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Fundamental Mistakes in Blackjack Math and Methodology in David McDowell's Blackjack Ace Prediction
By Arnold Snyder
(From Blackjack Forum Spring 2005)
© Blackjack Forum 2005
David McDowell’s Blackjack Ace Prediction is not a book I recommend for any blackjack player who wants to learn to track aces for profit. Despite the author’s claims, you cannot learn to track aces profitably from the information in this book. The author provides a modicum of the theory of ace location or prediction, but his understanding of casino shuffles, tracking methodology, and ace prediction in the real world is seriously flawed, and his blackjack math is replete with errors.
I usually ignore blackjack books that are worthless. I see little point in trashing some unknown author whose lack of credentials will ensure him a place in obscurity.
But I cannot ignore this book. I believe McDowell attempted to invent a valid acelocation method that could be used by serious blackjack players. I believe McDowell attempted to run his ideas by a number of notable blackjack experts to get their input on his methods. I do not believe he was trying to pull a scam on players and sell a phony ace prediction system. I think his heart was mostly in the right place.
Unfortunately, this is not just some unknown nobody that I can send a polite personal note to and tell him his system is all wet. McDowell’s book has been published with endorsements on the back cover from half a dozen notable blackjack authorities, and my own name is invoked throughout McDowell’s text in a such a way that my own writings seem to be lending credibility to his false conclusions.
I have been getting emails from players telling me that they are actually considering using McDowell’s ace prediction techniques in casinos. One email is particularly disturbing to me because it is from a very knowledgeable card counter whom I have known for many years, and whom I know has recently lost a substantial portion of his bankroll due to miserable negative fluctuation. He is hoping that the 4% edge McDowell has calculated for his ace location techniques will restore his bankroll a lot quicker than the 1% count game he is more accustomed to.
So, rather than concern myself with the feelings of a wellintentioned but misguided author, I will be blunt in my remarks on this book. I will not make any attempt myself to provide a comprehensive critique of this book. McDowell’s conclusions and methods have so many flaws that I could write a book myself just on the mistakes in his book! Instead, I will point out one of the key errors in the math. I will also publish a review later by experienced trackers that addresses some of the book’s worst tracking methodology flaws, as well as any further response that seems to be needed.
You may be wondering, as another player put it in an email to me, if in fact McDowell has at last told the big secrets of the ace trackers, and whether I, in fact, might not be just trying to "put the lid back on" the pros’ secrets in order to protect them. "How could all those other experts who endorsed McDowell’s book be so wrong?" this player asked me.
The fact is that the other blackjack experts who endorsed the book either didn’t actually read it or didn’t do the math. So, here’s my suggestion if you believe I’m just trying to cover up the professional gamblers' secrets.
I am going to provide a simple mathematical analysis in this article. McDowell claims roughly a 4% advantage for his methods on what he describes as a tworiffle R&R shuffle. Snyder claims McDowell’s advantage is roughly 0%. This isn’t a judgment call based on opinions. It’s math. Either do the math yourself, or take it to another expert for help.
A Simple Tell that McDowell Doesn't Know What He's Doing in Blackjack Ace Prediction
This example of one of the blackjack math errors in this book, the main one I am going to address, can be found on page 114, where the author describes how he estimates his advantage from tracking aces. I’m using this example because he explains that he used "Snyder’s rule of thumb" to develop the formula, so I fear that readers might conclude that McDowell’s findings would reflect my own.
This rule of thumb, as McDowell describes it, says that if the player is playing heads up, and he bets on only one hand when a key card predicts an ace, then over the long run any keyed aces that appear will be split 50/50 with the dealer.
Let me take a moment to point out that this is in no way my "rule of thumb" or overall recommendation for the best way to approach acesequencing. I specified in my Blackjack Shuffle Tracker’s Cookbook that my coverage of acesequencing was a cursory treatment that covered only a few of the basic theories, and that I considered ace location via a single key card, and playing a single hand, advisable only in a specific type of game primarily as a "side" strategy to be used in conjunction with other tracking/counting techniques.
But since the player McDowell describes is using no technique to "steer" the aces (that is, he is not playing multiple hands as necessary in an attempt to keep any keyed aces away from the dealer), I will go along with his assumption that what he terms my "rule of thumb" would fit this situation. Since the key card on the prior round that signals an impending ace would have an equal likelihood of being dealt as any card in that round, then I would agree that the ace the key card signals would as likely go to the dealer hand as the player hand on the next round where the player is betting on the ace.
Using McDowell’s singlekey/singlebet method, in the shuffle he describes, he estimates that for every 100 times he bets on the ace because he saw his key card, he will actually be dealt an ace on that hand 13 times. He estimates that this is about 6 extra aces per 100 bets. He then assumes, via "Snyder’s rule of thumb," that the dealer will also get 6 extra aces "by accident" (his words). And he points out again, as per Snyder, that the value of the ace to the player is much greater than the value of the ace to the dealer (51% versus 34%, to use his numbers), so that even if the player is splitting the extra aces with the dealer, the player will enjoy a substantial win rate. Using all of these assumptions, McDowell calculates his win rate on these hands as 4.2%.
I have known a number of successful ace trackers who have used various methods to locate aces, none of which are described by McDowell, and most have told me they estimate their overall advantage over the house at about 2% to 4%. So, McDowell’s estimate of the potential advantage from acelocation is not in itself inordinate.
What I could not fathom, however, was how he came up with this advantage if he was only hitting the ace on 13 out of every 100 times he bet on its coming. Most of the ace trackers I have known tell me they expect to hit the ace 40% to 60% of the times that they bet on it, depending on the shuffle, the number of hands they are betting, the number of keys they are using, etc.
The higher percentages of hit rates assume the player is betting on multiple hands to catch the ace and—especially—to act as a "buffer" against the dealer getting the ace.) But McDowell comes up with this 4.2% win rate when he is failing to get an ace on his bet 87% of the time!
So, without stopping right now to show all of the places where his math and methodology went wrong, I will first show how we figure out what the player’s edge would really be using McDowell’s hit rate. To keep things simple, I will also use the same numbers he used with regards to the value of an ace: 51% advantage if it hits the player’s hand, and 34% if it hits the dealer’s hand.
To estimate the value of this hit rate, the first thing to do is figure out how many times per 100 hands the player would be dealt an ace as the first card at random. Since there is one ace per 13 cards, this is a simple calculation. If a blindfolded monkey were placing these bets, he would hit a firstcard ace 7.7 times per 100 hands. (McDowell estimates this number as an even 7 times, but I prefer to use the exact number, 7.7.) Therefore, if the player is hitting 13 firstcard aces per 100 times that he bets on hitting one, he is hitting 5.3 extra aces per 100 times he bets on one coming.
To keep things simple, I will also use McDowell’s assumption that the player is betting only one spot, and that the keyed aces that appear are being split with the dealer. So, the dealer is also getting an extra 5.3 aces per 100 hands (not 6, as per McDowell).
To calculate the value of this hit rate, I first figure out the value of the extra aces when they land on the player’s hand (assuming for simplicity $1 bets each time an ace is predicted):
5.3 x 0.51 = $2.70 per 100 bets on the ace.
I then calculate the cost to the player when the dealer gets the extra aces:
5.3 x 0.34 = $1.80 per 100 bets on the ace.
I then calculate the player disadvantage on all of the remaining hands on which the player placed a $1 bet for the ace, assuming the player is facing a standard house (dis)advantage of 0.50%. This would occur on 89.4 hands (subtracting from 100 the total number of hands, 10.6, where either the player or dealer got the keyed ace).
89.4 x 0.005 = $0.45 per 100 bets on the ace.
So, the player’s dollar win per 100 ace bets is:
$2.70  $1.80  $0.45 = $0.45
Which is 45 cents profit per $100 bet ($1 x 100), or a percentage win rate of 0.45%.
Obviously, a win rate of only 0.45% is quite a bit different from the 4.2% win rate provided by McDowell. In fact, McDowell’s number is off by almost a factor of ten!
And there is another very important point that must be clarified here: This 0.45% win rate is the player’s win rate only when betting on a predicted ace. This is not his overall predicted win rate on the game if we assume he has any "waiting bets" while he is playing hands and watching for his key cards. Instead, this is what his expectation would be if he was betting only on the keyed aces, and he (and the dealer) were each getting an extra 5.3 aces per 100 hands.
And note that this is the player’s percentage win rate on these bets. (It makes no difference if the player is betting $1 or $1000 on these bets, his percentage win rate would be 0.45%.) I will address the cost of the waiting bets, and the effect of the player actually raising his bet when the ace is predicted, below.
For now, let’s return to McDowell’s calculation of his advantage to find out why there is such a huge discrepancy in our results. It turns out there is a rather gross error in his math. He assumes that the player gets a total of 13 firstcard aces, but that the dealer gets a total of only 6 firstcard aces. In other words, he gives the player both the random aces (7.7) and the extra aces from tracking (5.3), but he assumes that the dealer only gets a total of 6 aces, fewer than even the random number expected per 100 hands, despite specifying a playing style where the dealer will be getting the same number of extra aces as the player, due to "Snyder’s rule of thumb."
There are various ways of doing the math for this problem. But you cannot calculate the player’s aces expected at random into the estimate of advantage. Or, at least, not the way McDowell has done it. Let me explain why...
We know that in a completely random game the player and dealer will each get 7.7 firstcard aces per 100 hands. Since the expected value of these aces to the player is 51% on the aces dealt to the player and 34% on the aces dealt to the dealer, there is a very strong player advantage on these 15.4 hands per 100—in fact, an average advantage per hand of about 8.5%.
If we then estimate that, on the remaining 84.6 hands per hundred when neither the player nor the dealer is dealt an ace, there is a standard house advantage of 0.50%, we would have to conclude that a blindfolded monkey could beat blackjack simply by betting big on every hand.
So, where is the error in this thinking? The error, and it is a serious one, is in believing that the house has only a 0.50% advantage on the 84.6 hands when no firstcard aces are occurring.
That 0.50% house edge assumes that we are off the top of a full 6deck shoe, and that an ace will be dealt as the first card on one hand out of every 13 for both the player and the dealer. McDowell’s formula removed all 13 of the player’s firstcard aces to calculate the player’s acehit advantage, but only used the 6 "extra" aces the dealer was dealt to calculate the player’s disadvantage on these 6 hands, leaving the dealer’s 7 "random" aces in the 81 remaining hands, where he then estimates the house edge at 0.50%.
If our assumption, however, is that the player will be dealt no firstcard aces in these 81 hands (because we have already accounted for the player’s share of both random and keyed aces), whereas the dealer will get his 7 "random" firstcard aces in these 81 hands (which is how McDowell does the math), then the house edge on these 81 hands is roughly 4.5%, not 0.50%.
You may be tempted to correct McDowell’s error by simply taking the player’s 13 firstcard aces x 51%, and the dealer’s 13 firstcard aces x 34%, and assuming that the remaining 74 hands in which neither the player nor dealer are dealt an ace as first card are played with a 0.50% house edge. Wrong. If we assume that no aces are dealt to either the player or dealer as a first card on their hands—as we must because both have now received their full share of both keyed and random firstcard aces—but that all other cards are dealt in their expected proportions, then the house edge against the player on these 74 hands is almost 1.5%.
Note that we are still discussing only the hands on which the ace tracker has bet on an impending ace, as predicted by his key card. If the ace tracker using McDowell’s method is able to bet on 3 to 4 hands per shoe, then on 3 to 4 hands per shoe he will be betting with an advantage over the house of about 0.45%. Now let’s return to the cost of the waiting bets. Subject to the rules, on the other 40 or so hands per shoe when no ace is predicted, the ace tracker will be playing against a house advantage of about 0.50%. So, if he flat bets $100 when no ace is predicted, then bets $1000 when his key card predicts an ace is coming, he will almost—but not quite—be playing a breakeven game.
In other words, the actual overall advantage of his system is not 0.45%, but quite a bit lower. You need a pretty big spread to break even, and you might get an edge of about 0.20% with a 1to20 spread; but can anyone afford to play with such a small edge over the house?
You cannot (in practical terms) beat a blackjack game via ace tracking if you are only successful at hitting the ace on a total of 13 out of 100 bets. You must use a method that will get you closer to at least a 40% hit rate, and 50+% would be much preferable.
To save myself from having to answer a million posts from fellow math geeks, let me say clearly that I know that this methodology I’m proposing is an oversimplified way of addressing McDowell’s win rate calculation. The actual win rate of an ace tracker is complicated by numerous other factors. For example, if we are successfully locating 4 aces per shoe via key cards, then the house edge on a hand where there is no key card predicting an ace should reflect the fact that we are playing in a sixdeck shoe game minus four aces, since the number of random aces available to us has been diminished by 4.
That would make our waiting bets more expensive than 0.50%. Some of this is explained in my Shuffle Tracker’s Cookbook, but I’m not going to go into it all here. I’m simply trying to point out a glaring error in McDowell’s methodology, not provide a comprehensive guide for ace trackers.
I have already said that I do not believe the author was attempting to pull a scam with this book, and that, based on the endorsements on the cover, it appears he made an effort to send the book to noted authorities for an opinion. Unfortunately, he did not send the book to any actual ace trackers.
It is even more unfortunate that within the text he seems to represent that he himself has used these methods with great success. I suspect that some (or all) of the authorities he sent his manuscript to believed that he had used these methods with success and was writing from personal experience. So, they didn’t question, or even look at, the math.
In fact, McDowell tells me he used other ace location methods on much simpler shuffles many years ago, but has never attempted to use the methods he proposes in this book in the shuffles he describes. And, unfortunately, the methods in this book will not succeed.
Another Doozy in McDowell's Blackjack Ace Prediction
There are many other examples of bad math in the book. I’m not going to waste my time going into all of them. But here’s another quick doozy:
On page 100, the author describes a player hand of hard 15 versus a dealer 4 upcard, and the hand is sandwiched between two hands that contain aces that would wind up being adjacent to each other in the discard tray if that hard 15 were not on the table. McDowell states that "an Ace tracker may deliberately hit the hand until it busts" so that the two aces on the hands on either side of it will be adjacent to each other in the discard tray.
Well, maybe. But then, he says that the cost of this play is "relatively small, about 0.20 of the bet." Hmmm... Since when does deliberately busting a hard 15 cost only 20% of the bet? According to my copy of Griffin’s The Theory of Blackjack, the player’s expectation when standing on a hard 15 versus 4 is 21% of the bet. Deliberately busting any hand provides an expectation of 100% of the bet, or an additional 79% in this case.
This, of course, is silly. Maybe McDowell didn’t mean to say that the tracker would "deliberately hit the hand until it busts." It’s kind of hard to imagine the looks he’d get from the dealer and other pit staff if he drew a 6 on his 15 for a total of 21, then insisted on hitting it again!
Dealer: But, sir, you have 21.
McDowell: I know how to add, damn it! Now hit that hand!
The problem, I’m sure, is that McDowell never actually did this stuff, so he didn’t think it through. He simply looked up the cost of violating basic strategy on 15 versus 4, which is about 20%, and used this cost as the cost of "deliberately busting" the hand. This is always the kind of problem that occurs when someone is thinking theoretically instead of realistically, because the person never actually did what they are proposing.
Anyway, I’m sitting here looking at the endorsements on the book, and I’m thinking, "Steve! Ed! Don! I know you guys have never tracked aces, but couldn’t you at least have taken out a calculator and spent ten minutes going over some of the math before jumping on this bandwagon? Does Snyder always have to be the bad guy delivering the bad news?"
There are at least a dozen more examples of bad math in Blackjack Ace Prediction, but this is all the time I’m going to spend on it. Anyone who understands gambling math can go over it and find the errors fairly easily. The problem is that if you correct the errors, there just isn’t much of a book left. As for the tracking methodology, I do not at all mean to imply that, because I’ve addressed a math problem, the other stuff is okay. The system this book touts doesn’t work, but those problems will be addressed in subsequent reviews.
Maybe someday Tommy Hyland or Al Francesco or another of the reallife ace trackers out there will write a book on this subject and really tell you how to do it. The top ace trackers are hitting the ace on 40% to 70% of their bets, not 13%. If you want to track aces and actually make a profit from the endeavor, David McDowell’s book is not what you’re looking for.
And if anyone cares to argue about "Snyder’s rule of thumb" on this website, please post your arguments in the Fight Club where I can invoke "Snyder’s rule of finger." ♠
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