Author: Arnold Snyder

by ETFan

See Article Here

Six Deck Unbalanced Red 7 Running Count Conversion to Equivalent Hi-Lo Balanced True Count and Sensitivity of True Count to Errors in Estimating Decks Remaining

by Conrad Membrino
(From Blackjack Forum Vol. XVII #4, Winter 1997)
© Blackjack Forum 1997

rc.u = 23456p + (7p/2) – TAp

Red-7 is almost equivalent to hi-lo count + counting all the sevens as (1/2) each.
rc.u = unbalanced running count = 23456+ (7p/2) – TAp
tc.b = balanced true count
n = number of decks
dp = decks played
dr = decks remaining
rc.u(tc.b) = unbalanced running count corresponding to a balance true count of tc.b
rc.hl = hi=lo running count = 23456p – TAp
rc.u = hi-lo + (7p)/2
if hi-lo has a true count of “t” then rc.hl = t*dr and

rc.u = rc.hl + ExpVal(7p)/2 = t*dr + 2*dp = (t+2-2)*dr + 2*dp = (t-2)*dr + 2*n

rc.u = 2*n + (tc.b – 2) * dr

Number of Decks = 6

Red-7 Running Counts Corresponding to Various True Counts for a Six Deck Game

Six Deck Game	rc.unbal
rc.unbal = 23456 + (7p/2) – TAp	decks played
tc.bal	rc.unbal	1	2	3	4	5
0	12 – 2*dr	2	4	6	8	10
1	12 – dr	7	8	9	10	11
2	12	12	12	12	12	12
3	12 + dr	17	16	15	14	13
4	12 + 2*dr	22	20	18	16	14

Sensitivity of True Count to Errors in Estimating Decks Remaining

Estimation of True Count Using the Red 7:

rc.r7 = red 7 running count	n = number of decks
tc = true count	dr = decks remaining

rc.r7 = 2*n + (tc – 2) * dr

Number of Decks = 8

Red-7 Running Counts Corresponding to Various
True Counts for an Eight Deck Game

Eight Deck Game	rc.r7
rc.r7 = 23456 + (7p/2) – TAp	decks played
tc	rc.r7	3	4	5	6	7
2	16	16	16	16	16	16
3	16 + dr	21	20	19	18	17
4	16 + 2*dr	26	24	22	20	18

Estimation of true count with the Red 7
in an Eight Deck Game:

1. Estimate decks remaining
2. Compare Red 7 running count with 16, 16 + dr, or 16 + 2*dr for true counts of 2, 3, or 4
3. Use calculated true count with High-Low strategy indices.*

(*Ed. Note: Membrino is suggesting here that you may use this true count method not only to estimate your advantage, but also to alter your strategy with all Hi-Lo strategy indices. This is not the way I have developed the Red 7 in the new Blackbelt, but if you used a Starting Count of 0, then you could use this true count methology with any standard set of Hi-Lo count-per-deck indices. –Arnold Snyder)

Sensitivity of True Count to Errors
in Estimating Decks Remaining:

1. The closer to the pivot point, the less sensitive the true count is to errors in estimating the decks remaining.
2. At the pivot point, the true count is independent of the decks remaining
3. Pivot Point of the Red 7: True Count = 2
4. Pivot Point of Hi-Lo: True Count = 0
5. At True Counts ≥ 2:
   (a) Red 7 is closer to its pivot point (tc=2) than the Hi-Lo is to its pivot point (tc=0)
   (b) Red 7 is less sensitive to errors in estimating decks remaining when calculating true count.
   (c) Red 7 gives more accurate true counts than Hi-Lo.

Example:

A = Actual	E = Estimated
dr:a = actual dr	dr:e = estimated dr
tc:a = actual tc	tc:e = estimated tc

Eight Decks

r7 = red 7	hl = hi-lo
tc.r7 = 2 + (rc.r7 – 16)/dr	tc.hl = rc.hl/dr

Eight Decks
dr:a = 4 and tc:a = 3

		Red 7			Hi-Lo
		estimated	error		estimated	error
dr:e	rc.r7	tc:e	(tc:e – tc:a)	rc.hl	tc:e	(tc:e – tc:a)
6	20	2.7	-0.3	12	2.0	-1.0
5		2.8	-0.2		2.4	-0.6
4		3.0	0.0		3.0	0.0
3		3.3	0.3		4.0	1.0
2		4.0	1.0		6.0	3.0

Comments on the Blackjack Shuffle Tracker’s Cookbook

by Arnold Snyder

If you think nothing new has happened in the world of winning blackjack strategies in the past couple of decades, read The Blackjack Shuffle Tracker’s Cookbook.

If you think you already know how to track shuffles, I’ll bet you don’t. Read the Cookbook.

Although the full 3-part Blackjack Forum Shuffle Tracking Series is contained in The Blackjack Shuffle Tracker’s Cookbook, the Cookbook also contains much more. I guarantee you will learn more about shuffle tracking from the never-before-published Parts IV and V of the Series than you ever dreamed possible.

This is some of the secret stuff I’ve been keeping out of print for years. To my knowledge, the only players who know some of this stuff are a handful of trackers that I trained myself. I’ve never even seen these concepts discussed by other shuffle trackers, not in print, not on the Internet, not anywhere. From what professional shuffle trackers have said to me through the years about tracking, I know they don’t know these concepts.

This is not rehashed crap about how to draw maps and size your bets. This is not just a bunch of boring theory and analysis that’s already been discussed to death on the blackjack Web sites.

This is the stuff that none of the other shuffle-tracking experts ever even thought about. This is a guide to making money by tracking shuffles. This is primarily a guide for professional gamblers who want to get two to six times the edge over the house at blackjack that they can get from traditional card counting.

If you want to beat the complex multi-plug, multi-pass, stepladder/R&R combo shuffles that most of the major casinos are using today, and if you want to know why these are the most profitable shuffles available for trackers today, read the Cookbook. The never-before-published Part IV and Part V of the Shuffle Tracking Series will open your eyes to a world of blackjack opportunity you never even knew existed.

---

More information on The Blackjack Shuffle Tracker’s Cookbook

(by Arnold Snyder, From Blackjack Forum Vol. XXIII #3, Fall 2003)

Heresy Today, Gone Tomorrow

This is not so much a Sermon as a blatant advertisement for my new book. As a man of the cloth, it is not only my prerogative, by also my obligation as your spiritual advisor, to use this pulpit for your enlightenment. I know you always read this column first, looking for my pithy, and often brilliant, analogies between pit bosses and various of the knuckle-dragging species; but this month, there is a deeper and more pressing topic. The Bishop has something to sell.

I have just republished my complete Blackjack Forum “Shuffle Tracking Series,” along with a lot of new, revolutionary, and never-before-published information about shuffle tracking, in a new report titled: The Blackjack Shuffle Tracker’s Cookbook.

“Arnold, why would you want to do this?”

“I’m a heretic.”

“You’re not a heretic. You’re an imbecile.”

Well, I guess that’s debatable.

This Sermon is to let you know that the Cookbook is to shuffle tracking what the Blackjack Formula (my first book) was to card counting when it was published in 1980. Many counters who were around at that time considered the Blackjack Formula to be something of an oracular revelation, as it explained for the first time ever how to judge the real value of a game.

Up until that book was published, card counting experts put a lot more weight on the system being used than they did on table conditions. The game factor considered most important at that time was the house edge off the top. A good set of rules was every serious counter’s prime concern. Counters who aspired to professional level play were advised to use multi-level systems (such as Uston’s level-3 APC, Wong’s level-3 Halves, Canfield’s level-2 Master Count (later reborn as Carlson’s Advanced Omega II), the level-2 Hi-Opt II with multiparameter tables, etc.

All of these professional-level counting systems included charts for adjusting play with a side-count of aces, and they included 150+ strategy indices. The multi-parameter approach was carried even further in many of the high-end professional-level systems. Pros had strategy charts available which allowed them to use as many as half a dozen side-counts with Hi-Opt I, Hi-Opt II, and the DHM Professional system.

It was widely believed among experts at that time that as the games got tougher (primarily, as more decks were added), the counting systems had to get more complex to beat the games. No attention whatsoever was paid to the importance of deck penetration, nor did counters have any idea of exactly how much of a betting spread they would need to beat any specific game conditions.

Only two authors at that time had workable approaches to beating shoe games. Stanford Wong, in his groundbreaking Professional Blackjack, advised players to table-hop shoes in order to avoid playing in negative counts. And, because Wong was not playing in negative shoes, he also provided the first intelligently abridged set of indices, as he tossed out most of the strategy changes that occurred at negative counts.

Ken Uston, in The Big Player and all of his books, discussed Al Francesco’s “big player” team concept for shoe games, which also kept players from betting in negative expectation situations in shoe games. Both Francesco’s and Wong’s approaches were adopted out of the necessity to camouflage card counting strategies, as just about all casinos had learned by the 1970s to identify card counters by their betting spreads.

But, camo or no camo, the approach of most pros was to play the game with the best set of rules, using the most complicated advanced system they could handle, and every index number they could squeeze into their heads.

So, in 1980, I began my career as blackjack’s official heretic. Over a period of three years, in three books, a couple technical reports, and within the early pages of this very quarterly, I proposed a lot of hare-brained ideas.

I told players that finding deep penetration was more important than keeping a side count of aces.

I said that most of the 150+ index numbers players used were virtually worthless.

I stated that a simple, level-one, unbalanced counting system could perform by running count with nearly the same power as a “professional level” true-count system in most shoe games.

And I got a lot of flack from many of the game’s experts until independent computer simulations bore out my claims.

Most players today, however, don’t think of me as a heretic. They weren’t around back then. I’ve become mainstream, stodgy, just another stick in the blackjack mud. So, simply to add a little more fun to my life, it’s time to hit the heresy trail again.

Shuffle trackers today are in the same boat as the pre-1980 card counters. Trackers look at all the wrong factors, and devise strategies based on their general misunderstanding of the opportunities. The approaches to tracking today are eerily similar to the old days of card counting, when teams of players were struggling to get an edge in shoe games with 65% pen, using 150 strategy indices and side-counting aces, when the game across the street, with a slightly worse rule set, had 85% pen, and could have been murdered with the simple Hi-Lo count and a comparative handful of indices.

There was something truly weird going on back then. The casinos with the less-attractive rules felt that they had protected themselves from card counters, oblivious to the fact that their deep penetration actually made them sitting ducks for any counters who understood the value of penetration. But since counters didn’t know the value of penetration, the less-attractive-rules countermeasure worked! The casinos with the truly best games were protected because card counters simply didn’t play there!

The old-time card-counting experts were not, of course, incorrect that the multi-level, multi-parameter, mega-index systems were the most powerful systems ever devised by man. But they were so enamored of accurate play (even when the game itself sucked!), and so satisfied with each other’s convictions, that they never looked for the strongest ways that a player could use the count in order to get the most money from the casinos.

Tracking experts have blundered just as badly. They have devised all of these incredibly complex methods for tracking casino-style shuffles, with no idea that there is a stronger way to get more money faster. Just as with card counting, they worked out the math on their old ideas to the nth degree, without ever seeing the strongest profit opportunities.

And, ironically, the casinos have responded in kind. Just as the casinos used to foil counters by putting in less attractive sets of rules, today’s casinos have put in shuffles designed to foil the types of tracking strategies that today’s tracking experts advise. In fact, the casinos do not know what constitutes a beatable shuffle! They simply know what the trackers are out there looking for, and they foil the trackers by offering something different. Lucky for the casinos! Like the counters of 25 years ago, trackers today are in the dark ages and they ignore the juiciest opportunities. The ignorance about shuffle tracking pervades both sides of the table.

Shuffle trackers today believe that the most profitable shuffles are the simplest shuffles. They believe they will find their best opportunities in the few remaining one-pass, riffle-and-restack shuffles, preferably with big grabs so the slugs are easy to follow and do not get broken up.

So, most of the big casinos today employ multi-pass shuffles with multiple plugs, small grabs, multiple piles, and usually at least one stepladder (dilution) pass. These complicated shuffles annoy the tracking experts no end, because they believe that the most profitable approach to shuffle tracking is to track the shuffles. In fact, the most profitable approach to shuffle tracking is not to track the shuffles, but to track the slugs. These are two entirely different approaches.

A shuffle tracker who looks for opportunities by looking for the “easiest” shuffles to track is like a card counter who looks for opportunities by looking for the lowest house edge off the top. The smart counter does pay attention to the house edge off the top, but he chooses playing opportunities by looking for the game which offers the most frequent, and strongest, player advantages. (This usually equates to deep penetration, and not necessarily a good rule set.)

Similarly, the “easy” shuffles, as a general rule, offer weak slugs. The more complex shuffles, on the other hand, offer strong slugs, and the most frequent, and strongest, player advantages.

In fact, the complex shuffles do protect the casinos from shuffle trackers, not because the shuffles lack tracking opportunities but because the tracking experts have analyzed these shuffles as “weak” and trackers avoid them. So, here’s a bit of heresy for you to sink your teeth into: These complex shuffles offer trackers the greatest slug tracking profit opportunities available in shoe games today!

Why should I publish this heresy at this time? (And believe me, the tracking experts will be as incredulous at this notion as Peter Griffin was in 1983 when I said the Red Seven Count would perform in shoe games nearly as well by running count as the full-blown Hi-Lo. I devised the Red Seven Count almost entirely from the data I found in Griffin’s book, yet he thought my idea of using an unbalanced point count system was a huge mistake.) Won’t the publication of these secrets hurt the shuffle trackers who already know this stuff and are keeping it to themselves? Well, I don’t claim to know every shuffle tracker on the planet, but my personal assessment of the situation is that there aren’t any trackers out there using this stuff. The only trackers who know this stuff, to my knowledge, are the handful of players I’ve personally trained.

The question of whether or not I should reveal this information at this time comes down to a question of whether or not the revelation will take money of my own pocket. The trackers I’ve trained are playing for me. Will I regret this decision because it will ultimately hit me in the wallet? I’ll take my chances.

Ah, the quandaries of life, made even more perplexing by my position as your religious leader, the man you trust to guide you on your path to wealth and spiritual fulfillment.

Should I have published the Blackjack Formula in 1980? Should I have told players at that time that deep penetration was the single most important factor in assessing a game’s value to card counters? Card counting had been around for almost 20 years at that time, but this was not known by players – pros or otherwise. Ken Uston did not know this. Lawrence Revere never knew this. Ed Thorp did not know this. Stanford Wong did not know this. Lance Humble did not know this. Ian Andersen did not know this. Peter Griffin had devised charts which showed this to be true, but he had never done any analysis to discover the practical applications of his findings.

Mathematicians almost never understand the meaning of their own work. In fact, I’ll let you in on a secret: my methods for analyzing shuffles have been derived, almost entirely, from information in Peter Griffin’s Theory of Blackjack. Yet, Griffin never even mentions shuffle tracking in his book, and the one time I tried to discuss tracking with him back in the mid-1990s, he told me he didnít know that much about it. And here he had the key to unlocking shuffles in black and white in a book he’d written 15 years earlier!

I have no regrets today about telling card counters back in the 1980s that deep penetration was the key to value, that simplified sets of indices can be powerful, that side counts are not necessary, etc. I enjoy making the casinos scramble, because they scramble so slowly and incompetently.

Smart players who apply themselves should have years to reap the benefits from the Cookbook. Twenty-three years after the Blackjack Formula, blackjack games with deep penetration are still out there, and pros are still making money from my heresy. The casinos are still in a jam, trying to protect their games while competing with each other and catering to their customers’ likes and dislikes.

The number of shuffle trackers, in fact, who will make money from learning how to track slugs, instead of how to track shuffles, is small. It’s even smaller than the number of card counters who currently make money from knowing the value of deep penetration. Most players just do not do their homework. A handful of serious pros will reap the rewards.

The casinos are already doing everything they can to convert their games to machine shuffles, but they can only do this as quickly as their customers accept the change. In Las Vegas, they keep trying machines, then going back to hand shuffles. In new locales, machines are often brought in as the norm from the start. The machine salesmen are fear-mongers, hyping their wares to a gullible industry.

The Cookbook, on the other hand, honestly describes the difficulties of tracking shuffles for profit. Intelligent game protection personnel who read this report will realize that they are, for the most part, wasting huge sums of money reacting to a phantom when they buy these auto-shufflers. The casinos make twice as much money from incompetent players who attempt to track shuffles as they lose to competent trackers. Tables with hand shuffles, attractive rule sets, and deep penetration will always make more money than tables with blackjack games that players see as unbeatable.

In any case, my faithful flock, if you think nothing new has happened in the world of blackjack strategies in the past couple of decades, read the Cookbook. If you read the BJF Shuffle Tracking Series back in 1994, but found the concepts too difficult to apply in the casinos, read the Cookbook. If you think you already know how to track shuffles, I’ll bet you don’t. Read the Cookbook.

Although the full 3-part BJF Shuffle Tracking Series is contained in The Blackjack Shuffle Tracker’s Cookbook, the Cookbook also contains much more. You will learn more about shuffle tracking from the never-before-published Parts IV and V of the Series than you ever dreamed possible. This is the stuff I’ve been keeping out of print for years, concepts I’ve never seen discussed in print, not on the Internet, nor anywhere else, not even spoken of in whispers over champagne at Max Rubin’s Blackjack Ball. If there are other trackers who know this stuff, they’ve been keeping a lid on it.

This is not rehashed crap about how to draw maps and size your bets. This is not just a bunch of boring theory and analysis. This is a guide to making money by tracking slugs. This is a guide for professional players who want to get two to three times the edge over the house at blackjack that they can get from traditional card counting, and who want to be invisible while doing it.

If you want to beat the complex, multi-plug, multi-pass, stepladder/R&R combo shuffles that most of the major casinos are using today, and if you want to know why these are the most profitable shuffles available for slug trackers today, read the Cookbook. The Cookbook will open your eyes to a world of blackjack opportunities you never knew existed.

And that’s heresy. ♠

Get The Blackjack Shuffle Tracker’s Cookbook. If you are new to shuffle tracking, there is an introduction to this professional gambling technique in Arnold Snyder’s Blackbelt in Blackjack.

Convexing Calculations for McDowell’s Blackjack Ace Prediction Or — I’m Sorry!

by ETFan
(From Blackjack Forum XXIV #2, Spring 2005)
© ETFan 3/9/05

    In the Spring issue of Blackjack Forum Online, Arnold Snyder published his critique of David McDowell’s book, Blackjack Ace Prediction, and put forth a correction of the now infamous EV calculation in Chapter 7. Since I wasted much of my youth pondering ace sequencing, I excitedly posted another approach to solving this EV, which I thought might be of interest to the few people involved in such an arcane endeavor.

    I had no idea my little post would cause such a furor. It didn’t occur to me there would be any controversy regarding Arnold’s review of the book. The book came out. Some people endorsed it. Then Arnold took the time to study it and found some problems — something which has happened over and over through the years. Generally, the reaction of the blackjack community has been to look more closely at the product, do some calculations, and close ranks behind the Bish, since he is widely acknowledged as a blackjack expert and all around sage, soothsayer, and holy man.

    But this time was different.

    Since I posted my little calculation, and a half dozen or so follow ups, I’ve been informed that I couldn’t possibly know anything about the arguments posed in the book, since I hadn’t read it. That I needed to look at Table 3-2. That the errata explained everything. That I was grievously ill-informed to believe Snyder, since he panned the book purely to embarrass another authority. That any word of defense for Arnold is like spitting on that authority, since Arnold has friends who say mean things about that person. That Snyder is [words too severe for your ear, gentle reader.] That I am dishonest. That all books have errors anyway. That I have lost all credibility. And that I have angered forces so powerful, that my personal, private communications are no longer secure.

    I’m sorry. I’d like to retract everything I said about the book.

    I’d like to, but I can’t. Once I had the book in my hot little hands, three things were quickly evident. 1) Everything Snyder said was accurate. 2) The reference to Table 3-2 was a sidestep; pure flimflam. And 3) The errata explains nothing. It actually provides more nails for McDowell’s coffin. Those who are deeply concerned about any of these three points, or my credibility, have permission to skip to the Infamous EV section. Those who passed Probability 101 and would like to see some quick samples of elementary errors in the book, may want to skip to The First Calculation and A Broken Calculation.

    I wish Blackjack Ace Prediction was the greatest book ever written on any exotic blackjack technique. Then dealers would have to expend tons of energy learning new shuffle routines, and that would divert attention from my own advantage play, which has nothing to do with ace sequencing. I wish ace sequencing was all it’s cracked up to be in the book and more. Then I could just post that Arnold had lost it, and my calculation had no relation to anything in the book. I would have loved writing that post. But I’ve owned the book for three weeks now and — sorry — no can do.

    The remainder of this article will attempt to explain why I don’t believe the pie-in-the-sky promises outlined in the first few pages of Chapter One. Why I turned my attention away from ace sequencing some 15 years ago, and toward a simpler, more rewarding ploy. And — sorry again — why you must take everything McDowell says about ace sequencing with two or three truckloads of salt.

The Promise:

Chapter 1, pg 18: “If you get it right, Ace Prediction in modern games will get you (conservatively) a 3-4% edge.”

    It’s clear, from the context, that McDowell is talking about overall edge — not just the edge on an occasional hand, or a half dozen times an hour. He compares it with card counting, but clearly, card counting provides you with an occasional 3-4% edge. The reason we are to take up sequencing is the far greater net edge — 3-4% compared with 1-1.5%. This 3-4% figure is not supported anywhere in the book. Sorry — I will show you it is insupportable.

Filler and More Promises

    Pages 1 through 16 simply contain cover pages, Table of Contents, a list of “Tables and Figures,” Acknowledgments, and the Foreword by Michael Dalton. Then on pg 17 we read: “huge advantages over the casino,” and on page 18: “Predicting Aces is a very simple idea and it’s easy to do at the table.”

The First Calculation

Chapter 1, pg 19: “To pull this off requires nerve, but getting big money on to the table when you have the advantage is the real key — not the size of your edge. Four bets at 2% is just as good as two bets at 4%, and it looks less suspicious.”

    Sorry, no. It isn’t. When you have a 4% edge, you can bet twice as much and keep the same risk of ruin. 4 bets of one unit at 2% are only half as good as 2 bets of 2 units at 4%. The first has an EV of 4 x 0.02 x 1u = 0.08 units, the second, an EV of 2 x 0.04 x 2u = 0.16 units. You can’t rehabilitate the argument by saying “we were going to bet table max in either case.” If you bet the same, with a higher edge, you’re rewarded by a smoother uphill climb in bankroll. Your situation improves by every risk-adjusted measure, such as Kelly’s G, N0, or SCORE. This should be reflexive knowledge for all advantage players. Note the formula for SCORE invokes advantage squared in the numerator, not just the advantage.

    It has little to do with the rest of the book, but it’s a note of caution that we should carefully chew any further math in the book before swallowing. This is no typo. He’s got the suspicion factor turned round 180 degrees. The doubling of EV allows extra room for all manner of cover play. The very first calculation in the book betrays a fundamental misunderstanding of advantage play math.

More Acknowledgments and History

    The rest of pg 19 and 20 through 37 contain brief synopses of prior works, each generally getting one paragraph; a bit longer for Golumb and Thorp. Some have said this is the best feature of the book — its thorough list of references. However, no original math in this section, and nothing of earth shaking importance for sequencers in the real world.

The Second Calculation

    On pg 38, McDowell correctly interprets what Epstein wrote about “highly expert dealers” — up to a point. He correctly plugs the numbers into Epstein’s formula and comes up with 8/9, 8/81, and 1/81 as the probabilities for 1, 2 and 3 card interlacings in a single riffle. But before we even have a chance to say “well done, McD” he tries to get creative.

    Here is the quote from the bottom of the same page: “When calculating n above, zero-card interlacings were grouped with one-card interlacings.” [No, they weren’t.] “Since, in practice, they are identical to two-card packets, we deduct 8/81 from r1 and add it to r2. The probability of one-card packets becomes 8/9 – 8/81 = 64/81 = 0.78; two-card packets = 8/81 + 8/81 = 0.20 …”

    Sorry, incorrect. Nowhere, in Epstein, is permission given for such an adjustment. In fact, part of the rationale McDowell gives — the centrality of “transition probabilities” in Epstein’s analysis — provides proof this was no part of Epstein’s premise. The very first transition probability Epstein gives is Pi(AAA) = 1, which means that a transition to the alternate packet (ie. the dealer’s other thumb during a riffle) is certain when three cards in a row fall from a given packet. This is clearly untrue when zero-card interlaces are permitted. All the other transition probabilities are also untenable with zero-card interlaces. Epstein was simply saying one, two or three cards could come down from a given packet with specified probabilities.

    Perhaps McDowell was confusing interlaces with what he calls “gaps,” later on. (A two-card interlace results in one-zero card gap, a three-card interlace results in two-zero card gaps, etc.) Or maybe he just wanted Epstein’s results to fall in line with Hannum’s and Curtis’ results for Table 2-5. Sorry — they don’t. Of course, making similar adjustments in Hannum’s or Curtis’ results wouldn’t have made such a nice, neat chart.

Moving On

    Pg 40 presents some more of Hannum’s results, without comment, and gives an encouraging quote from a 1988 paper entitled Non-Random Shuffling Strategies in Blackjack. No practical data.

    Pg 41 contains an explanation of Shannon’s formula for information and entropy. I am not sure why base 2 was chosen, but using another base (eg. natural logs) would simply change the unit of information measure, so I have no comment. I’m not an expert on information theory.

    But pg 42 — here we go again. McDowell presents — with an air of authority — Table 2-7, “Information Loss in Card Shuffling,” which he attributes to Trefethan, Lloyd N., AND Lloyd M. Trefethan. He concludes “After the first riffle … 52 bits of information about deck order are destroyed (23.05%) and 173.58 bits remain (76.95%) … After the fourth riffle, only 12% of the original information remains … After ten riffle, I = 0 bits and U … = 225.58 bits.”

    Sorry. You don’t need to be an expert to see this for the nonsense that it is. On pg 167 of The Theory of Gambling and Statistical Logic, Epstein presents similar formulae and immediately concludes: “The transition probabilities for a 52-card deck … are far beyond the reach of any practical high-speed digital computer.” No, computers aren’t that much faster today than they were when Epstein was around.

    McDowell makes it seem these information figures apply to any dealer. To any riffle! There’s no report on the experimental procedure used, no transition matrices, no specification of how the various Pi values were obtained. The Pi values (the various probabilities ascribed to every possible deck state) depend critically on the dealer’s individual riffle.

    I have a copy of the paper by the Trefethens. A computer shuffle was used with the property that “one shuffle is equivalent to separating the n cards into two subsequences at random, then concatenating the subsequences, a process that comes very close to moving the deck to one of 2^n possible configurations with equal probability.” Thus, the “riffle” used moved the deck into any one of 2^52 = 4.5 quadrillion possible states with “close to” equal probability. Sorry — such a riffle, while interesting from an information theoretic point of view, has virtually nothing to do with a riffle we’re likely to encounter in a casino.

Chapter 3

    This is where he lays out his methodology (which he credits to Thorp) for counting “gaps” between initially adjacent cards. Before presenting the tables, he writes “1530 observations of Di (n = 1530) were recorded for a one and two-riffle shuffle, and … 5100 observations (n = 5100) for a three-riffle shuffle. Therefore, the statistics presented can be looked upon as good estimates of the parameters of the parent group for all card shuffles …”

    Sorry — no way. First of all, he’s assuming the gaps are distributed identically through all parts of each riffle — start, middle, and finish. You can’t just do that. Even if you feel you have a very smooth, even shuffle, you need evidence for that in the form of statistics. You can’t just magically transform a sample of 30 into a sample of 1530 based on what you think should happen.

    Consider what happens when you riffle a deck of cards. Don’t 3-5 cards sometimes come out together at the beginning of the riffle? Isn’t there usually a packet of 2-6 “leftovers” at the end? McDowell may, indeed, be one of the smoothest rifflers in Kingdom Come, but identical distribution, from bottom to top??

    Moreover, McDowell assumes his shuffle is a fine sample for all shuffles. Ultimately, his sample size is one (1). And I honestly believe it’s an inappropriate sample, because his shuffle is apparantly quite unusual. More on this later.

From pg 47: “The exact number of single-card separations may vary from person to person but, over a large number of trials, the most frequently occurring distance between cards after one riffle always will be one card.” [Emphasis in the original]

    Sorry — wrong. The most common “distance” (gap) for me, by far, is zero cards. Remember, a zero card gap, under his definition, occurs when two initially adjacent cards stay adjacent through the shuffle. I would challenge you to pull out a deck of cards, put it in any well-defined order, riffle once, and check to see how often precisely one card is interleaved between initially adjacent cards. I just did it now, and I got 9 one-card gaps and 22 zero-card gaps.

    And the number of separations may vary from person to person? May vary?? Think about skin tone and the thickness of a single card. Think about what’s required to make precisely one and only one card drop before action switches to the other thumb. Think about dealer’s with long fingernails!

    Well, on the same page he presents his table of gaps, based purely on his personal shuffle. Then he does a little massage on the numbers: “The percentage of sequences broken was … 1.96%. Before dividing the total for single-card separations, which includes broken sequences, the raw value was reduced by the number of broken sequences.” I was a little perplexed when I read this. Then I looked at his table of adjusted percentages vs. raw percentages. In the Adj. column, the percentage for a 1-card gap (his mean) goes down and all the other percentages go up! A little study led me to discover he had subtracted all 30 of the broken sequences (one for each riffle) from the mean as well as the total sample, but none of the broken sequences from any of the other card gaps.

    Is there some reason to believe 0-gap sequences can never be broken? Sorry. None that he explains, and none that I can imagine. 2-gap sequences — unbreakable? Not. He might claim he’s trying to be conservative, since bets will be placed based on the mean (which is also the mode — the most common — in his charts) gap. But he bases his mean and standard deviation statistics on this adjusted sample!

    It would have been easy for McDowell to subtract out broken sequences from all gap sizes on a prorated basis, but, for reasons known only to him, he chose to distort the percentages.

    But wait — I skipped a page. From pg 46: “This distribution has an almost symmetrical “bell shaped” normal curve. This makes the arithmetic mean and standard deviation appropriate measures of location and dispersion …”

    Well, the mean and standard deviation are always good measures of, well, the mean and dispersion about the mean. But he’s basing his normal distribution theory on four data points for the one-riffle shuffle. The only reason it looks normal is because he’s connected the dots with a nice, bell shaped curve. And the curves for two and three riffles, with a few more data points, look less and less normal.

    This is a common mistake, even among experienced statisticians. People tend to assume raw data is normally distributed, when in fact, it’s quite an unusual distribution in nature. But — sorry — four data points?!?

    It was very important to McDowell that his one-riffle gaps conformed to a normal distribution. You see, the central limit theorem tells us that any large set of independent random variables (typically 20 or more, but certainly 5 or 6 at minimum), when added, have a normal distribution. But McDowell has page after page of calculations with the normal distribution on two and three riffles. Normal distribution calculations are very simple. All you need are two parameters — mean and variance — and you know everything about that distribution. But if one-riffle gaps aren’t very near normal, there’s no reason to assume two or three-riffle gaps are normal either.

    Let’s see what results from this “normal” distribution. On pg 48, McDowell gives the mean = a gap of 1, with standard deviation = 0.61 of a card. Using the usual formula, the mean should be represented at 0.5 to 1.5, or 1.22 standard deviations on either side of the arithmetic mean. From my trusty table of areas under the normal curve, I get 78% compared to McDowell’s recorded 69%. That leaves 11% for everything to the left of the mean and 11% for everything to the right of the mean. McDowell has 18% of his data to the left of mean, and 13% to the right — a lopsided curve.

    I’m sorry. I don’t believe McDowell has enough evidence to declare that a “typical” dealer’s riffle has gaps which are normally distributed. Let’s look at some facts.

Item #1: Here are my results for 30 rif-rif shuffles, corresponding to McDowell’s Table 3-2.

     Gaps          Percentages:
      0             16.40523%
      1             17.71242%
      2             16.53595%
      3             15.42484%
      4             9.542483%
      5             7.45098%
      6             4.575163%
      7             1.960784%
      8             1.764706%
      9             .9803922%
      10            .7843137%
      11            .3267974%
      12            .3267974%
      13            .130719%
      14            .130719%
      15            6.535948E-02%
      16            0%
      17            0%
      18            0%
      19            0%
     Broken:        5.882353%

    I defy anyone to fit this to a normal curve. Now people will point out that my riffles can’t compare to the riffles of someone who deals for a living. However, for several years I handled and shuffled cards nearly every day, playing gin rummy with friends and relatives. I was considered overly precise — overly fussy. Every time I walk into a casino I see dealers with riffles that make me look like Steve Forte. But if nothing else, my results show that different dealers have very different signatures. McDowell’s mean for two rifs was gap = 3 at 36% compared to my 15%, and at gap = 0 he had 5.1% compared to my 16.4%.

Item #2: On pgs 90 and 91, McDowell discusses the Bayer-Diaconis formula: A/(A+B) for determining the probability the next card will come from a given thumb in a riffle. If a riffle follows this distribution, the gaps after one riffle will be approximately 0.5, 0.25, 0.125, 0.0625 … not even vaguely approaching normal. After two rifs, it’s easy to show zero sized gaps will come in at approximately 25% (compared to McDowell’s 5.1%). Again, not a normal distribution.

Item #3: McDowell also mentions Curtis’ interleave results where interleave = 1, 66% of the time. This means gap = 0 well over 30% of the time. Not normal. After two rifs, gap = 0 approximately 10% of the time.

Item #4: Finally, let’s look at riffle results for someone who was a professional. For those who have Wong’s Professional Blackjack, take a look at Table 89 “Post-Shuffle Gaps Between Initially Adjacent Cards.” This is Wong’s record of ten rif-rif-strip-rif shuffles on a single deck by a professional dealer who worked in Las Vegas for five years. Point #1: The most common gap for this dealer was one (1) card, compared to three (3) cards for McDowell’s rif-rif-rif. The extra strip made initially adjacent cards come closer together?? Sorry — afraid not. Point #2: There are 23 data points to the left of the mode in Wong’s data, and 430 points to the right of the mode. Not even remotely a normal distribution. Even if we try to shoehorn Wong’s data into McDowell’s model, there are 120 points to the left of 3 card gaps, and 347 to the right.

    I’m not even sure Wong’s chart represents what we’ll see on an average day in your corner casino. I think dealers who know they are being tested are likely to be more fastidious than dealers in a casino, even if they’ve been instructed to behave “normally.” But if a dealer riffles with anything like the distributions in items 1 through 4, it destroys McDowell’s frequency calculations in Chapter 8, because if the mode clusters toward small gaps, such as zero or one-card gaps, instead of three, the frequency of betting opportunities is cut 50% or more.

    Maybe the normal curve only applies (very loosely) to “highly skilled” dealers, as Epstein described them. Maybe. But how many times have you seen experienced casino dealers riffle 30 cards in one hand and 40 cards in the other? You can’t have that and get results like McDowell’s. How many times have you seen a group of 4 or more cards slap down together at the end of a riffle? If the piles were even to start, that means there’s 4 cards somewhere else that didn’t fit neatly into the A|B|A|B … pattern, and I promise you, you’re going to come in below his 69% per riffle mean.

    McDowell must have practiced riffling long and hard. It takes tremendous effort to interleave cards one to one over twice as often as any other combination, as showed in his one riffle statistics. Sorry — I don’t believe very many dealers train hard to make their riffles predictable for all the would-be sequencers in the world.

    Do you begin to get the picture? On nearly every page where McDowell isn’t giving a history lesson, or quoting an authority, or telling a cute story, I find deep and troubling evidence that he’s in far over his head. Fundamental errors in math and methodology. Buy it for the history, or the list of authorities, or the beautiful cover, or for the cute stories. But don’t buy it if you need someone to hold your hand through the math, or because you believe sequencing is an easy road to quick riches.

    I won’t go through every page. Let’s get some background, then skip to the good part.

A Very Exclusive Club

    On pg 14 of Epstein’s Theory of Gambling and Statistical Logic we find something learned by every school boy who’s made it through week one of Probability 101. Not to get too technical, I’ll put Epstein’s Axiom III into my own words: If you have a group of events which are mutually exclusive, which means that no two of them can occur together, the probability that any of these events will occur is the sum of the probabilities that each one of them will occur individually. People who understand this rule, along with the rule for multiplying independent events, are in a very exclusive club, since they are able to solve a whole class of interesting problems in probability.

    Epstein goes on, with equation 2-2, to extend the rule for events which are not mutually exclusive. Again, in my own words: If A and B are any two events, the probability A and/or B will occur is P(A+B) = P(A) + P(B) – P(AB), where P(AB) is the probability that both A and B occur.

    What we can learn from equation 2-2 is that people who ignore the mutually exclusive rule are doomed to fail. If events are not mutually exclusive, there is, by definition, a non-zero probability they can occur together. P(AB) <> 0. Therefore, probabilities obtained by simply adding or subtracting non-exclusive events are always wrong. You need extra terms to subtract out all the ways the non-exclusive events can happen simultaneously.In addition to the words “independent” and “mutually exclusive,” first year probability students usually hear the word “exhaustive,” which means, simply, that you have to make sure you enumerate all mutually exclusive events you are interested in before adding.

A Broken Calculation

    On pg 59 of BJAP, McDowell presents Table 3-4, of four different two-card sequences, together with their probabilities of being broken (4/51 in each case). He also presents the total — 16/51 — for some reason. Note that if there had been 13 or more sequences (certainly conceivable) the total would be >1. You can’t have a probability greater than one. Of course, these are not mutually exclusive events, so we shouldn’t be adding them in the first place.

    McDowell repeats this mistake in Tables 3-7 and 3-8, and in the errata, we learn he believes these totals are probabilities. “The total probability for “Stripping” (12/31) should be replaced with 12/51″ Sorry — McDowell is apparantly not a member of the club.

    On to pg 60. Quoth McDowell: “The average probability p1 of one sequence being broken is 4/50 = 0.08. The probability p2 of two sequences being broken is 0.08 X 0.08 = 0.01, (p3 and p4 ~= 0). Well, this assumes the four sequences are independent. I won’t object. Close enough. But then we have: “Finally, 1 – (p1 + p2 + p3 + p4) gives the probability p0 for zero sequences.

    Sorry — no. For this to be true, it would follow that p0 + p1 + p2 + p3 + p4 = 1. In other words, he’s saying it’s certain that either no sequences are broken, or one specific sequence is broken, or two specific sequences are broken, or three or four. Do you see the problem? He has a list of things that can happen, but it’s not exclusive, and it’s not exhaustive. There are many other terms needed in ths calculation.

    When McDowell writes 4/51 = the probability of a specific sequence being broken, he does not exclude the possibility that another sequence may be broken at the same time. In fact, he has the non-zero probability that two sequences will break together right in the calculation!

    McDowell’s is also not an exhaustive set of events. For example, if two sequences are broken, that doesn’t prove that some other two sequences aren’t also broken.

    A mutually exclusive list looks something like this:
(The probability that no sequences are broken) + (The probability that any one of the four sequences are broken alone, with no others) + (The probability that any two of the four sequences are broken alone, with no others) + (The probability that any three of the four sequences are broken alone, with no other) + (The probability that all four of the sequences are broken) = 1

    Rather than write out all the possible combinations of four events, to use the above formula, there is a simple way to compute this probability. Since the sequences are assumed to be independent, it follows that their negations are also independent. In other words, the probability of the first sequence not being broken is 1 – 4/51 = 47/51. Similarly the probability of the second, third and fourth sequences not being broken are each 47/51. So the probability that no sequence is broken = p0 = (47/51)x(47/51)x(47/51)x(47/51) = 0.72129. Ambitious students are encouraged to work out the probability for the long list I provided above to see that the first term works out to this exact same result.

    This is quite different from the 0.91 given in the book. It’s such a common, elementary mistake, I believe I noticed it within ten seconds after I flipped to page 60. Within a minute or two I cranked out the correct answer on my trusty TI-89 calculator. And I’m no math professor. My point being, sorry, but to anyone who’s taken elementary probability, this is not rocket science.

    But I’m afraid it gets worse. On the next page, pg 70, McDowell does a little calculation using his formula 3-1: B = (ih – ic) + bh, for a more complex rif-rif shuffle he analyzed on computer. He arrives at 0.096 for the probability of a broken sequence, which, he says, “is in close agreement with the 0.9 estimate calculated using pencil and paper above.” I believe he means the 0.09 calculated in Table 3-5 from 1 – 0.91 — ostensibly the probability that one or more sequences will be broken in a rif-rif shuffle. This is the only interpretation that makes sense, since we know that 0.096 is not in close agreement with 0.9. (Where probabilities are concerned, these are about as disparate as it gets.)

    But we now know the 0.91 given in the book for p0 was incorrect, so we know the 0.09 for sequences > 0 was also incorrect. The correct probability for sequences > 0 is 1 – 0.72129 = 0.27871, which can hardly be taken as corroboration for the 0.096 produced by McDowell’s formula!

    How can the formula be so wrong? Guess what — same problem. Once again, he’s adding and subtracting probabilities without the critical assessment that they represent mutually exclusive events. You simply can not take a bunch of probabilities and say “let’s add the ones we like and subtract the ones we don’t like,” but it honestly seems this is how McDowell thinks.

    Formula 3-1 also provides the answer to why McDowell did the little “massage” I mentioned on pg 47. He subtracts out the broken sequences earlier, so he can add them back in using his pet formula 3-1 later on. So (sorry!) he does an invalid transformation on his raw data, which messes up his standard deviation calculations on a normal curve which isn’t normal, so that he can retransform later using his invalid formula.

    At a certain point you just have to laugh. It’s really rather charming, this valiant attempt to slay the great sequencing Goliath. But at least that David had a sling-shot. 😉 😉 Let’s mush on to…

Page 82

    I just wanted to pause here and congratulate David. This may be the only page in the book filled with calculations that are all correct. I don’t know where he got the formula for the probability of two consecutive aces, but I think it’s 100% accurate, as are all the numbers and the chart. Nice job.

The Infamous EV Calculation

    This is where all the fun started. But before we get to the EV calculation proper, let’s look briefly at the preceding page 113, where we find the author’s formula 7-2 for P(h) — the probability the ace will “hit the money.” His formula: a – (b + f), again violates the rule about adding probabilities that aren’t mutually exclusive. I just wanted to mention that it seems reasonable to assume these probabilities are independent, so a better approach might be:
a(1-b)(1-f), giving 0.29 compared to McDowell’s 0.13, for a = 0.38, b = 0.15 and f = 0.10. I haven’t tested it; I just wanted to throw it out there. But before anyone decides to plug 0.29 into the equation, they might want to study Radar and Snyder’s work on the proper estimation of false keys.

    Now the formula:

E(X) = E1h + E2d + E3m

Where,
E1 = player’s expectation if the Ace hits the money
E2 = player’s expectation if the dealer gets the Ace by accident
E3 = player’s expectation if the Ace misses the money
h = probability that the Ace will hit the money
d = probability that the dealer will get the Ace by accident
m = probability that the Ace will miss the money

    Note that he’s dropped the P() from P(h) for this formula.

    Now, there is an interpretation under which this formula is correct! If the Ace in question refers to the one and only one ace being sequenced, then I have no problem with the formula as written. Note that E3 doesn’t say expectation off the top, precisely (though he uses -0.5% in the example, which is the prototypical off the top edge). Unfortunately, in the preceding paragraph, McDowell writes “In this case that means the dealer gets six additional Aces.” Aces, plural. If there is more than one ace floating around, then there’s a perfectly good chance one of them will hit the player, and another one will hit the dealer, and maybe a third will “miss the money” altogether. (Note: I am using McDowell’s terminology, though I may not always like it.) Thus, McDowell may be adding nonexclusive events once again.

    Now, although the formula is not adding probabilities per se, the prohibition against adding nonexclusive events applies to prorated EVs as well as probabilities. If you don’t watch out for this, you are practically guaranteed to get the wrong answer.

    In addition to this ambiguity, McDowell clearly messed up in his invocation of “Snyder’s rule of thumb” which, according to his own definition, splits aces equally between player and dealer. Yet, in his example, he sets h = 0.13 and d = 0.06. This is such an obvious goof he had to write up a correction in his errata.

    Also, McDowell gives us no hint on how to calculate E3, the expectation if the ace “misses the money,” sharply curtailing the usefulness of the formula, save for the (not too useful) case of an infinite deck, where E3 would just be the edge off the top.

    Desperately, we seek out the errata, in hopes it will clarify the ambiguities and save this centrally important formula, which, we were promised back in Chapter 1, is “as good as the accuracy of the figures plugged into it.”

    Sadly, I quote the errata: “Page 114, Line 15: At this point we invoke Snyder’s rule of thumb — the player and the dealer share the Aces 50/50. In this case that means the player and the dealer get three additional Aces each. The probability of the Ace “hitting the money” P(h) and the probability of the dealer getting the Ace by accident P(d) become 0.10 while P(m) is reduced from 0.87 to 0.80.” Thus, McDowell is using h and d in the formula as an aggregate probability that any ace will hit one of these two spots.

    In confirmation of this interpretation, McDowell uses d = 0.07 in the errata for the case where “the dealer can be prevented from getting the ace,” 1/13 ~= 0.077. If we were talking exclusively about the tracked ace, d would be zero for this case. But, sorry, since we are talking about more than one ace, we no longer have mutually exclusive events.

    My friend Zenfighter repeats this same mistake in his “final take” calculation posted on www.bjrnet.com. He derives different values for E1, E2 and E3, but then he simply plugs those values into the formula as restated in McDowell’s errata. To present the formula in the best possible light, I want to use the best possible version. Here is Zenfighter’s improved version of the McDowell formula:

E(X) = 0.10 * 50.79 + 0.10 * (-34.17%) + (-1.5246%) = 0.1374%

where -1.5246% represents “Exact cost for the 80% of the hands where neither the player nor the dealer gets a first card ace.” Let’s rewrite:

E(X) = 0.10 * 50.79% + 0.10 * (-34.17%) + 0.80 * (-1.90575%) = 0.1374%

so we can clearly see: E1 = 50.79%, E2 = -34.17%, E3 = -1.90575%, h = 0.10, d = 0.10, and m = 0.80

    Here are some paradoxes which result if we accept this formula as gospel:

Paradox 1) Assume all the aces are distributed as in a regular shoe (ie. no sequencing). Then we have h = 1/13, d = 1/13, and m = 11/13.

E(X) = (1/13) * 50.79% + (1/13) * (-34.17%) + (11/13) * (-1.90575%) = -0.3341%

    -0.3341% is different from the – 0.4069% computed by CA, and quoted by Zenfighter on 3 Feb 2005 at www.advantageplayer.com .

Paradox 2) 0.1374% is substantially different from the 0.03% I calculate below using nothing but elementary probability. If Zenfighter believes his number is correct, he needs to show the error in my calculation.

Paradox 3) Suppose we have the best of all possible sequencing opportunities. We’re tracking a single deck where the dealer only does one rif. With single deck we can forget about false keys. Let’s further assume dealer always breaks right at 26 cards, and we saw the key-ace combination go into the discards at 10 and 9. Now we can forget about a broken sequence. Finally, we have this dealer’s signature down pat, and he never interlaces more than one card from his right thumb. The bottom of the pack — with our key-ace combo — was in the dealer’s left hand. Now, we know without doubt there will be exactly zero or one cards between the key and the ace.

    We’re heads up, one hand, against the dealer. We get lucky. The last card of the first round is the key! Under the McDowell/Snyder rule of thumb, the probability the tracked ace will be player’s first card is 0.50, and probability it will be the upcard is also 0.50. There is also a positive 1.5/51 chance one of the other three aces will hit either spot. h = 0.50 + 1.5/51 = 0.5294, d = 0.50.+ 1.5/51 = 0.5294. Calculating m with the McDowell/Zenfighter method, we have m = (1 – 0.5294 – 0.5294) = -.0588. Ladies and gentlemen, we have a negative probability!

    Should we throw out three and a half centuries of probability theory and accept a negative probability? Should we just plug it into the formula? Or should we begin to suspect there’s something seriously wrong with this formula? At this point, I hope you can guess my vote.

    A brief aside, here, to a brilliant programmer who shall remain nameless. The Correct Calculation below (which first appeared on advantageplayer.com) very definitely pertains to the book, as well as the errata. Table 3-2 doesn’t enter into the discussion here, because we (Arnold and I) are accepting the numbers from Table 3-2 as inputs. The formula gives incorrect answers no matter what numbers are input, so Table 3-2 is irrelevant. And the errata doesn’t change the formula, but instead, further undermines its validity by pinning down the meaning of some of the inputs.

The Correct EV Calculation

    Here is the right way to find this Ev, given all the inputs required by McDowell’s formula. We will follow the one tracked ace over the various positions, since one ace can’t be in two places at the same time. Thus we can list EVs and probabilities for mutually exclusive events. Therefore we will redefine h, d, and m as probabilities for the given ace to land as the player’s first card, the dealer’s upcard, or somewhere else in the shoe respectively, and we’ll redefine E3 as the expectation when the tracked ace goes somewhere other than the first two cards. In addition, I’ll show a simple way to calculate E3 given the E1, E2 and the off the top expectation.
    We’ll assume 6dks, das, spl3, nrsa as Zenfighter proposed on advantageplayer and bjrnet, so we can borrow some of the numbers he has kindly provided. The premise is: we’re tracking an ace, and based on this information, we know there is a 0.10 probability any ace will be dealt to the player’s first spot, and a 0.10 probability any ace will be dealt to the dealer’s upcard. If the tracked card does not go to one of those two spots, it’s assumed to be in some other completely random position. It may not be realistic, but that is the premise. All other cards are assumed to be randomly dispersed as well, in the remaining positions.

I. First off, let’s look at the player’s first card. We know there is a 0.10 probability this card is an ace. Therefore there is a 0.90 probability it is a non-ace. There are 288 non-aces in the shoe. Therefore, the probability any one of those non-aces will hit the player’s first spot is: 0.90/288 = 0.003125

II. Since all untracked cards are evenly distributed, it follows that all the aces but one also have a 0.003125 probability of hitting the player’s first card. There are 23 such aces. We therefore know: 23 x 0.003125 + h = 0.10, where h = the probability that the tracked ace will hit the player’s first card. Solving, we find: h = 0.028125

III. By premise, the distribution for the dealer’s upcard is the same: d = 0.028125

IV. We now have the probability that the tracked ace will hit either the player’s first card or the upcard, with associated EVs (E1 = 50.79% and E2 = -34.17%) provided by Zenfighter. Since the tracked ace has to land somewhere in the shoe, we know the probability the tracked ace will hit any spot other than those two spots is: m = 1 – 2×0.028125 = 0.94375

V. If we now had the EV associated with a hand where all we know is the tracked ace did not hit the first two spots, but could have hit anywhere else at random, then we’d have three mutually exclusive EVs covering all possibilities (the ace can only go to one spot at a time, but it has to go somewhere) which we could add to find our total EV for this problem.

    But we don’t have that EV. Or do we …

    … Set up a hypo. Suppose we’ve tracked this ace, and come to the conclusion there is a 1/312 chance of it hitting the player’s first card, a 1/312 chance of it hitting the upcard, and a 310/312 chance of it hitting anywhere else. All other cards are randomly dispersed just as we said earlier. Hey! This is normal, off the top distribution! Using the overall off the top EV (-0.406923%) provided by Zenfighter, we can write: 1/312 x 50.79% + 1/312 x (-34.17%) + 310/312 x E3 = -0.406923% Or solving: E3 = -.463161%

VI. Now we can substitute this EV into our original set of facts: 0.028125 x 50.79% + 0.028125 x (-34.17%) + 0.94375 x (-.463161%) = 0.0303% Total EV on the one tracked hand = +0.03%

    It’s a very straightforward problem in probability. But, just to be sure, I had a PhD who teaches a course in probability review it. This person occasionally plays blackjack, but has not been involved in any of the controversy surrounding this book. (S)he states that it is accurate, given the premises (the same premises Zenfighter used in his “final take” calculation). At every stage we’re dealing with mutually exclusive events. The final result is completely reliable, assuming the premises and the EVs are accurate. (Actually, one more digit of accuracy in the EVs would be nice, to assure the .03% isn’t actually .04% or .02% due to cumulative roundoff error.)

    Note this is just your EV when you’re “lucky” enough to track an ace with this (weak) degree of accuracy. It goes without saying that no matter the bet spread, your waiting bets are going to wipe out any potential profit.

    If we start with the premises laid out in the errata: a 0.10 probability any ace will be dealt to the player’s first spot, and any ace will be dealt to the dealer’s upcard, with E1 = 51%, E2 = 34%, and assuming the ambiguous -0.5% refers to expectation off the top, the calculation becomes:
1/312 x (51% – 34%) + 310/312 x E3 = -0.5%
E3 = -0.5580%
.028125 x 51% + 0.028125 x (-34%) + 0.94375 x (-0.558%) = -0.0485%, compared to McDowell’s +0.0130%

Simulation Verification

    I ran four simulations by rewriting tracking simulation software written by me (remember my misspent youth?) and available only to vetted APs:

6D S17 DAS SP3 NRS NS, fixed number of 33 rounds per shoe:
Round 500000000 was completed at: 03-02-2005 11:38:38
Dealer garnered 4025786
Players accumulated:
Player 1 : -4025786 / 1000000000 = -.4025786 % av. bet = 2
A maximum of 221 cards were dealt from the shoe.

Same as above, but with an ace removed from the shoe and dealt to the player’s first card every time:
Round 500000000 was completed at: 03-04-2005 13:36:04
Dealer garnered -253927397
Players accumulated:
Player 1 : 253927397 / 500000000 = 50.7854794 % av. bet = 1
A maximum of 189 cards were dealt from the shoe.
Simulated E1 = 50.7854794%

Same as above, but with an ace removed from the shoe and dealt to the dealer’s upcard every time:
Round 500000000 was completed at: 03-05-2005 14:21:57
Dealer garnered 170791276.5
Players accumulated:
Player 1 : -170791276.5 / 500000000 = -34.1582553 % av. bet = 1
A maximum of 190 cards were dealt from the shoe.
Simulated E2 = -34.15822553%

1/312 x 50.754794% + 1/312 x (-34.15822553%) + 310/312 x E3 = -.4025786% Or solving: E3 = -.458713198%

   EV on the tracked hand = 0.028125 x 50.754794% + 0.028125 x (-34.15822553%) + 0.94375 x (-.458713198%) = +0.03387%

    If Zenfighter’s equation is correct, the EV should either be the 0.1374% quoted above, or very slightly higher, since my simulated edge off the top was slightly higher (though well within one standard deviation) and E1 + E2 — the positive contribution of the ace — is 0.007% higher.
   So we’ll test it …
Same as above, but with an ace removed from the show and dealt as the first card whenever (total aces to first card)/rounds < 0.1, or if that doesn’t occur, to the second card whenever (total aces to upcard)/rounds < 0.1, or else to one of the other 310 positions in the shoe (dealt or undealt) chosen at random.

Round 500000000 was completed at: 03-08-2005 02:53:04
Dealer garnered -150428.5
Players accumulated:
Player 1 : 150428.5 / 500000000 = .0300857 % av. bet = 1
A maximum of 227 cards were dealt from the shoe.
Aces to Player’s first spot: 50000000
Aces to dealer’s upcard: 50000004

    Now, if we put an ace on the first spot 1/40 times, and deal from a full shoe the other 39/40 times, we’d get an ace on first spot 10% of the time as in the sim. This suggests adjusting the variance per hand to (1/40)x1.495439 + (39/40)x1.34 = 1.35. [The 1.495439 is from the Grosjean/Mankodi article.] But this doesn’t take into account the effect of the extra aces to the upcard. This will have little effect, since splits and doubles are unusual against an ace, but there are also fewer pushes with an Ace up. But heck, since I’m in a generous mood we’ll nudge the variance all the way up to 1.4. Standard deviation for the 500M hands is then SqRt(1.4/500,000,000) = 0.0053%. So my predicted 0.03387% is 0.71 standard deviations from the sim, while the best McDowell/Zen prediction to date (0.1374%) is over 20 standard deviations from the sim.

More Than One Ace

    If you have reliable figures for EV when the dealer’s holecard is an ace, the above method of calculation is easily extended to any distribution of one tracked ace into the first four cards in a heads up game. Also, small variations in the distribution of the tracked ace to possible hit cards should have little effect on the EV.

    However, it must be noted that you normally need to track more than one sequence per shoe to have a viable advantage. Unfortunately, when you have more than one sequence, the EV on each key is reduced. Each time you see a given key, the other tracked aces — not associated with that key — are basically “unsequenced.” You may have noted my EVs all generally come in lower than McDowell’s. Tracking multiple sequences brings the EV down lower still.

    It may be possible to work up a formula for the multi-sequence approach, but at a certain point writing a simulator begins to look easy in comparison. ;-Q

A Word About Chapter 8

    Chapter 8 of BJAP is devoted to determining how much ace sequencers should bet. The first section is entitled “Expected Return” wherein he calculated (somehow — he doesn’t make all his variables explicit) that a sequencer can lay down 4 bets per hour with the positive EV from the previous chapter (obviated by the errata) of +4%. After much arithmetic sorcery (and a few sprinkles of magic dust) he winds up with the figure 2051 hands for a one-third Kelly bettor to double his bankroll.

    Let’s grant the four x 4% bets per hour. Although the rest of the book doesn’t tell you how to get such an edge, or even how to calculate it if you’ve got it, four x 4% is not an unachievable goal. One small fact he neglects to mention (I’m so sorry): The whole scenario he lays out to get those four bets per hour requires controlling four spots at all times, at a table with 7 spots, and assumes 60 rounds per hour. Thus, the number of negative EV waiting bets you need to make per hour = 236. None of the growth rate conjury takes this into account! And the 2051 hands he calculates actually represent 2051 positive EV hands plus 121,009 negative EV bets (no, I’m not kidding.), after which your bankroll will very definitely not be doubled, nor anywhere close to doubled, since McDowell’s growth calculation doesn’t subtract the drain from the negative EV waiting bets.

    Also remember, the negative EV waiting bets will be more negative than the regular off the top expectation. Since the waiting bets have no associated keys, they are, in effect, “unsequenced,” and have a lowered probability of catching an ace on either the first or second card.

    Now it has been pointed out that McDowell’s techniques may work better in European casinos, where back-betting is common. This skirts the problem of waiting bet drain, but it leaves several other problems. 1) Even if you can get down four +4% bets per hour down, your time to doubling will be approximately 2051/4 = 513 hours or half a year if you play 20 hours per week. (Note this is a Kelly growth calculation, so it involves continuous bet resizing, which is something most APs don’t like to do.) 2) If you have competition for back-bet spots, this cuts your EV even more. 3) You are relying on the basic strategy of European strangers — cut the 4% down to about 3%, and 4) All this assumes European dealers have riffles as neat and precise as McDowell’s charts. Hey, maybe they do — I’ve never played in Europe.

    Also, McDowell’s risk calculations use the oft-quoted standard deviation of 1.1 units for a blackjack hand. Since substantial bets will be placed on the assumption that an ace will appear, this needs to be prorated and adjusted along the lines mentioned by Grosjean and Mankodi in their article “42.08%: More on the Ace in Hand.” Blackjack Forum Winter 2003/04, Vol XXIII #4. There’s no mention in BJAP of adjusting strategy in order to reduce variance on the hand.

    In sum, I’m afraid I must counsel you that Chapter 8 offers very little clue on how much to bet — even if you are a math whiz, with the ability to develop valid EV formulae to replace the invalid formula in Chapter 7.

    Also note that tracking four sequences at a time involves juggling up to 8 keys in your head at a time. You need to be learning new keys at the same time you’re remembering, and using, keys from the previous shoe. You also need the ability to forget old keys very rapidly, to make room in your head for fresh ones.

Realistic Expectations

    I mentioned an EV of 4 x 4% with 236 waiting bets was an achievable goal, so I owe you this calculation. Assuming an off the top edge of -0.5%, we can find the EV of waiting bets (call it E4) as follows: 4 x 4% + 236 x E4 = 240 x (-0.5%), E4 = -0.57627%. With a 20 to 1 bet spread, your expectation is: 20 x 4 x 4% – 1 x 236 x 0.57627% = 1.84 units per hour. Total action = 20 x 4 + 1 x 236 = 316 units per hour. EV = 1.84/316 = +0.58% with the huge risk that always goes with a huge bet spread.

    I’m not going to tell you what to look for — Arnold isn’t paying me enough for that article — but let me say I think it’s possible — with a lot of hunting, and studying of dealer signatures — to do better than +0.58% with ace sequencing. In fact, I think +1% may be achievable in a few select games around the US. But this involves constantly juggling a dozen or more keys in your head at all times — much more difficult than counting, which only requires you to track one number. I can’t categorize this as “huge advantages over the casino” nor “easy to do at the table,” and obviously it isn’t close to approaching the 3-4% — conservatively! — promised by McDowell on pg 18.

Why am I so sorry?

    I’m sorry that there is no free lunch. I’m sorry it’s so difficult to carve out an edge with the voracious double-bust sinkhole sucking on us hand after hand. I’m sorry so many gamblers tell tall tales. I’m sorry the numbers don’t sit up, roll over, and bark at our command.    I, personally, have no stake in the worth (or lack thereof) of this book. I have no enmity for any of the parties involved on either side of the issue. I wish David’s work really was “the final chapter in advantageous blackjack play.” I’m sorry it isn’t.

    I’m sorry I think very few dealers have riffles as predictable as McDowell’s. I’m sorry most dealers have shuffles much more like Wong’s dealer, or like mine, or Curtis’. And I’m very sorry several august authorities continue to defend this dangerous, unsound work with vengeful attacks on people they once proclaimed “brilliant.”

Summary

    To the best of my knowledge, never before in the history of blackjack literature has such a thoroughly flawed work — flawed in both math and methodology — received so many accolades from highly respected authorities. It’s not that the book isn’t perfect. It’s that there’s almost nothing of value (from the point of view of beating the casinos) in the book. An errata for all the errors in the book would be nearly as long as the book itself. ♠

Don’t Toss Out Those Coupons!

by Dog-Ass Johnny
(From Blackjack Forum Vol. XI #4, December 1991)
© Blackjack Forum 1991

[Editor’s Note: How can it be that years after Dog-Ass Johnny’s untimely demise, Blackjack Forum continues to publish articles by him? Let’s just say we have a backlog of his one time “works in progress” which we will continue to decipher and edit as the need arises.

There are many reasons, to be sure, why Dog-Ass Johnny’s writing should never be published. And certainly not at Christmas time. Dog-Ass Johnny was noted for his bad attitude. His grating personality endeared him to few. But he was the only person I ever met who literally made his living playing coupons.

He once told me that he had valid I.D. for forty different states. I don’t know if that was true. He also told me he hadn’t worked an honest job since 1967. I believe that. He felt that casinos were built to provide free food, free drinks, and free money to the unemployed. “Working,” he used to say, “is un-American. It conflicts with the pursuit of happiness.” Or something like that.

So, despite the inevitable letters of complaint I get any time I publish anything with Dog-Ass’s name on it, I feel that this is an article that I have an obligation to publish. In these hard economic times, the wisdom of Dog-Ass Johnny just might help one of Blackjack Forum’s less fortunate subscribers to put a little food on his table. This article is my gift to my homeless readers. Remember, when the chips are down, the spirit of Dog-Ass Johnny is there for you, a shining light of hope for the wretched. — Arnold Snyder]

Say hello to Bill and Fred, a couple of low-rollers who’ve quit their jobs at the boiler factory to become professional gamblers. Bill is a card counter. He’s not a bad player, but he’ll probably never be able to earn a living at the tables—counting errors, preferential shuffling, and sitting through too many negative decks all grind him down. Fred plays video poker; he makes errors too, and when he has trouble finding big jackpots, he plays anyway.

It seems clear that Bill and Fred aren’t going to strike it rich as pro gamblers, yet the hapless duo just can’t bring themselves to give up their dreams—the lure of living off the casinos is too strong. Mediocre gamblers, unwilling to work, without discernible moral qualities, Bill and Fred are well-positioned for the big move up to Coupons.

You Can Live on Coupons

If Bill and Fred had spent last summer doing coupons in Lake Tahoe (one of Nevada’s least profitable coupon venues), they could have averaged about $50 a day each with a lot less effort than they put into their gambling.

That’s $350 a week tax free, plus complete personal freedom: Instead of inhaling asbestos fibers at the plant, Bill and Fred would get to enjoy the beautiful lakes and mountains. They’d lounge around in the sports books, watching five ball games at once while sucking down free casino alcohol. They’d dine on all-you-can-eat buffets at great prices, pocketing extra portions for later to keep those food bills down. And, of course, they’d be a big hit with the ladies, because nothing attracts a beautiful woman like a guy with a fistful of lucky bucks.

You might even say that coupons turn every casino in the world into your private bank account. Granted, you can only withdraw a dollar or two at a time from these private bank accounts, but you can make up for that by visiting the bank a lot. That’s what this article is about.

Disclaimer

Do not think of Dog-Ass Johnny as a couponomist. Couponomists are yuppie intellectualizers. A little book-learning is fine, but the essential romance of coupon redemption completely eludes these people. They are all talk and no action.

Can they calculate the exact expectation of any casino promotion you show them, right down to the last tenth of a cent? Absolutely. Do they really bet their lucky bucks at the tables, do they actually eat coupon food? Maybe. Will you find them shaking down frightened tourists for their coupons, or dredging discarded fun books from the unspeakable sludge of snot and vomit at the bottom of some municipal garbage can? I don’t think so.

No, couponomists are too good for that. After a hard day at the calculator, they just want to relax with their trendy foreign beers, maybe catch that new Elvis tribute at the Trop. They even have regular jobs! Couponomist Peter Griffin is a mathematics professor; couponomist Anthony Curtis is a publisher. It’s hard to believe what some people will do for a dollar.

Well, to hell with these raised-pinky dilettantes and their distorted values. Dog-Ass Johnny lives in his car and has a chronic cough. Dog-Ass Johnny drinks generic beer, and the only dinner show he ever catches is watching the other derelicts gum their free meals at the Reno-Sparks Gospel Mission (Rev. Howard Cannon, Pastor).

However, despite these obvious disadvantages, Dog-Ass Johnny doesn’t need a calculator to tell him when a promotion is worth cashing in on, and neither do you. To succeed in the redemption biz—really succeed, on the streets and not just on paper—what you need to do is get your hands dirty. Dog-Ass Johnny wants to help you get started.

Obtaining The Vouchers

The first thing you have to do is collect a lot of vouchers. Vouchers are the coupons you get at motels. You can’t play them at the tables; instead, you turn them in at some special casino location, usually called the Fun Center, for coupon booklets that contain lucky bucks, food discounts, and the like.

Having plenty of vouchers is the basis of your life as a coupoñero, and the search for them should underlie everything else you do. Vouchers transform you from a bum hustling lucky bucks into a businessman, continually liquidating and replenishing his inventory. When good vouchers become scarce, as they inevitably will, you mustn’t give up. Work harder! Never doubt that the coupons are out there; all you have to do is lower your standards to the point where you can get them. Here are a few of the methods Dog-Ass Johnny uses.

Find Coupons

As a professional casino chiseler, you’ll be patrolling the casinos constantly, looking for careless dealers, big jackpots, and money left in slot trays. As you walk around, grab everything that might have value; you can sort it out later. Learn the size and color of every voucher and lucky buck in town, so you can recognize them anywhere. Inside the casinos, those big sand-filled ash trays are always a good source; outdoors, try the streets around coupon-dispensing casinos, motel dumpsters, and municipal trash barrels. More on these later.

Bum Coupons

Sometimes you can get other blackjack players to give you their lucky bucks, especially if you’ve already been conversing with them at the table. When the guy next to you pulls out his coupon booklet, express intense interest in it. Act as though you’ve never seen such a thing before. Appear to accord him status and importance because he possesses it. When he plays his first lucky buck, watch the hand as though hypnotized. Then, affecting a wistful tone, say something like “Gee, that looks like fun. Sure wish I had one.”

Now you’ve turned the gift of a lucky buck into the means by which he can confirm his own personal worth. Your benefactor feels good about himself as he guides you through the mysteries of playing lucky bucks. You feel good too, as you rake in the free dough.

Ask for Coupons

Many casinos will give you their coupon booklet if you ask for one. If the casino handles its giveaways at a place like the hotel desk or cashier’s cage, ask there. Avoid special redemption centers, where you need to limit your exposure.

Once you find the right place to ask, tell them you’ve just arrived in town with your whole family for a two-week stay, and you want to be sure to cash in on every freebie the casino has to offer. Unless you look like a professional coupon sleazeball, they’ll usually load you right up. Of course, you can’t do this every day, so save it for periodically checking what each casino is giving out. That way, you’ll always know what to look for on the street.

Hang Around the Players Club or Fun Center

The pickings are always good around the area where the casino gives out their fun books. Not only is there a large amount of material available, but you get the actual lucky bucks instead of just the vouchers. Lurk nearby when conditions are busy; grab any discarded coupons as quickly as you can without appearing unseemly, because there’s always a lot of competition in these locations. Even amateurs know enough to pick up lucky bucks.

Platform the Trash Barrels

A lot of coupons and vouchers end up in the municipal trash barrels on the streets outside the casinos. You absolutely must maximize this rich and valuable source if you are to succeed. Unfortunately, image-obsessed casino towns have taken to emptying their barrels with such alarming frequency that the garbage rarely has a chance to accumulate. Instead of piling up in easily collectible mounds around the rim, the coupons fall to the bottom of the barrel, where they tend to degrade rapidly (and where you can’t reach them without an embarrassing head-first plunge). Platforming is the answer.

What you do is simple. First, identify the most profitable locations and observe their maintenance schedule. Then, just after the sanitation workers empty the barrels, you stuff them full again, with those promotional newspapers most casinos leave lying around. You don’t need to waste time and paper by filling the whole barrel; just wad up a good firm plug of Nevada Nite Life or Sports Form or Showroom Superstars and wedge it tightly into the container’s mouth. Now you’ve created a sturdy platform atop the otherwise empty barrel. Dress it up a little with some candy wrappers, a crumpled cigarette pack or two, and your private coupon collection center is ready to go.

Return later to sort through the now-accessible trash for vouchers and lucky bucks. If you want to avoid looking like a bum, act like you’re trying to throw something away that doesn’t quite want to go down (because of your platform, probably); meanwhile, mix the garbage around with a rolled-up newspaper, scanning for the good stuff. When you’re finished, push your platform into the barrel and brush most of the trash down after it, so as not to alert the sanitation workers (who, if you’ve timed your collection properly, will be along any minute). Wait until they leave, then put a new platform in place and move on to the next barrel.

Soft Platforming

If the idea of flagrantly subverting municipal policies upsets you, well, you might not have what it takes to live wild and free like Dog-Ass Johnny. But don’t despair; you can still enjoy the platforming experience with soft platforming. Instead of plugging the opening with a carefully wrought wad of newsprint, you just cover it up with a newspaper. Most people will still drop their trash on top of the pile where you can get at it.

Compulsively neat types and people with bottles or cups of liquid will reduce your profits, true, but in return, you get reduced effort and reduced detectability: you can apply the newspaper quickly and casually, plus you leave no concrete evidence of your intentions behind. And, whether plugging or covering, don’t worry about the paper falling in; the mouths of public trash barrels are always unnaturally small, because the municipal agencies that maintain them want to discourage potential depositors from the industrial and household sectors. Another big advantage for the alert coupoñero.

Visit the Dump

Platforming is an advanced technique that requires a lot of energy and endless diligence. If it proves to be more than you can handle, don’t give up; you still have a shot at those coupons. First, you have to find out where they empty the trash barrels. A call to the local Department of Sanitation will usually do the job — tell them you accidentally threw away some important papers. Then visit the dump and pick through the garbage at your leisure. This is an especially apt tactic for the elderly and physically-infirm, who can go through a large volume of material without a lot of walking around.

While at the dump, be sure to take care of the attendants. Like dealers at a blackjack table, these people can make you or break you. If you’re able to establish a close working relationship with an attendant, you’ll have a valuable ally who will direct you to the richest plots of garbage, tip you off when an especially promising load comes in, and even skim some of the premium stuff for you on his own. Don’t let self-respect stand between you and this important resource. You’ll find the Reno Dump at 63029 Wasatch Flats Road, about ten miles southeast of the airport. Ask for Ernie.

Steal Coupons from Motels

One casino town, Stateline, Nevada, is so hip and trendy that you can’t find any coupons on the street — Northern Californians would rather set themselves on fire than litter. In Tahoe, therefore, you must rely on motels as your prime source of casinos vouchers. Unfortunately, most motels hide their supply of coupons behind the counter, rationing them stingily to their customers in a shameful display of mistrust and suspicion. What a sad commentary on the state of American business!

Happily for readers of this publication, however, Dog-Ass Johnny knows of at least eight Tahoe motels that leave the vouchers out where you can get at them. These benevolent establishments are: Trout Creek, Pinewood Lodge, Ski Bird, Midway, Elizabeth Lodge, Thunderchief, El Nido, and Cal-Va-Rado. You should be able to discover similar locations for yourself in other casino town. Of the above, Trout Creek is the cheapest, and Pinewood Lodge is probably the best value. Cal-Va-Rado is right across the street from Harvey’s, so Dog-Ass Johnny recommends that you reserve it for emergency resupply rather than sleep there.

Note that you do not need to check into one of these motels to get the vouchers. Just walk into the office. If the manager is present, ask briefly about the rates or whatever, then leave. Otherwise, grab as many vouchers as you need. A bag can be a big help here, as things often get awkward when you’re hastily trying to cram a huge sheaf of paper into a pocket that turns out to be too small (Dog-Ass Johnny uses a special rubber bag sewn inside his shirt, an outfit he put together years ago for stealing food from buffets).

Best time to show up is between 10 AM and noon, when the managers are usually helping clean the rooms. Above all, don’t spoil things for everyone else by emptying the rack; know in advance how many vouchers you can reasonably pass off, and take pride in being able to net just the right number in one swipe.

Be a pro.

Bus Coupons

Bus riders are knowledgeable about what they’re getting. They rarely throw away anything of value, so there isn’t much you can do to capitalize on bus coupons.

One exception involves deferred vouchers common in Atlantic City. If you frequent the bus platforms long enough, you’ll eventually run into someone selling these turn-visit coupons at a discount. Get to know such a person if you can; with a goodly supply of these $5 beauties, you can pump up your rebate beyond the cost of the trip.

This in turn allows you to live on the bus. Instead of paying outrageously for an apartment, you spend your waking hours in the casinos; at bedtime, you just get on a bus to Boston or Washington or wherever, and curl up on that spacious back seat. When the bus arrives, you buy another round-trip ticket, get another set of coupons, and sleep your way back to the casinos.

In the morning, you’re a buck or two richer than when you went to bed, you didn’t have to spring for a motel, and you’re still in Atlantic City — a clear net gain! After cashing those vouchers, you have the whole rest of the day to spend looking for money on the floor, and probably make a bundle.

Redeeming the Vouchers

Most of the stuff you collect will have to be redeemed in the casinos before you can cash in at the tables. This stage is the bottleneck in the coupon process. Your approach here, your guiding light, your overriding goal, can be summed up in one word: multiple redemption. Okay, two words.

Multiple redemption means you show up more than once a day. More than once a shift, in fact. The $50 a day figure mentioned earlier assumes six cash-ins daily, two per shift, of the standard set of motel vouchers. The value of this set last summer was about $8.50. It takes about 45 minutes to redeem the whole Lake Tahoe set, and only about 20-25 minutes if you don’t visit the distant (but always profitable) Lakeside Inn. That works out to between $10 and $25 per hour, even more when you consider the tax advantages. Of course, if the casinos realize how much they’re paying you, they’ll take you off the payroll. Your job is to keep them from finding out.

Note Personnel Rotation

The most basic thing you can do is to take advantage of shift changes. At casinos where the coupon personnel work three shifts, you get three redemptions per day right off the top. If those employees get time off for lunch, you can get another three from their replacements. That gets you the desired six redemptions daily without ever seeing the same face twice. Concentrate on the casinos using the most people at their Fun Centers, and be sure to get at least one redemption per day from each employee.

Employ Master Techniques of Disguise

To get more than six cash-ins per day, or to work one area for longer than a month or two, you’ll probably have to disguise yourself. Your disguise does not have to be complex or expensive; the Fun Center drones will always give you at least a glance, but they’ll only check you out carefully if they’re already suspicious. You can usually get away with simple props like hats and glasses. The trick is to give them something to remember you by, like an unusual hat or shirt, an attitude, a strange facial expression, whatever, then show them the opposite the next time — young/old, happy/grumpy, smart/stupid, etc. Polarity is the key.

If you are middle-aged, for example, you can go for a young/old duality. First, give them a younger look; cover your balding head with a wig or a baseball cap, and carry a Walkman. Wear a button supporting euthanasia.

After a few weeks of that, just when they might begin to notice you, change to the older look. Comb your hair back, and give it a few streaks of grey (drug stores sell special hair paint for this purpose). Wear a button supporting the draft. You might even put sharp stones in your shoes, to keep that youthful spring out of your step.

But the very best way to look older is to soak your face in brine for a few hours. Breathe through a straw while submerged. This treatment (immersion in salt water, not breathing through a straw) will dry and harden your skin, giving it that hagged-out look for several days. A copy of Retirement Times under your arm, and you’re all set.

At the Redemption Center

Optimally, there will be a crowd; if so, just blend in quietly. If you’re the only customer, try to avoid speaking or making eye contact (but don’t appear furtive). Accept your coupon book as though receiving an honorary diploma. Examine it carefully right there at the Fun Center, as if stunned by your good fortune. Then shamble away, still completely entranced. Wait until you are on the street to rip out the usable stuff and throw the husk away (Dog-Ass Johnny tears all his empties in half and tries to dump them in the same general area, so as to avoid picking them up again later).

If You Have to Show a Motel Key

Many casinos want to see a room key before they’ll come across with the goods. They want to know that you’re an actual gambler, not just some scumbag off the street. Therefore, whenever you stay at a motel or casino hotel, steal the tag from your key. Acquire as large a collection of key tags as possible. Return the key itself by leaving it in your room, or on the counter when the manager is away from the office. The motel people won’t be too upset, so long as you don’t take the actual key; they’ll think you wanted their tag as a valued souvenir of your wonderful motel experience.

Besides, you checked in under a fake name, didn’t you? Attach some random key to the tag, and you’re in business.

If You Have to Show a Motel Receipt

Many casinos want to see a room receipt before they’ll come across with the goods. They want to know that you’re an actual gambler, not just some scumbag off the street. Therefore, whenever you stay at a motel or casino hotel, save your room receipt. Be careful not to fold or crumple it. When you encounter a promotion that requires such a receipt, paste a piece of blank paper over the date. Then visit a copy store and have them run off a few hundred copies using the highest quality color process available; ask the copy people to match the motel’s paper as closely as they can. Fill in the current date on a fresh copy, and you’re in business.

If You Have to Show Out-of-State ID

Assuming you live outside Nevada, showing id is no problem. However, some casinos record your driver’s license number on their computer, effectively limiting even the most skillful coupoñero to one redemption per day. For this reason, you should obtain multiple fake driver’s licenses. Never carry more than one license with you at a time, and only drive with your real license; there will be unpleasant consequences if the authorities discover that your papers are not in order.

Rehabilitating Pebbled Coupons

One type of voucher you should not try to redeem is the pebbled voucher, also known as the braille voucher. These are vouchers that have been stepped on or driven over while lying in the street. They have the texture of the pavement imprinted on them, and usually a footprint or tread mark as well. When you try to redeem them, the casino people will recognize them at once as street vouchers, and mark you as a would-be pro. You cannot afford to let one of these persons peg you as a coupon hustler.

If you have a pebbled voucher that you absolutely cannot live without redeeming, Dog-Ass Johnny suggests you give it the old steam and press treatment. Hold onto the target voucher until you find yourself in a motel. Use a gum eraser to remove the footprint or tread mark, then take the voucher into the bathroom and suspend it somewhere near the ceiling.

Now turn on the hot water taps in the sink and shower, leave the room, and close the bathroom door. After a few hours, retrieve the steamed voucher and immediately press it between the pages of a large, heavy book. The preferred choice is the standard motel Bible — that high page count per bound inch does a great job, and the top-quality paper quickly conducts excess moisture away from your valued coupon. Maximizing pressure by wedging the book under one leg of the bed until ready to check out. Badly damaged items may require more than one treatment.

Pebbled Lucky Bucks

As opposed to pebbled vouchers, pebbled lucky bucks are no problem. Play them as you would any other lucky buck. The dealers touch them only briefly, and don’t much care anyway, since it’s not their job to monitor your behavior. If a dealer does give you a hard time, you can just avoid that dealer in the future.

Cashing In

Now that you’ve finally got the actual lucky bucks in hand, you’re almost home free. Should you run right over to the blackjack tables and cash in? Absolutely not. After all, you’re a pro, and you do things the professional way.

Be Inconspicuous

Because there are so many dealers in a casino, it won’t hurt much if one or two become aware of your activities. Still, you should do what you can to avoid recognition. For example, save up your stash of lucky bucks for a week or two, then play them all off on a crowded weekend or evening. It’s more efficient, plus it’s much easier to blend in and go unnoticed among the throngs of tourists. (Don’t hold onto coupons for too long, though, as they can suddenly become obsolete if the casino changes to a new promotion.)

Play a maximum of two lucky bucks per table, four per pit. This will leave you well below the amount of coupon activity most casinos will tolerate, but you must accept this inefficiency as the price of longevity. Play standing up, and avoid interacting with the dealer if possible; have your coupon book (not just the lucky bucks) plainly visible, and make it clear by your body language (checking watch, looking across casino as if trying to find wife or friend) that you’re only there for a couple of hands. Bosses just don’t see you until you sit down or buy in.

Finally, to limit your exposure to blackjack dealers, play off your lucky bucks at the craps tables sometimes. Betting the Don’t Pass line (against the shooter) gives best results, and also has an antisocial quality that resonates nicely with your parasitic lifestyle.

Play Multideck Games

Don’t wait for a positive count at a single-deck table to play your lucky bucks — it looks bad. You’re playing for the 50% return on your coupon, not the 1% return on your bet, so you can afford to protect your enterprise by playing where they won’t notice you. That means multiple-deck games. An additional advantage of multideck games is that, when the shoe becomes favorable during your two-hand visit, you can sit down and get five or ten profitable blackjack hands without looking like a counter.

When you must play single decks, do so only off the top, rather than risk looking like a Wonger. Stay for a second hand when you have a second coupon, but don’t throw out a big bet if the deck goes plus. Remember, this isn’t card-counting, where you just want to conceal your abilities; with coupons, you don’t even want them to know you exist.

Dog-Ass Johnny must now close his big book of secrets until next time. Some of the techniques he has revealed here may at first seem fanciful, but do not underestimate their power; Each is completely legitimate, and Dog-Ass Johnny has profited handsomely from all of them. Yes, thanks to coupons, Dog-Ass Johnny is sitting pretty.

Things might have turned out well for Bill and Fred too, if only they’d given coupons a try. Bill busted out at the blackjack tables, and had to take part-time work at a Reno convenience store. Over the years, he and Fred gradually lost touch; Bill eventually married a lap-dancer from one of those topless joints out in Fernley, and developed an interest in monster truck rallies.

Fred’s story is more sobering: Last year, he was arrested during a vice raid on a kiddie-porn operation up in Winnemucca. While subsequently serving a lengthy prison term, Fred was stabbed to death by another inmate, in a dispute over a Pez dispenser. Fred’s cashing his coupons in Hell now. ♠