	-500
			Day 8	-1245
9	Harold’s Club	1	1-20	-50
			Day 9	-50

Summary of Wins and Losses

Day	Win	Loss
1		-93
2	+235
3	+738
4	+499
5	+510
6		-142
7		-110
8		-1270
9		-50
Totals	+1982	-1665

And as 1,982 dollars minus 1,665 dollars equal 317 dollars, that was our final total win for our nine-day trip to Nevada, excluding expenses: +$317.00.

[Note: Read more about Ed Thorp and his pioneering research in blackjack card counting in our article on the Blackjack Hall of Fame. In addition to card counting, Ed Thorp is famous for his success as an investor in the stock market.] ♠

How to Stack the Deck with Stock Control Shuffles

by Sam Case (with photo by Ron Hunter)

(From Blackjack Forum Vol. II #4, December 1982)
© 1982 Blackjack Forum

I must be the world’s pickiest card player. I always study a dealer before sitting down. Although there are more ways to cheat while shuffling than there are cards in a game of Blackjack II, there are several important things to watch for to avoid most cheating poker or blackjack dealers.

One thing that I look at is the quality of the shuffle. I don’t want a perfectly even shuffle, but neither do I want to see clumps of cards falling together. In either case, the dealer may be false shuffling. That clump that you see may be a card stock, selected cards that the cheating dealer wants to put in or keep out of play.

This type of cheating was documented way back in 1902, in S.W. Erdnase’s Expert At the Card Table (p. 34-39). I’ll explain the methods of stock control shuffles and the uses of this type of cheating technique.

Stacking the Deck in Your Neighborhood Poker Game

Suppose you’ve stacked the deck in your neighborhood poker game, and you do not want those stacked cards shuffled with the rest of the deck. Or, suppose a blackjack dealer decides to keep several tens on the bottom of the deck. This would have a very nasty effect on your game.

Tip-off #1: If you’re a card counter, you’ll keep getting high counts near the end of the deck. However, poker players won’t get this tip-off, and waiting to find this out in a blackjack game can be expensive. I’ll explain how this stock control shuffle is done, and the only other tip-off you can watch for.

Place three or four tens on the bottom of the deck. Cut the top half to the right, and get ready to shuffle. Lift the inner long sides slightly, then run those three or four cards from the left-hand side before dropping any from the right-hand pack. Continue the shuffle in a normal riffle fashion. Square the deck. A cheating dealer who does this three times before offering you the cut keeps that card stock intact on the bottom. All he has to do is nullify the cut and you’re in trouble.

This running of cards can be used to control a stock at any point in the deck, but the bottom of the deck is the easiest place to use. The top of the deck is the next easiest place, but since the top of the deck is the easiest place to watch, some cover is needed. Now try this: put the card stock on top of the deck, and cut the top half to the right The idea is to do a similar shuffle, except start off shuffling evenly, but drop the top stock last to keep it in place.

Notice how easy this would be to spot (see photograph above, exaggerated for clarity). No dealer would leave this uncovered. What a dealer might do is hold back the top card of the left-hand packet (as well as the right-hand card stock), drop the right-hand stock and then the left-hand top card.

From almost any position, this makes the block of cards more difficult to spot, especially when you consider the speed with which most dealers shuffle. If you suspect an uneven shuffle you might try sitting low in your seat to view the deck edge on.

Why doesn’t that top card get in the way? By using a crimped card, say the bottom one of the stack, the dealer could bring the stack to the bottom with one cut (see “The Gamblers Crimp”, link at the left, for details).

The dealer, however, would probably want to keep a top stock on top. After you cut, in a head-on blackjack game, if the cards were in the following order: ANY CARD, ANY CARD, 10, ANY CARD, ACE, after the burn the dealer would beat you with an ace-down blackjack.

The following stack—ANY, ANY, 5 or 6, ANY, 5—would be a very nasty percentage play in blackjack (dealer 10 or 11 with low card showing). A larger stack could force you to split 8s vs. a dealer 21, etc., depending on the dealer’s stacking abilities and possibilities. And I’m sure you can imagine the possibilities in poker.

So, the moral of the story is to watch for even shuffles, especially at the top and bottom of the deck. Next time I’ll explain how a card stock is set up during a riffle shuffle. Till then remember: you want to play with stacked women, not stacked decks! ♠

An Ace Tracker’s Lament

By Darryl Purpose

(From Blackjack Forum Volume XVII #3, Fall 1997)
© Blackjack Forum 1997

[Note from Arnold Snyder: Since the writing of this article, Darryl Purpose has been elected to the Blackjack Hall of Fame. He is also a successful folk singer and songwriter. To check out his latest musical release, see DarrylPurpose.com.]

I’m playing some blackjack around St. Louis. Scouting and playing small stakes, trying to dig out this 50k bank that is now 10k. They have some special gambling law in Missouri designed to protect the players – you can only buy in $500 on any one “cruise.” Before I learned to be careful about that I had to walk away from a +20 shoe for lack of chips. Arghh.

At one point the dealer was a tad sloppy and exposed the card on top of the pack just before the cut. Clear as day, it was a big ol’ ACE. Also clear as day was a big ol’ Queen of Diamonds on the bottom of the pack.

There was one other player on the table and she cut the cards about 1 ¾ decks from the top. I’m thinking – if that Queen of Diamonds comes out as the last card of a round, this could be real interesting…

Tracking the Ace: Strategy

It’s a quarter game, and my strategy to this point, on a quarter game, has been to leave the table at any negative, but when this one goes negative, I stay. The woman sitting in the #2 spot is large, taking up a little of the #1 & #3 spots as well. This doesn’t leave much room for me to squeeze into first base, so I wait to grab the spot, because, after all, it’s a long shot that I’ll need it.

I’m hoping this woman leaves the table because that would increase the chances of the Queen of Diamonds falling as the last card of the round. Sure enough, my prayers are answered. She taps out, gets up… then with a grunt and a mumble reaches between her sizeable breasts and pulls out two sweaty hundred dollar bills, and the game continues. As we get to the middle of the deck, she says, “Mind if I play two hands?” That would certainly reduce the chances of the Queen falling on the last card!

I conjure up a sincere confidence and say emphatically, as if I know that this would bring us all kinds of bad luck, “Play one.”

The deck is down to about the two-deck level. I bet $50. I figure, that Queen might come out in the hit cards leaving me with the option of doing something really interesting and really profitable, like… doubling on a hard 20?! Maybe not, but certainly an A,9.

The possibilities are swimming through my head. I don’t bet too big because I only have $250 in chips and the chance to buy only $500 more, and if that Queen of Diamonds comes out as the last card of the round, I’d like to have some money to bet the ACE!

The dealer has a 6 up, 5 in the hole, hits it with a little card and Boom, the QUEEN OF DIAMONDS! Exactly what I was looking for! Exactly the place in the deck expected!

But…I didn’t really plan this whole thing out… I’ve got to get over to first base!!! I’ve got to buy some more chips!!! I’ve got to figure out how much to bet!!!

I say to the dealer, “Hold it up!” Then, turning to the other player, I say, “Mind if I play first base on this one?!” I speak confidently, just a little like a crazy person, a superstitious person, like a person who doesn’t care what the world thinks, but has a notion that this or that will bring him luck and is hell-bent on carrying that out. (Like your average casino-goer, maybe?)

I give the dealer $200 and announce, “I’m going to play first base!”

As I move over I’m quite relieved to see that the large woman isn’t miffed. She thinks it’s kind of amusing. I think she even feels a little sorry for me. The whole thing must seem a tad pathetic at this point.

How much to bet?! I’m thinking—save enough to double down. Bet half of what I have available (about $350). But as the dealer is changing my $200, I realize that the advantage from doubling (8%??) is nothing compared to the 50+% that I have for getting the ace, besides, since I’m getting the ace, it’s only soft doubles that we’re missing anyway.

Right?!

1/3 Kelly would call for a $3,000+ bet.

Right?

I bet everything.

It’s a big stack ’cause it’s got lots of $1 and 50 cent chips in it.

The floorman comes over and asks, “Is this a bet?!”

I haven’t bet over $100 in this place to this point… I figure the whole scene is already over the top, so I say with conviction, “Yes! I’m going to get a blackjack!! Give me my blackjack!”

I’m thinking hard about the appropriate reaction on my part when I do get my blackjack. I hadn’t really figured it out when the first card was dealt… the ace… but what’s this?!… it’s not the ace… It’s the FIVE of CLUBS!

A Bad Time for a Sad Lesson in Ace Location Technique

In that crack in time between worlds when everything is in slow motion, you have all the time in the world to understand.

I fully understand that I’ve got 7% of the bankroll out there, I can’t double down, and I’ve got the FIVE of CLUBS! The next card falls…

QUEEN of DIAMONDS to the player next to me!… and even before the next card comes out I understand even more… I’ve got 7% of the bankroll on the table and I’ve got the Five of Clubs and the dealer has the big ol’ ACE…

Of course, y’all being professional and all, you know that it doesn’t really matter how the hand turned out… ♠

Why Skillful Satellite Play Cuts Your Bankroll Requirements More Than You Think

by Arnold Snyder and Math Boy

(From Blackjack Forum , Vol. XXVI #1, Winter 2007)
© Blackjack Forum Online 2007

This article will look primarily at how you should estimate the dollar value of a poker tournament satellite. It will look at the factors that make a satellite a good investment, and will discuss how skill at satellites can lower the bankroll you need to enter bigger, more costly poker tournaments, or lower your risk of ruin (RoR) for any given bankroll. We’re going to focus on specific satellite strategies in a later article. In this article, we’ll look at the dollar value of satellites to a player who already has the strategic skills necessary to beat satellites.

First, a definition: A satellite is a tournament that does not award money to the winner(s), but instead awards an entry to a tournament that has a bigger buy-in cost. In some satellites, a small amount of money is awarded in addition to a seat or seats into a bigger event, but this money is a secondary prize, awarded either to cover the winners’ travel expenses or to “round out” the prize pool.

It is the seat in the bigger tournament that is the main prize. There are some satellite pros who play satellites primarily for money, not seats into a bigger event. Many satellites, especially for big multi-tournament events, award tournament chips rather than seats in specific events, and these chips can be sold to other players who are buying into the big events. Our primary focus in this article, however, will be on players who want to use satellites to lower their entry costs into main events.

A single table satellite will usually start with ten players, and will typically award one winner a seat in a tournament that has a buy-in of about ten times the cost of the satellite. Some single-table satellites award seats to the top two finishers. A “super satellite” is a multi-table satellite that will award multiple seats into a major tournament, with the exact number dependent on the number of entrants in the satellite.

The “House Edge” on Poker Tournament Satellites

The first thing we need to consider when analyzing a satellite’s value is the house edge. By this, I mean the percentage of the total cost of the satellite that the players are paying to the house for hosting the satellite. Figuring out the house commission is pretty straightforward. We need only two factors: the total combined dollar cost to all of the entrants, and the total dollar value of the satellite prize pool.

For example, a popular World Poker Tour (WPT) single-table satellite has a buy-in of $125. One of the ten entrants will win two $500 tournament chips plus $120 in cash. The other nine entrants receive nothing. For big multi-event tournaments like the WPT or the World Series of Poker (WSOP), it is not uncommon for the hosting casino to run satellites that award tournament chips that may be used in bigger events, instead of seats into a specific event.

Although the series of tournaments may conclude with a “main event,” usually the event with the highest buy-in, there will be many events with smaller buy-ins that precede the main event. A player who wins two $500 tournament chips may use those chips to enter two $500 events, a single $1000 event, or as partial payment into an event that has a buy-in greater than $1000.

In this particular WPT satellite example, the house collects $125 x 10 = $1250, while paying out $500 + $500 (two chips) + $120 (cash) = $1120. So, the house profits $1250 – $1120 = $130 every time they run this satellite. The house edge from the players’ perspective is: $130 / $1250 = 10.40%. Which is to say, the house keeps 10.4% of all the money they collect from the satellite entrants.

At the 2006 Mirage Poker Showdown, a WPT series of tournaments with a $10K main event, there were seven different single-table satellite formats. All were ten-player/one-winner formats. The least expensive had a $60 buy-in, with the winner receiving a single $500 chip plus $50 cash. The most expensive had a $1060 buy-in, with the winner receiving twenty $500 chips (a total of $10K in chips) plus $340 in cash. The chart below shows the buy-ins of these seven satellites with their payouts, the house commission (in $), and the house edge (in %).

Mirage Poker Showdown (WPT) Single Table Satellites, one winner format
Buy-in	Payout in Chips	Payout in Cash	Total Payout	House Commission $	House%
60	500	50	550	50	8.33%
125	1000	120	1120	130	10.40%
175	1500	120	1620	130	7.43%
225	2000	120	2120	130	5.78%
275	2500	120	2620	130	4.73%
810	7500	340	7840	260	3.21%
1060	10000	340	10340	260	2.45%

One obvious trend is that, in general, the greater the buy-in cost, the lower the house edge. The only exception among this group of satellites is that the house edge on the least expensive ($60 buy-in) satellite is smaller than the house edge on the second cheapest ($125 buy-in) satellite.

Poker Tournament Satellite Value for the “Average” Player

As a satellite player, what does the house edge mean to you? Consider your result in these satellites if you play with an “average” level of skill that gives you neither an advantage nor a disadvantage against the field. Some may be better players than you, and some worse, but in the long run, you will average one satellite win for every ten satellites you play.

This chart shows the value (in $ and %) of each of these seven WPT satellites for an “average” player:

Player Wins One Out of Ten Satellites Entered
Buy-in	Dollar Value	Player Adv. in %
60	-5.00	-8.33%
125	-13.00	-10.40%
175	-13.00	-7.43%
225	-13.00	-5.78%
275	-13.00	-4.73%
810	-26.00	-3.21%
1060	-26.00	-2.45%

Let’s first note that in all cases, the dollar value is negative. “Average” players will lose money in satellites over the long run. Computing the dollar value is simple. Consider the $60 satellite. If you play ten of these satellites, you will invest $60 x 10 = $600 in these ten plays. If you win one, you get paid $500 (chip) + $50 (cash) = $550 in dollar value. Collecting $550 for every $600 you invest is a loss of $50 for every ten satellites you play, which is an average loss of $5 per satellite. So, the dollar value of this satellite to the average player is -$5.00.

Likewise, the “Player Advantage in %” is negative for the average player in the other satellites, as we would expect. Again, the computation is simple. If I lose $5 for every $60 I invest, then my result (in percent) is: -5 / 60 = -0.0833, which is -8.33%. You might also note that if you look at the prior chart which shows the “house edge” on these satellites, the “player advantage” for an “average” player is always the negative of the house edge.

From the professional players’ perspective, this means that to play satellites with average skill is a waste of money. In the long run, it will cost the average player more to enter tournaments via satellite than it would cost him to simply buy-in to the big events directly.

A player who is short on funds might argue that he would never pay $10K to enter a major tournament, but that he is willing to gamble $1060 on a long shot to get into such an event. In fact, this is a good argument if you accept the fact that you are gambling on a negative expectation game. One of the reasons that tournaments have become so valuable to professional players is that so many amateurs are willing to gamble at a disadvantage on satellite entries.

Satellite Value for the “Better-than-Average” Player

Now let’s look at how satellite skill affects the value of satellites to the player. Let’s say you can win one out of every nine of these ten-player satellites that you enter. How would this affect both the dollar value and your advantage (in %) in these same seven WPT satellites?

Player Wins One Out of Nine Satellites Entered
Buy-in	Dollar Value	Player Adv. in %
60	1.11	1.85%
125	-0.56	-0.44%
175	5.00	2.86%
225	10.56	4.69%
275	16.11	5.86%
810	61.11	7.54%
1060	88.89	8.39%

Big difference. The cheapest satellites are still not worth the trouble, due to the high house edge. Few players at this modest level of satellite skill should be interested in entering the $60 satellite, as the $1.11 value is a pretty low payout for a lot of work. And the $125 satellite (which you may recall has a higher house edge) is still a negative expectation gamble. The player who can win one in nine of these has a notably superior result to the “average” player who can win only one in ten, as the more skillful player will be losing only 56 cents per satellite played, as opposed to losing $13. Still, in the long run, the player who expects to win just one in nine of the $125 satellites would be paying less to enter the bigger tournaments if he skipped the satellites and just paid the full buy-in price of the event.

With the $1060 satellite, the one-in-nine player advantage is 8.39%, which is a dollar value of $88.89. For many players, this modest level of satellite skill might make these satellites worth the effort.

The Satellite Professionals

As your satellite skill increases, the value of playing satellites goes up dramatically. A more talented satellite pro can do quite well in ten-player/one-winner satellites if he can win just one of every eight satellites he enters. Here’s how he would do in these seven WPT satellites with this level of skill:

Player Wins One Out of Eight Satellites Entered
Buy-in	Dollar Value	Player Adv. in %
60	8.75	14.58%
125	15.00	12.00%
175	27.50	15.71%
225	40.00	17.78%
275	52.50	19.09%
810	170.00	20.99%
1060	232.50	21.93%

At this skill level, there may be sufficient dollar value to warrant playing these satellites at any buy-in level. With the smaller buy-in satellites, you must decide if the expected dollar return is sufficient to spend, on average, about an hour of your time in satellite play. Many of us wouldn’t work for $8.75 an hour, which is the dollar value of the $60 satellite for the one-win-in-eight player.

But we must also consider that for every satellite we play, we gain more satellite experience, and this should translate sooner or later to greater satellite skill. Like all other forms of poker, you can’t increase your satellite skills without playing them. And this becomes more important as the dollar value increases with the satellite cost. For example, with a dollar value of $232.50, the $1060 satellite will gain you entry into a $10K event, on average, for a cost of just over $8K. That’s a substantial discount.

The Top-of-the-Line Poker Tournament Satellite Pros

The top satellite pros win, on average, about one out of every six to seven ten-player/one-winner satellites they enter. They accomplish this with a combination of skill at fast-play strategies, skill at short-handed play, and skill at choosing weak fields of competitors. No pro wants to enter a satellite and find himself facing a table full of other pros. The value of satellites to a pro is as much a function of his competitors’ lack of skill as it is of his own skill.

Let’s compare the dollar values of these seven WPT single-table satellites for players who expect to win one of every ten, nine, eight, seven, six, and five satellites they play:

Dollar Value per Satellite, if Player Wins Once per:
Buy-in	10	9	8	7	6	5
60	-5.00	1.11	8.75	18.57	31.67	50.00
125	-13.00	-0.56	15.00	35.00	61.67	99.00
175	-13.00	5.00	27.50	56.43	95.00	149.00
225	-13.00	10.56	40.00	77.86	128.33	199.00
275	-13.00	16.11	52.50	99.29	161.67	249.00
810	-26.00	61.11	170.00	310.00	496.67	758.00
1060	-26.00	88.89	232.50	417.14	663.33	1008.00

Let’s also look at the various players’ advantages in percent for these frequencies of wins:

Player Advantage (%), if Player Wins Once per:
Buy-in	10	9	8	7	6	5
60	-8.33%	1.85%	14.58%	30.95%	52.78%	83.33%
125	-10.40%	-0.44%	12.00%	28.00%	49.33%	79.20%
175	-7.43%	2.86%	15.71%	32.24%	54.29%	85.14%
225	-5.78%	4.69%	17.78%	34.60%	57.04%	88.44%
275	-4.73%	5.86%	19.09%	36.10%	58.79%	90.55%
810	-3.21%	7.54%	20.99%	38.27%	61.32%	93.58%
1060	-2.45%	8.39%	21.93%	39.35%	62.58%	95.09%

The final column in both charts, which shows the player’s expectation if he is skillful enough to win one out of every five of the satellites he plays, is more theoretical than realistic. This may be possible for a skillful satellite player who always manages to face a very weak field, but most satellite players today are not this weak. Many players are aware of the necessity of taking risks in satellites as the field diminishes and the blinds increase.

Nevertheless, we can see from the charts that a satellite player who is skillful enough to win one out of every six or seven of these satellites will have an advantage in the neighborhood of 30% to 60%, and that is a big enough edge to interest any professional gambler.

Other Poker Tournament Satellite Formats

As mentioned at the beginning of this article, not all satellites are single-player one-winner formats. The two-winner format is also quite common. Typically, a player might pay $200 plus the house fee to win one of two seats into a $1K event. With this format, the average player would expect to win two out of every ten satellites entered, as opposed to one in ten. Likewise, a win of two in eight with the two-winner format would be equivalent to winning one in eight with the single-winner format. And the top satellite pros would expect to win two out of every six or seven played in the two-winner format.

The two-winner format is generally advantageous for both the players and the poker room. Satellites played down to two winners finish faster than satellites played down to one winner. This means that more satellites can be played prior to a big event, with twice as many main event entries generated per satellite. The two-winner format also reduces fluctuations for the players.

If you’re good with spreadsheets, you can easily set up a spreadsheet to calculate the dollar return and house/player advantages for the two-winner format based on the buy-in costs and payouts.

Multi-table satellites, often called super-satellites or mega-satellites, are also very common, especially for major events. For example, the WSOP typically has a $1060 super-satellite for the $10K main event. One seat to the main event is awarded for every ten satellite entries. Here’s a chart that shows the dollar values and player advantages based on the frequency of player wins:

WSOP Mega Satellite, $1060 Buy-in, One Seat Awarded per Ten Entrants Dollar Value and Player Adv. If Player Wins Once per:
	10	9	8	7	6	5
$ Value	-60.00	51.11	190.00	368.57	606.67	940.00
Adv. (%)	-5.66%	4.82%	17.92%	34.77%	57.23%	88.68%

Note that the house edge on this event is 5.66%. These multi-table satellites are a good value for a player on a budget who can only afford to enter one satellite for a shot at the main event. And this is especially true if the player is a good tournament player, but not really all that skilled at single-table satellite play. (And there are many players who are skilled at multi-table tournaments who do not fare well in single-table satellites, primarily because of a lack of experience with making the quick adjustments necessary for the speed and short-handed play.)

For a skillful single-table satellite player, however, super satellites have less value than single-table satellites. The higher-priced single-table satellites often have a lower house edge, and they play out much faster. Big multi-table satellites often take many hours to determine the winners, as opposed to the typical 60-90 minutes a single-table satellite lasts. Time is money.

Using Satellites to Lower the Buy-In Costs of Major Poker Tournaments

Let’s say you’ve been playing a lot of small buy-in tournaments and your tournament skills have increased to the point where you want to start playing bigger events where you can make more money. You don’t feel ready for the major $5K and $10K events that the top pros dominate, so you want to start playing in $1K events as a stepping stone to the majors.

Let’s also assume that you’ve been beating the small buy-in tournaments at a rate of well over 200%, and you believe you would have an advantage of at least 100% in these bigger $1K events. You’ve been building your bankroll with these small buy-in tournaments, and your plan is to start using the money you’ve won to advance. The main question you have: Is your bankroll really big enough to withstand the greater fluctuations you’ll encounter in these $1k events?

If you do not have a sufficient bankroll to enter $1K events at this time, that does not necessarily mean that you must resign yourself to smaller buy-in tournaments. In fact, if you are a skillful satellite player, you can start entering $1K tournaments via satellites with a smaller bankroll. As the charts above show, satellites can very effectively lower the buy-in costs of bigger events. And a lower buy-in cost means a smaller bankroll is required. But how much smaller?

Before we can figure out how much satellite skill can lower your bankroll requirements, however, let’s quickly review what your bankroll requirements would be for a given type of tournament if you pay the full buy-in cost.

Let’s say you would like to play a hundred $1K tournaments in the next year. If you live in Las Vegas, this is easily accomplished, as Bellagio has two $1K buy-in tournaments every week. $1K events are also popular preliminary events for WSOP Circuit series, WPT events, and many other special tournament series that poker rooms run throughout the year. In order to estimate the bankroll requirements for entering a hundred $1K events, assuming a player has a 100% advantage on the field, let’s use a real-world example.

At the recent WSOP Circuit Events that were hosted by Harrah’s Rincon in San Diego, the $1k event on February 13, 2007, had a total of 89 players who paid $1060 for a seat. Nine spots were paid. This was the payout structure:

Place	Payout
1st	$31,079
2nd	$17,266
3rd	$9,496
4th	$6,906
5th	$6,043
6th	$5,180
7th	$4,317
8th	$3,453
9th	$2,590

Now, we have something to work with. Obviously, every $1K tournament you enter will not have this payout structure, but for our purposes, we’re going to assume that you want to know the bankroll requirements for entering a hundred of these specific tournaments. Since we already said that you estimate that you have a 100% advantage in these events, we next have to make some assumptions as to how that 100% advantage will be realized.

When we say that you have a 100% advantage, we mean that for every $1K you pay to get into these tournaments, you will cash out $2K. That cash out will pay you back your $1K buy-in and provide a $1K profit, which is a 100% advantage. From looking at the payout schedule, we can see that there is no payout of exactly $2,120 (twice the buy-in/entry) for any finishing position. Even 9th place pays $2,590, which is a 144% profit. In order to realize a 100% advantage, we will accomplish this by having many finishes with no return, but a number of finishes that return greatly in excess of 100%. (And note that many tournament pros enjoy advantages in excess of 200% and even 300%. We are using this 100% example for a player who is just moving up to $1K events from smaller events, and who is still sharpening his skills.)

Assuming you play a hundred of these tournaments, let’s create a win record that would provide an advantage in the neighborhood of 100% overall. To do this, I’ll assume that you finish in the money 16 times, or about once every 6 to 7 tournaments. Here’s a set of finishes that includes 3 first places, 3 second places, 3 third places, 2 fourths, 2 fifths, 1 sixth, 1 seventh, 1 eighth, and 84 finishes out of the money, and would earn you 100.35% on your total investment:

Finish	Payout	# Cashes	Total
1st	$31,079	3	$93,237
2nd	$17,266	3	$51,798
3rd	$9,496	3	$28,488
4th	$6,906	2	$13,812
5th	$6,043	2	$12,086
6th	$5,180	1	$5,180
7th	$4,317	1	$4,317
8th	$3,453	1	$3,453
9th	$2,590	0	$0
10-89h	$0	84	$0
Total Cashed: $212,371
Total Invested: $106,000
Total Win: $106,371
Win %: 100.35%

That’s close enough to 100% for our purposes. Obviously, no player could estimate that these will be his exact finishes in 100 tournaments. We’re just creating a set of finishes that would return approximately 100% profit to the player. There are many other ways that a 100% advantage could be realized. There could be fewer than 16 money finishes, but more with the higher payouts, or there could be more than 16 money finishes, but fewer with the big payouts and more low-end finishes. The above set of payouts is just one way that a player with a 100% advantage might realize this profit.

In fact, despite the assumed 100% advantage, over the course of these 100 tournaments, a player in real life would be highly unlikely to show a result this close to an actual 100% profit. His real-world result in a series of 100 consecutive tournaments would be subject to what statisticians call standard deviation. He may have an exceptionally good run of tournaments, or an exceptionally bad run, just due to normal fluctuations in the cards and the situations he encounters.

In the Appendix to The Poker Tournament Formula, pages 328-340, there is a discussion of what standard deviation is, what it means to a gambler, and how you figure it out for poker tournaments. I’m not going to reproduce that discussion here, so if you do not understand the concept of standard deviation, and especially how it applies to poker tournaments, read that chapter the book. Also, different payout schedules caused by different field sizes will have a major effect on standard deviation, so don’t assume that the discussion about this specific $1K tournament would apply to all $1K tournaments. This tournament is just an example from real life.

Using the method described in The Poker Tournament Formula, and applying it to the player with the 100% advantage in the $1060 tournaments described above, we find that although our expectation is to win (profit, after subtracting our buy-in/entry fees) a total of $106,371 over the course of 100 tournaments, one standard deviation on that result is $62,853. So, if we finish two standard deviations below our actual expectation (and 2 SDs = $125,706), we could actually finish these 100 tournaments with a loss of $19K, despite our 100% advantage!

This may sound impossible, but keep in mind that we only expect to win $106K and that $93K of our total return comes from just three first place finishes. A few bad beats and cold decks at crucial times at the final table, or before we get to the final table, can wreak havoc with our overall results.

Since we could conceivably suffer a net loss of $19K over the course of these 100 tournaments, and still be within the realm of what a statistician would consider a “normal” fluctuation, we might conclude that a bankroll of $20K would be sufficient to finance our play, though we could conceivably lose it all. In fact, a bankroll of $20K would usually be more than sufficient to finance this level of play, assuming we are correct about our 100% edge. The standard deviation on our expected results does give us a pretty good idea of the kinds of fluctuations that are possible due to bad luck in tournaments of this level over this period of time.

Technically, your $20K bankroll would probably be very safe if you cut back on the cost of the tournaments you entered if you started experiencing significant negative results. In other words, if you finish out of the money in the first ten tournaments you enter—and this is entirely possible—then you really would be wise to start entering $500 tournaments until you hit a few wins (or just one good win) to build your bankroll back up. Fluctuations of greater than two standard deviations happen all the time.

(Cutting back on the size of your bets after bankroll reductions is a method of “Kelly Betting,” another term that would be known to any serious blackjack player, but few poker players. I’ll discuss Kelly betting approaches for tournament players in more detail later in this article.)

If you are familiar with the statistical concept of standard deviation as discussed in The Poker Tournament Formula, you are probably aware of the fact that a statistician expects a fluctuation of greater than two standard deviations 5% of the time. Which is to say that if 20 players were playing these tournaments with a 100% advantage, 19 of them would expect to finish 100 tournaments within two standard deviations of their expectation, but one of them would expect to experience a fluctuation of greater than two standard deviations from his expectation.

A fluctuation of 3 standard deviations is extremely rare, as results this far from expectation have only about a 1/300 chance of occurrence, so it’s not really necessary to maintain a bankroll to withstand this much of a fluctuation. But to be safe, just based on the standard deviation, I’d advise a bankroll closer to $30K for these $1K events, assuming you have a 100% advantage.

But What About Different Levels of Aggression in Poker Tournaments?

Let’s consider the fact that we’ve created our hypothetical 100% advantage in this tournament by devising a specific set of in-the-money finishes that would result in this win rate. In the real world, there are many different ways a player could end up with a 100% advantage. He could be a very aggressive player who had fewer cashes but more top-end finishes, or he could be a more conservative player who had a greater number of cashes, but more low-end finishes. Might not the bankroll requirements for these player types differ from each other?

Let’s analyze and compare the requirements for these different player types.

The More Aggressive Hypothetical Tournament Player

To provide this player with a 100% advantage, let’s say he finishes in the money only 10 times in 100 tournaments (instead of 16 times as in our prior example), but with more high-end finishes. Here’s the aggressive player’s chart:

Finish	Payout	# Cashes	Total
1st	$31,079	4	$124,316
2nd	$17,266	4	$69,064
3rd	$9,496	2	$18,992
4th	$6,906	0	$0
5th	$6,043	0	$0
6th	$5,180	0	$0
7th	$4,317	0	$0
8th	$3,453	0	$0
9th	$2,590	0	$0
10-89h	$0	90	$0
Total Cashed: $212,372
Total Invested: $106,000
Total Win: $106,372
Win %: 100.35%

Conveniently, this very different set of cashes provides this aggressive player with the same 100.35% win rate as the player in our first example.

The More Conservative Hypothetical Tournament Player

For further comparison, let’s also create a sample player who has more final table finishes (25, instead of 16 or 10), with more low-end finishes, but still, with a 100% win rate. Here’s this more conservative player’s chart:

Finish	Payout	# Cashes	Total
1st	$31,079	2	$62,158
2nd	$17,266	2	$34,532
3rd	$9,496	3	$28,488
4th	$6,906	3	$20,718
5th	$6,043	3	$18,129
6th	$5,180	3	$15,540
7th	$4,317	4	$17,268
8th	$3,453	3	$10,359
9th	$2,590	2	$5,180
10-89h	$0	75	$0
Total Cashed: $212,372
Total Invested: $106,000
Total Win: $106,372
Win %: 100.35%

In order to compare the bankroll requirements of these three different player types, who all have the same overall win rate in the same tournament, let’s use a different statistical method of analysis, the Gambler’s Ruin Formula, or, as gamblers today more often call it, Risk of Ruin (or RoR).

First, here’s an explanation of what we’re trying to figure out here. A player wants to play tournaments that have a specified buy-in/entry cost, say, $1060. He knows from the above discussion on standard deviation that he could conceivably lose his bankroll due to negative fluctuations, even if he has a 100% overall advantage. The player wants to minimize his risk, so he wants to know how much of a bankroll he’d need to insure himself of a 90% chance of success, or 95% chance of success, or even 99% chance of success. Also, if the player could maintain his 100% win rate while using either a more aggressive or more conservative strategy, how would this affect his chance of success?

The Risk of Ruin Formula that was used to analyze the three sample players described above was first published by Math Boy and Dunbar in the Fall 1999 issue of Blackjack Forum. The article is titled, “Risk of Ruin for Video Poker and Other Skewed Up Games”. You may follow this link to get to the article in our online library, so we’re not going to print the Risk of Ruin formula here.

Let’s just look at the RoR data on our three player types. Remember, the conservative player makes it to the final table 25 times in 100 tournaments, but wins the fewest top-end prizes. The middle-of-the-road player has 16 money finishes out of 100 tournaments, with more at the top-end. The aggressive player has only 10 money finishes, but takes four firsts, four seconds, and two third-place payouts. These are their bankroll requirements for various levels of risk:

RoR	Conservative Bankroll	Middle-of-Road Bankroll	Aggressive Bankroll
1%	37,213	45,449	55,853
5%	24,208	29,565	36,333
10%	18,607	22,724	27,926

We want to emphasize here that the “conservative” and “aggressive” styles we’ve created are for purposes of analysis—they are not meant to be realistic. With regards to the aggressive player, it’s unlikely that a player who was skillful enough to always make the top three when he finished in the money (with 80% of those finishes in the top two) would never finish in any other final table position. Even the most aggressive and skillful players will suffer bad beats and cold decks and hit lower payouts occasionally.

It’s really more likely that such a player would have a range of money finishes at all levels, even if he had more than his share of the top prizes. The problem we faced in creating this player, however, was that we were attempting to maintain that 100% win rate for purposes of risk of ruin comparison, and if we start scattering a more realistic set of smaller wins among his finishes, his win rate will climb dramatically (as it tends to do in real life with the best aggressive players).

The conservative player’s results were similarly skewed. It is highly unlikely that a player with so few top-end finishes would be able to hit the money often enough to have a 100% win rate. But the purpose of the example is simply to show that playing style does have an effect on a player’s bankroll requirement, all other things being equal. If we look at the middle-of-the road player’s bankroll requirement for a 5% RoR (95% chance of success), we see he needs a bankroll in the neighborhood of $30,000. The conservative player might get away with a bankroll of about $5000 less than this, but the aggressive player might need $5000 more.

These numbers may surprise many tournament players who have not read The Poker Tournament Formula, as there is widespread ignorance among poker players with regards to bankroll requirements. On one of the WPT shows, for example, a sidebar feature showed professional poker players being asked for advice on bankroll requirements for tournament players, and wound up providing the specific recommendation that tournament players should have a bankroll of ten times their buy-in cost. Don’t we wish…. In fact, that was potentially disastrous advice for serious poker tournament players.

One other thing we must note here is that this sample tournament we’re analyzing had a prize pool based on a total of 89 players. In fact, if you are entering $1K tournaments with 300 players, or 2000+ players as in some of the 2006 WSOP $1K events, the top prizes will be much bigger, and so will your fluctuations and bankroll requirements. We don’t want you to think that a $30K bankroll is sufficient for all $1K tournaments. Again, we ’ll refer you to the detailed discussion in The Poker Tournament Formula on how field size affects bankroll requirements.

Now let’s look at how we can use satellites to lower this bankroll requirement.

Lower Buy-In Costs Mean Lower Poker Tournament Bankroll Requirements

Since we’re discussing a specific $1K WSOP Circuit Event, let’s look at one of the actual satellites that was being offered at Harrah’s Rincon that allowed a player to win entry into this event. They ran ten-player satellites that cost $240, and the satellites paid two winners. Each winner received two $500 chips and $100 cash. So, if you were one of the two satellite winners, you’d have been able to cover your $1060 buy-in/entry to the $1K tournament, and still have $40 cash to put in your pocket.

Here’s a chart that shows this satellite’s dollar value and player advantage, based on a player’s expectation of winning twice out of every 10, 9, 8, 7, 6, and 5 satellites entered:

WSOP Circuit Satellite, $240 Buy-in, Two Winners Each Get $1000 (chips) plus $100 cash Dollar Value and Player Advantage If Player Wins Twice per:
	10	9	8	7	6	5
$ Value	-$20.00	$4.44	$35.00	$74.29	$126.67	$200.00
Adv. (%)	-8.33%	1.85%	14.58%	30.95%	52.78%	83.33%

Now, let’s say the same three hypothetical tournament players (conservative, middle-of-the-road, and aggressive), all with the same 100% advantage in the $1K events, are also skillful satellite players, and they each decide that they will always enter these $1K events through satellites. And, let’s assume their levels of satellite skill provide all of them with an expectation of winning twice for every seven (or, essentially, one for every 3.5) two-winner satellites they enter. How does this affect the actual cost to them of entering the $1K events? Here’s the math:

3.5 of these satellites will cost $240 x 3.5 = $840. For that $840 expense, the player gets $1K in chips plus $100 in cash. Since the buy-in/entry for the $1K event is $1060, the player can buy-in to the $1K tourney and pocket the extra $40 cash, leaving him with a total buy-in/entry cost of $800 even, a discount of $260 from the full tournament price.

This lowered cost per tournament drastically reduces the risk of ruin for each player. You might guess off the top of your head that paying $800 per tournament on average, as opposed to $1060, might cut your bankroll requirements proportionately. Since 800 / 1060 = 75.5%, you might be tempted to take the 1% RoR bankroll requirement of $45,449 for the middle of the road player, and multiply it by 75.5%, for a bankroll requirement of $34,300.

But, in fact, the effect of the satellite discount entry is quite a bit stronger than that. What you must also consider is that although you are entering these $1060 tournaments for only $800, the prize pool does not change. This means that the same win record will increase your percent advantage from 100% to 165%. This increased percent advantage will lower the bankroll requirement for a 1% risk of ruin for the middle of the road player from $45K to $30K.

(Unfortunately, figuring out the RoR for entering these tournaments through satellites is not quite as simple as just using $800 as the entry fee. The situation is similar to a parlay bet at a sports book. The satellite has its own flux, and that must be included with the flux on the main tournament in order to correctly calculate overall flux and risk of ruin.)

Let’s look at the RoR bankroll requirements assuming these three players always enter these $1060 tournaments through these $240 two-winner satellites, with each winning two out of every seven of the satellites:

RoR	Conservative Bankroll	Middle-of-Road Bankroll	Aggressive Bankroll
1%	19,358	30,380	42,905
5%	12,592	19,762	27,910
10%	9,679	15,190	21,452

What Risk of Ruin Should a Poker Tournament Player Consider Safe?

Many players might say off the top of their heads that they’d be comfortable with a 90% chance of tournament success. But bear in mind that ruin means ruin. If you have a $30K bankroll, and you find yourself in that unlucky 10% of players who would expect to lose it all if playing at this risk level, then you are flat broke. So, if this $30K represents your life savings, that would be a foolhardy way to play. If it represents money that is easily replaceable (say by cashing in a few CDs when they mature), then it may not be a bad gamble.

Most professional gamblers prefer to use a “Kelly betting” system. Those of you who have read any of my blackjack books know what this is. If you are not familiar with the term, it essentially means that you always bet proportionately to your bankroll in order ensure that you never go broke.

For example, if a player was playing these $1K tournaments with a 5% RoR, he could do so with a $30K bankroll. If this same player decided in advance that he would enter tournaments with smaller buy-ins than $1K if he lost a significant portion of his bankroll (until he built it back up), then his actual RoR would be quite a bit lower than 5%. This is why, in The Poker Tournament Formula, I advise: “…Should your bankroll go into a nosedive, be willing to start entering tournaments with either smaller buy-ins or smaller fields of players, until you rebuild your bank.”

In an ideal world, a player would create a chart of optimal entry fees to pay, based on his current bankroll, that would virtually eliminate any Risk of Ruin. For example, If he lost 40% of his starting bank, he would play in $600 events. If he lost 50% of his bankroll, he would play in $500 events. With a 75% loss, he would play in $250 events, etc. Unfortunately, in the real world, is that in the real world we can’t always find tournaments priced to our needs. If I lose 10% of my bankroll, can I find a $900 tournament?

For any poker tournament player on a limited bankroll, however, it makes sense to follow such a plan as closely as possible. If your bankroll drops by 10% from the amount sufficient for $1k tournaments, you have to decide if you are willing to accept the increased risk of ruin inherent in continuing to play $1k events, or if you had better drop to $500 events, if that is all that’s available below the $1k buy-in level.

Also, be honest with yourself about the actual size of your tournament bankroll. Your playing bankroll should not include your rent money, car payments, living expenses, credit card or installment loan payments, or the like.

Poker Tournament Satellite Frequently Asked Questions

Q: If I’ve already played a few satellites without winning, should I still buy-in to the main event if there are no more satellites?

A: If you have the bankroll to enter the main event at full price, and you have the skill to make money in these types of events, then by all means, buy-in for the full price. If the reason you are playing satellites is to cut your costs because you can’t afford to play the bigger events, then do not buy-in for the full price. Just keep developing your satellite skills and play only in the big events when you win your seat through a satellite.

Q: If I’m just an amateur but I really want to get into a WPT main event just to play with the pros and take my shot at fame and fortune, but that $10K buy-in is a bit steep for my wallet, should I set a limit to the number of satellites I’ll play in an attempt to enter the event?

A: If you are not a skillful satellite player and you are simply attempting to enter a major event cheaply on a long shot gamble, then you should definitely decide beforehand exactly how much you’re willing to spend on satellites, and quit if you hit your limit.

Q: If I’m skillful at both satellites and regular tournaments, should I limit the number of satellites I’ll play for any one event?

A: For a skillful player, it’s a different situation. First of all, if you fully intend to enter the main event regardless of your satellite result, then there is the practical consideration of time. If an event starts at 2pm, and satellites start at 8am, it may not be in your best interest to play satellites for six hours prior to starting day one of a tournament that might go 12-14 more hours.

Some pros always play a satellite or two before major events, even when they can afford the full buy-in price, because the satellites are a good value and can be used to lower their overall tournament expenses. If a pro can win one out of seven $1K satellites in order to enter $10K tournaments, and if he plays just one satellite before each major event, then six times he’ll be paying $11K for his seat, and once he’ll pay just $1K. If he plays 21 of these $10K tournaments per year, his three satellite wins will lower his overall tournament cost (and raise his overall profit) by $9K.

If time constraints and guarding against fatigue aren’t part of the equation, and you have an edge at satellite play, there is no reason to limit the number of satellites you’ll play to enter any one event. During big multi-event tournaments like the WSOP, there are satellite pros who virtually camp in the satellite area, playing one satellite after another, day after day.

On some days, they may play ten satellites without a single win, then they’ll win three or four the next day. They use the tournament chips they win to buy-in to the events they want to play, and sell the rest to players for full value. Obviously, these pros pay no attention whatsoever to limiting the number of satellites they’ll play for any one event. Nor should they. They’re in it for the long run, and if they have the skill to beat the satellite fields and the house edge, they’ll come out way ahead in the long run.

Conclusion

If you are serious about playing in major tournaments with big buy-ins, there is a huge value to developing satellite skills. At the 2006 WSOP, I overheard one player commenting to another that he was surprised at how many of the big name pros were entering satellites, since they could so easily afford the full buy-ins. If you look at the value of satellites as shown in the charts provided in this article, you can see why many pros are attracted to satellites.

If you play a lot of major events, and a lot of satellites to enter these events, you can substantially lower your overall annual tournament costs while increasing your percentage return. I don’t care how much money you have. If you’re paying $10,000 for an event that you could get into for $8,000 (or less), you may be a great poker player, but you’re not that great at financial planning.

Notes and Acknowledgments

Much of the material in this article will be unfamiliar to poker players who have not read The Poker Tournament Formula, because bankroll requirements are estimated using statistical methods that are not taught in your everyday high school math courses. Professional gamblers really need to understand this math, or they will condemn themselves to many years of going broke repeatedly, no matter how skillful they are. As I put it in The Poker Tournament Formula (Chap. 28, “How Much Money Do You Need?”):

It’s the rare blackjack book these days that doesn’t provide at least some information on such topics as standard deviation, the Gambler’s Ruin formula, risk-averse betting strategies, the Kelly criterion, and various related topics, in addition to simplified charts of data that card counters can use to estimate their bankroll requirements.

Most poker books, by contrast, stick to strategic advice exclusively. Blackjack players learn early how to manage their bankrolls; poker players learn early how to hit up their friends when they go broke.

Thankfully, some five months after I published those lines in The Poker Tournament Formula, another poker book has been published that does deal seriously and intelligently, and in much greater depth for cash game players, with the topic of bankroll requirements. This book is The Mathematics of Poker , by Bill Chen and Jerrod Ankenman. This book is to poker what Peter Griffin’s The Theory of Blackjack is to that game.

Unfortunately, like Griffin’s book, much of the material in the Chen/Ankenman book is not readily accessible to a player who has not taken some college level courses in probability and statistics. I still urge any serious poker player to get this book, just as I have always recommended Griffin’s book to all serious blackjack players. The Chen book is essentially a book for math heads, but there’s a lot of discussion on a wide range of topics I haven’t seen elsewhere for serious players. He even delves into risk of ruin when you don’t know your advantage.

Risk of ruin is a statistical measure that blackjack players would be familiar with, but that has been long absent from the poker literature. There is an excellent explanation of RoR in Dr. Allan Wilson’s classic text, The Casino Gambler’s Guide (1965). But neither Wilson’s description of risk of ruin, nor any of the descriptions that have been published in many blackjack texts since then, allow for the formula’s use in a game where there are multiple possible payouts, ranging from a loss of the bet (buy-in), to a modest win (low-end finish) to a very large payout compared to the size of the initial bet.

The first published discussion of RoR that I know of for games with multiple payout possibilities was an article by Russian mathematician Evgeny Sorokin that appeared in the March 1999 issue of Dan Paymar’s Video Poker Times newsletter. In response to Sorokin’s article, professional gamblers Math Boy and Dunbar developed an Excel spreadsheet method of applying Sorokin’s generalized risk equation to virtually any game with a skewed payout structure, and I published their method in the Fall 1999 issue of Blackjack Forum, in their article, “Risk of Ruin for Video Poker and Other Skewed Up Games”. Now, Math Boy has helped me to adjust his method for analysis of poker tournaments (or any other gambling tournaments).

I am especially indebted to Math Boy for creating an Excel spreadsheet for me that would not only estimate a tournament’s standard deviation and risk of ruin, but would automatically recalculate these values based on entering the tournament via satellite, at any satellite cost, and with any selected percentage of satellite wins. The spreadsheet is not currently user-friendly for anyone who is not familiar with some of Excel’s advanced statistical functions, but Math Boy is working on a simpler version, similar to his Patience Factor Calculator, that he plans to make available to players at this Web site in the near future.

Incidentally, a number of players have asked me where I came up with the standard deviation formula that appears in The Poker Tournament Formula as they had never seen a method for calculating standard deviation for a game with a tournament payout structure. My method was simply to modify a formula originally created by Doug Reul, which first appeared as the “Volatility Index” in one of Dan Paymar’s 1996 issues of Video Poker Times, and can currently be found in his book, Video Poker: Precision Play.  ♠

FREQUENTLY ASKED QUESTIONS

by Arnold Snyder
© 2005 Arnold Snyder

Q: Is it legal?

A: Online gambling and poker are legal in most countries. In the U.S., the current administration is very hostile to online gambling and poker. However, though the Congress recently passed a law that may make transferring funds to and from online casinos and poker rooms less convenient, it did not pass a law that makes playing in online casinos and poker rooms illegal in the U.S.

According to gambling attorney I. Nelson Rose, the 1961 Federal Wire Act made betting over the telephone wires on races and sporting events illegal in the U.S., but federal courts have repeatedly ruled that the 1961 Wire Act applies only to sports and race betting, not online casino or poker play. The new law does nothing to change that.

Some online casinos and poker rooms have decided to stop accepting U.S. players for now, while they work out other deposit and withdrawal arrangements. A number of others have contacted us to tell us that, after a thorough review of the new law, they have decided to continue accepting U.S. players, and have already worked out new deposit and withdrawal methods.

See our Recommended Deposit Methods and Offshore Banking Options for more information. Deposit methods accepted at various online casinos and poker rooms are included in the casino listings on our Top Rated Online Casinos and Bonuses page and our Online Poker Bonuses and Deposit Methods page.

We will report on further developments.

U.S. players should also check their local state law before playing online. There’s a lot of legal debate on whether state law applies to online gambling and poker, since the actual betting occurs outside of the state. We don’t really know the answer to that, since no player has ever been charged. For players’ information, the states that have passed anti-online-gambling laws are: Illinois, Michigan, Wisconsin, Washington, Indiana, Nevada, Oregon, Louisiana, New Jersey, New York and South Dakota.

Free games are available at every casino and poker room we list for any players restricted by law from playing with real money.

Q: How much money do I need to do this?

A: Anyone with $500 that he or she can afford to gamble with can afford to get into this venture. $1000 would be better, for a quicker start, but $500 you can afford to risk is sufficient. Most of the deposits you’ll make will be smaller than this, usually $50 to $200. You need the extra funds because you may wish to have money on deposit at more than one Web casino simultaneously. It sometimes takes days (and even weeks) to withdraw your funds from a Web casino. Ideally, it would be best to have $5,000 to $6,000 in order to take advantage of some of the more lucrative bonus offers that allow larger deposits and pay bigger bonuses. But you can build up your bank as you play and take advantage of these bigger bonuses when you have the funds.

For more information on bet sizing and bankroll management for Internet casino bonus play, see Blackjack Betting and Risk for the Basic Strategy Player in the Blackjack Forum Gambling Library.

Q: Do I need to know any special strategies for the games?

A: Absolutely, but the strategies are not that difficult. One nice feature about gambling on the Internet is that you can have your strategy right out on your desk as you play. For blackjack, you simply need to know basic strategy.

Q: How much time is required for this?

A: That’s another nice feature about Internet gambling. The Web casinos are all open 24/7, and you can play sessions of any length you desire, whether a few minutes or a few hours. The actual amount of time it will take you to earn your bonuses will depend on various factors that I can’t analyze precisely for all players. If you have a dial-up modem, the games will be slower than if you have a DSL connection. If you already know the basic strategy for a game, you will play faster than someone who must consult a chart to make decisions. Some casinos’ software is fast, and some software is slow. How much you are betting per hand to meet your WR is also a factor. Depending on all of these factors, you may spend from 30 minutes to many hours playing for each bonus you collect.

Q: How much will my hourly win rate be?

A: Your rate of profit will depend on all of the time factors described above. You will also be spending time on the clerical chores of reading the bonus offers and T & C, figuring out the bonus values, copying the necessary information, and bookkeeping.

Q: If I have $1,000 exactly to try this out, what is the chance that I would lose my whole $1,000 just due to bad luck?

A: I would put the chance of this at slim to none providing you are careful about choosing reputable casinos and bonus offers (lists of reputable Internet casinos are provided through the links at the left), you play the accurate basic strategy for the games you enter, and you follow the advice on this page and on the Traditional vs. Sticky Bonuses page for sizing your bets according to your bankroll and the type of bonus you are playing.

With a small bankroll, you must play more conservatively—which will lower your hourly win rate—in order to protect yourself from simple bad runs of cards (which occur in all casino games). You should never deposit all of your bankroll in one casino. If your bankroll is small, you should be making many small deposits and withdrawals at numerous Web casinos over time.

Players do lose money on individual bonus plays even when the bonus provides them with a significant advantage over the house. If you use an extremely conservative betting strategy—say making all bets of $1 to $2—then I would have to say that your chance of losing all your money would be close to zero if you avoid playing in rogue Web casinos with cheating software. With such small bets, assuming you are playing for the most generous bonuses, you would be extremely unlikely not to show a decent profit for your dollar investment. The bonuses really are that advantageous. But you must decide how much risk you want to assume by playing at a higher level, and what your time is worth to you if you play at a lower level.

I advise any person considering this venture to not play with money that is dear to her or him. You do not gamble with the rent money. If you have only $500 or $1,000 that you can afford to gamble with, then I would suggest starting out very conservatively. Go ahead and play with $2 bets for your first few bonuses. Just take the hours it takes to do this and keep your peace of mind while you’re learning and building confidence. When your bankroll gets up to $1500 or $2000, and you can see the process is working for you, then get a bit more aggressive. Make bets of $4 or $5. Always play at your own comfort level.

For conservative players on small bankrolls, there are certain types of bonuses (called “sticky” bonuses) that will have little value to them until their bankrolls have grown to at least $1500 or $2000. This is because the sticky bonuses must be played more aggressively in order to extract their value, and aggressive play is always more risky. Wait to play these after your bankroll has grown a bit. The traditional (non-sticky) bonuses, and “pseudo-sticky” bonuses, however, will still have value for you, and need not drain your funds with bad fluctuations. For specific advice on bankroll requirements for sticky bonuses, see the Traditional vs. Sticky bonuses link at the left. ♠