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David McDowell'sBlackjack Ace Prediction: A False Key to Ace Sequencing
FROM ET FAN:
David McDowell's Blackjack Ace Prediction: A Problem with False KeysBy Radar O'Reilly
(From Blackjack Forum XXIV #2, Spring 2005)
© 2005 Blackjack Forum
David McDowell states his hopes for his book Blackjack Ace Prediction on p. 17, the first page of his actual text. He says: "For the first time there is a lot of detail about methods for working out if a shuffle is predictable before you play it—and how much you can expect to earn."
McDowell says elsewhere on the same page that "analyzing shuffle predictability is typically the type of thing big money pros do that non-pros don’t do, so it depends on how serious you are about winning."
I am concerned that these statements, along with others in the book similar to them, may lead players to believe that the shuffle analysis, shuffle tracking, and ace tracking methods McDowell provides are what "big money pros" do. I want to try to show you why nothing could be farther from the truth.
This article will provide specific examples of important errors in McDowell’s tracking methods. Then I will discuss why visual tracking is, in most cases, not experienced players’ preferred method of chasing aces.
There are mistakes in almost all of the author’s analyses, but this article will focus on his analysis on pp. 67-77 of what he calls a two-pass "combo shuffle," that is, a 4-pile stepladder followed by a 2-pile riffle and restack. I don’t know why McDowell chose to analyze such an unusual shuffle (in many years of shuffle-tracking I’ve only run into such a shuffle in two casinos, and both of those casinos used multiple plugs of the cutoffs). I’m choosing to address it because the mistakes in this analysis can be used to address important tracking issues that have a high potential cost to players.
The shuffle McDowell describes has no plugs. Instead, he avoids many difficult tracking issues by specifying that cutoffs are topped (or implying that there are no cutoffs). So, all the analysis has to deal with is 312 cards (6 decks) in a stack, numbered and arranged in order so that card #1 is the first card sitting on the table and card #312 is the last card on the top.
McDowell specifies that the dealer breaks this pre-shuffle stack into 4 piles. Details of these breaks are not given.
The dealer next grabs half-deck, or 26-card, grabs from two of these piles, riffles them together, and places them on the table to start a fifth pile (called the "final stack").
The dealer next grabs 26 cards from another of the four original piles (which pile is not specified—only that it is "alternating"), riffles it once with a half-deck grab from the "final stack," and places these 52 cards on the "final stack."
McDowell says: "This is repeated until the first pass is complete."
McDowell then specifies that the "final stack" is broken into two three-deck piles to begin the second pass of the shuffle. In this pass, a half-deck grab is made from each of the two piles, the two grabs are riffled together once, and then are placed on the table to make a new "final stack." This is repeated until the second pass is complete, and the cards are offered to a player for the cut.
The methodology McDowell uses to analyze this shuffle is provided on p. 72. What he does is run through this shuffle four times, each time graphing the location of each numbered card in the post-shuffle stack. Based on these four repetitions of the shuffle, he provides four pages of proof that this shuffle did not completely randomize the order of the pre-shuffle cards. Fine. No experienced tracker will disagree with him there.
Instead, the real problems begin back on p. 72, where the author goes into a methodology for identifying and exploiting shuffle weaknesses based on his analysis of his four executions of this shuffle, and immediately, in his very first example, goes seriously wrong. How wrong?
Well, to start, McDowell mis-maps his target slug. To be specific, on p. 72 he has used faulty methodology to identify cards 50 through 62 (pre-shuffle) as a "shuffle weakness" (or particularly trackable slug) and supposedly shows, based on "detailed statistical analysis," that these cards will be located in the second deck from the top after the shuffle.
But, given McDowell’s own specifications for this shuffle this cannot possibly be true. Now, I realize I’ve already lost all the hole-card boys, but I hope the rest of you will bear with me while I get nitpicky for a moment, because the details to come get more important when it comes to larger tracking issues. Although the error will look small and meaningless to you, I will try to show you why it is not small when you're dealing with locating an ace to bet.
McDowell has specified a 2-pass shuffle with no plugs and 26-card grabs on each pass. The shuffle is being done on 6 decks. That means it is impossible, on a proper idealized shuffle map, for cards 50, 51, and 52 (supposedly in slug C of his diagram on p. 73, but actually in slug B, at the top of the first deck dealt) to ever wind up in the same post-shuffle location as cards 53 through 62. Why? Because 26-card grabs place card 53 at the bottom of one of the dealer’s grabs (the grab of half-deck C), and cards 50, 51, and 52 at the very top of another of the dealer’s grabs (the grab of half-deck B). Cards at the top of a grab will repeatedly get included in the following grabs from the "final stack," and will evaporate upward in the stepladder into unpredictable (in practical reality) post-shuffle locations.
(Please hold back, for a moment, your objections that no dealer can be expected to grab precisely cards 53 through 65 as the bottom half of a grab either, because that is part of the larger point we will soon be getting to. I’m perfectly aware that dealers will sometimes grab cards 50-62 as the bottom half of a grab, just as they will sometimes grab cards 55-67, or 55-70, as the bottom half of a grab. The problem is that McDowell doesn’t seem aware that there is no such thing as "the house shuffle." Instead, in any casino there are a constellation of shuffles, all relating more or less to a central recipe. McDowell fails to address the errors that will ensue if the dealer does not have precisely his grab size or consistency, and he doesn't seem to realize that his map is only right for dealers who share his precise grab error.)
So, how did McDowell come up with graphs and statistical analyses that "prove" cards 50 through 62 consistently wind up together in the 2nd deck from the top after the shuffle? There are only a couple of possibilities. Either McDowell initially "broke big" in his four kitchen table repetitions of the shuffle—that is, broke the initial stack in such a way that the pile containing cards 50-62 was smaller than the others by three cards--, or his first 26-card grab on this pile was actually consistently a few cards bigger than he thought.
And why does this matter?
McDowell is proposing that a player visually track a 13-card slug through a difficult shuffle in order to use one key to locate an ace within one post-shuffle deck. But any aces that appear in location 50, 51, or 52 cannot be expected to show up in the anticipated deck. Therefore, 3 out of every 13 times, or roughly 23% of the time, the ace McDowell is tracking will not turn up where expected.
And exactly how costly will this 23% disappearance rate be?
For one of every three of our disappearing aces, its actual key will wind up in the expected post-shuffle deck to trigger a bad bet. In addition, the normal number of other false keys will trigger bad bets on this ace that will never appear. For the other two of every three of our disappearing aces, the normal number of false keys can be expected to double, because the real key and its ace will never appear in the deck to stop you from betting false keys after them.
On p. 101, McDowell calculates the probability of one or more false keys in a half-deck slug as 18%, or presumably 36% in a full deck (which is the number we must use since thus far all assumptions are that we have been tracking our slug to a full deck). To keep this paper from straying too far from its central point, let’s just use McDowell’s false key probabilities without quibbling for now. So:
23 out of every 100 times we will have "disappearing" aces
7.7 of these 23 times, the true key will be in the deck without its ace, causing a bad bet. (On these 7.7 times of 100, you will also see the normal number of "normal" false keys.)
For the other 15.3 of these 23 times, you will have an additional 36% chance of betting a false key, since your true key and ace will not be in the one-deck post-shuffle slug to prevent you from betting false keys after them. This means roughly an additional 5.5 bad bets per 100.
7.7 + 5.5 = 13.2 additional non-hits per 100 bets.
Since McDowell, on p. 111, only has us hitting at a rate of 13 ace bets per 100, removing roughly 13 hits per 100 leaves us with:
0 ace hits per 100!
Or something like that. I don't see much point in going further with the math when McDowell's underlying formula is wrong.
(Note: if you use McDowell's 10% false key number on p. 111 as the correct one [I can't find how he came up with that], you may give yourself credit for something like 2.5 hits per 100 aces bet after accounting for tracking error. And there are other details we’d have to account for to be precise, yada yada yada…)
Okay, Arnold Snyder has already shown that McDowell’s math for calculating his win rate is all wrong. And I do realize a player will still get 7.7 hits per hundred on "accidental" aces. I am making no attempt to provide an overall correct hit rate estimate here. I only want to make the point that, even if McDowell’s math was correct, he would have to seriously lower his win rate estimate because of mistakes in his mapping and tracking methodology.
Now, McDowell has taken care to hedge his bets by saying, on p. 87, that "the diligent Ace tracker eyeballs a shuffle to make sure Ace-rich segments are tracked to their actual final locations rather than just where the map says they will be after the shuffle." But, given the problems he has apparently had eyeballing grab size on his own kitchen table, to prepare shuffle maps for a book he knows expert players will be reading and talking about, exactly how does he propose other trackers go about doing these "eyeball" confirmations?
The problems that real trackers will be having are, after all, going to be much more difficult than the challenge McDowell faced. At least McDowell got to control his own grab sizes, and his grabs in his four shuffle repetitions were consistent. In real casinos, trackers will be watching dealers grab all over the place.
This is particularly true for any grab near the top of a shuffle pile, such as the grab near the card 50-62 slug that McDowell has us tracking. These slugs are more vulnerable than many others to displacement because of uneven pile breaks and bigger-than-recipe grabs. How confident are any of you that you will be able to visually detect that a grab in this shuffle is off by three cards? And the top 3 cards of any 13-card slug are certainly as vulnerable as the bottom 3 even for the most skilled trackers. And what if the dealer is off by three cards in the other direction? Now you are missing almost half of this slug from its post-shuffle destination. As for trackers of more typical skill, nothing in cards 50-62 is safe. The wrong grab will completely explode them out of their expected post-shuffle destination and fling them all through the rest of the shoe. If you’re going to try to attempt McDowell’s methods, you must account for the extra likelihood of errors in your win rate estimates. Experienced players know this, and that is why they don’t use McDowell’s methods!
McDowell does seem to recognize some of the problems in his methodology, because on p. 80 he advises players to "pinpoint" the locations of their sequences with extra precision by "counting the number of cards dealt as they go into the discard tray," which he assures us "is easily done."
First, speaking as someone who has actually tried this (as all players do while still in their newbie period), it is not easily done. Not only do actions at the table screw up your count by causing cards to go into the discards in unexpected order (because of blackjacks and busts), but you have to retain your count (and card location numbers) while also retaining key card info and also tracking and talking with the dealer, pit boss and host and playing your hands.
Second, what good does any of this pinpoint precision do if you are not able to eyeball card locations with equal precision during the shuffle? It’s not like you can stop the dealer and count the number of cards in each grab.
Skilled slug trackers know they have to do a lot of edge work in any shuffle like the one McDowell has specified. They have to retain information on edges because they know they’re going to screw some of them up, and they have to know what dangers are lurking out there if they are unsure of their edges. They limit their slug tracking to areas of the stack where they will have visual aides to eyeballing accuracy. They don’t just pluck out any slug that some map tells them is a "weakness" and hope for the best from that!
And slug trackers are tracking an overall rich slug with many high cards and decent borders. They aren’t generally trying to visually account for the location of two cards anywhere within these 13. And if they are, they are limiting themselves to doing this on one slug. Has McDowell ever even looked at the "final stack" of a stepladder shuffle? It’s a mess. It’s constantly getting jostled by the repeated action of adding and picking up cards. If you look away from any slug in the middle of that mess, you’ll never find it again.
There are a lot of other problems with McDowell’s analysis of this shuffle. For some examples that won’t cause your eyes to glaze over, McDowell makes the beginner’s mistake of drawing conclusions about post-shuffle slug locations without providing the details of grab order necessary to justify his map. Beginners lose a lot of money this way and, instead of making this elementary mistake, he should be pointing out the extreme importance of such details.
McDowell also assures us that "it is enough to identify one or two reliable [shuffle] weaknesses," when all of his win rate calculations require identifying four.
And, I do not mean to imply, by addressing only one shuffle, that the rest of McDowell’s shuffle analyses are free from error. I’m only trying to keep my comments down to a manageable bite for most readers.
In any case, experienced players do not use visual tracking for ace location where other methods would work much better.
For an idea of the type of situation where a skilled player will use visual tracking to help locate an ace, go read Jim Taylor’s article "St. Louis Blues" in the Professional Gamblers at Work section of this Web site’s library. Taylor is using visual tracking methods in this situation because the only information he has is one key and he has no other choice. Other trackers have used visual tracking for ace location in exceptionally dreamboat shuffles they happened to stumble upon in Podunk, in situations where their other sequencing skills were rusty from non-use.
The fact is, many players turn to ace sequencing specifically because it seems to offer methods that don’t rely on the individual skill and judgment involved in visual slug tracking. They want to get away from NRS mumbo-jumbo and self-doubt and actually be able to count the aces they hit. Unfortunately, they find that some skillful judgment is still involved.
If you want to sequence aces, the best way to prepare is to train yourself to be able to memorize multiple keys for as many aces as you can handle per shoe. If you do everything right, and you’re playing heads-up, you should be able to sequence 8-12 per shoe and bet 3-4 on a 2-riffle shuffle that isn’t too problematic, with a dealer who isn’t too awful.
If you’re playing at low-stakes, in the U.S., the more you learn from observation what to do and the more skilled you get, the more frustrated you will get because you will so rarely get an ace you can actually bet on. You may even give up on ace location because you’re not making enough money from it to justify all your work.
Still, I do not mean to dissuade you from trying. I believe every skill a player acquires will lead to other skills, one of which may even eventually pay off. In the meantime, if you go into serious training to memorize multiple sequences, it will be good for your brain. ♠
Spring 2005 Blackjack Forum
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||McDowell's Blackjack Ace Prediction is Unfortunately Full of Major Mistakes
Although David McDowell's Blackjack Ace Prediction book is full of major mistakes in everything from blackjack shuffle tracking and ace sequencing technique to basic blackjack math, this discussion of the errors in the book should help players learn to think through how to actually carry out blackjack ace sequencing and better shuffle tracking. The other thing we hope you get from the article is that it's important to test blackjack advantage play techniques like ace prediction thoroughly, and at low stakes, under real life casino conditions, before you ever put any serious money behind them. Until you work out the practical details of any advanced blackjack play, and have statistically significant win results, do not assume that you have an edge at the play!