Effects of Shuffle Tracking Errors on False Key Rates
by Radar O’Reilly
(From Blackjack Forum XXIV #2, Spring 2005)
© 2005 Blackjack Forum
In his book Blackjack Ace Prediction, David McDowell has made a serious error in his ace prediction hit rate calculation that has not yet been addressed. I am addressing it because I expect it to jump out at readers some time soon, and it may lead them to overestimate their advantage using his methods.
On p. 111 of Blackjack Ace Prediction, McDowell provides ace hit rate calculations as part of calculating an overall win rate estimate for players. He estimates, based on shuffle studies, that an Ace can be expected to land on the predicted betting spot 38% of the time.
I am not, at this time, going to address his estimate of 38%.
McDowell goes on to explain in Blackjack Ace Prediction that this 38% hit rate assumption must be reduced by the probability of broken sequences and false keys. There are serious problems with the probability he provides for false keys. I will address these in a moment. For now, I want to point out that the way he makes his adjustment for broken sequences and false keys is wrong. McDowell subtracts his overall probability of broken sequences (.15) and false keys (.10) from his .38 hit rate instead of multiplying and then subtracting the product. This is wrong because a share of the broken sequences and false keys properly belong to the aces that land on the other betting spots.
By subtracting .15 and .10 (.25) from .38 he comes up with an estimated 13% hit rate on his ace bets.
Instead, he should have multiplied .38 by .25.
.38 x .25 = .095
Then, he should have subtracted .095 from .38.
.38 – .095 = .285
If McDowell’s false key probability were correct, this would give him a 28.5% hit rate on his ace bets instead of 13%.
Since he specified that he is splitting the aces with the dealer, that would have given both the player and the dealer 20.8 additional aces each beyond the 7.7 accidental aces they will receive.
20.8 x .51 = 10.608 expectation for player when he gets the keyed ace on his ace bet
20.8 x .34 = 7.072 negative expectation for player when the dealer gets the keyed ace when the player has an ace bet out.
10.608 – 7.072 = 3.536% expectation for player on hands where keyed ace hits either dealer or player hand.
To this you must add the expectation on the 58.4 hands where the 7.7 per 100 accidental or random aces will be appearing (7.7 each for both dealer and player). This means 58.4 hands per 100 ace bets played at the house edge of roughly .5%.
58.4 x -.005 = -.292
3.536 – .292 = 3.244% edge on player bets for the keyed ace.
Again, this is assuming that the author’s false key probability is correct, which I will be disputing below.
But this is still not an overall win rate. Assuming the player is able to bet 3 aces per shoe (another assumption that needs to be challenged) in the 66% penetration game the author specifies, where 1/3 of the cards are cut out of play, the player will be playing roughly 33 hands per shoe heads up at the house edge. If he uses a spread of 100 to 1000:
$3000 x .03244 = $97.32 per shoe on his ace bets
$3300 x -.005 = -$16.50 per shoe on his waiting bets
97.32 – 16.50 = 80.82 player profit per shoe
80.82/6300 action = 1.28% win rate.
So, if McDowell’s false key assumptions were correct, and if his assumption that you could use visual tracking to locate three aces per shoe were correct, on this heads-up game where he split aces with the dealer, and used a 1-10 spread, he would have an overall win rate of around 1.28%. It’s less than 1/3 of the 4% win rate McDowell provides, and it would require more than three times the bank, but at least it would be a respectable card counter level of win rate.
Unfortunately, McDowell’s assumptions are not correct. I have already addressed the problem of visually tracking three aces per shoe in my Spring 2005 Blackjack Forum article. To summarize, I make the case that McDowell cannot visually track three aces per shoe in the shuffles he describes. A more realistic assumption is one ace per shoe in the more difficult shuffles (a highly skilled tracker might be able to visually track two per shoe in the simpler ones). I also show that McDowell needs to allow for a visual tracking error rate that depends on the size of the slug he is attempting to track. McDowell allows for no visual tracking error at all.
McDowell’s assumptions about false keys are also a serious problem. McDowell’s shuffle analysis on p. 72 has us tracking cards 50 to 62 in the preshuffle stack to positions 200 through 250 in the post shuffle stack. He also uses his analysis to track cards 80 through 92 to a full deck in the post shuffle stack. On p. 77, he says that “the ordinal post-shuffle position of a card is not easily obtainable in actual play,” and says “simply knowing where this slug of thirteen cards ends up after the shuffle is enough.” These statements are accompanied by a chart showing that a pre-shuffle slug of thirteen cards will end up spread over a full deck post-shuffle.
Yet, on p. 101, when it comes to the important calculation of the probability of false keys, McDowell suddenly has us visually tracking our ace to a half deck, rather than a full deck. He does this specifically to cut his false key probability in half. Then, on p. 103, he cuts his false key probability in half again, to one quarter of what it should be, by claiming that we can use “pointer cards” to halve the number of false keys we bet. Specifically, he claims that trackers can use asymmetric face designs (the number of pips facing up versus down) or the space between the index on a card and its edge to distinguish between a false key and our real key.
Let’s look harder at this assumption.
Regarding the 22 cards that have asymmetrical pip patterns, this accounts for 42% of the possible key cards (22/52). Assuming that false keys would be turned half one way and half the other, we could eliminate 21% of the false keys (those that were turned the wrong way). This would assume that the player is able to follow the orientation of the pip pattern through the dealer’s pick up, placing the discards into the discard tray, removing the discards from the discard tray, turning one half of the cards before the shuffle, as is the procedure in the majority of U.S. casinos, then continuing to follow the orientation through the tip-over, cut, and replacing the cards into the shoe. And this is assuming the player can remember the orientation of multiple key cards to begin with.
Regarding the asymmetrical gap between the index and the edge, unless there is a huge cutting error in the manufacturing process, most decks will not show any easily detected difference from the index on one corner to the index on the other corner. The prospect of a player having six or eight decks that are all this badly miscut is very slim. There may be a card here and there noticeably miscut, but there is only value if one of these cards happens to fall as a key card, and again, the player must follow that card orientation through the entire pick-up, shuffle, and placement into shoe procedure.
Then, on p. 104, McDowell says “Predicting Aces at the table is easier than the theory makes it look.”
McDowell’s automatic assumption that players will reduce their false keys by a full 50%, and his use of this number in his win rate calculation as standard, is naive at best.
Anyone planning to actually try McDowell’s ace location methodology at an actual casino table should use a false key probability of roughly four times the number he provides (roughly .40 instead of .10). Remember, McDowell is using only a single card to key each ace, not a two- or three-card sequence, which most ace trackers use in order to greatly reduce false keys.
What does a false key probability of .40 do to the expected win rate?
First, multiply the .38 theoretical hit rate by .55 (.15 probability of broken sequences plus .40 probability of false keys).
.38 x .55 = .209
br />Subtract the probability of broken sequences and false keys from the 38% hit rate.
.38 – .209 = .171
This means a 17.1% hit rate on his ace bets.
Since we have been analyzing a heads-up game, and McDowell specified that he is splitting the aces with the dealer, this would give both the player and the dealer 9.4 additional aces each beyond the 7.7 random aces they will receive.
9.4 x .51 = 4.794 expectation for player when he gets the keyed ace on his ace bet
9.4 x -.34 = -3.196 expectation for player when the dealer gets the keyed ace when the player has an ace bet out.
4.794 – 3.196 = 1.598% expectation for player on hands where keyed ace hits either dealer or player hand.
To this you must add the expectation on the 81.2 hands where the 7.7 per 100 accidental or random aces will be appearing (7.7 each for both dealer and player). This means 81.2 hands per 100 ace bets played at the house edge of roughly .5%.
81.2 x -.005 = .406
1.598 – .406 = 1.192% edge on player bets for the keyed ace.
And this is assuming that the player never makes a visual tracking error, and never has a shoe in which the tracked ace does not make it to the expected post-shuffle deck.
But this is still not an overall win rate. Assuming the player is able to bet 3 aces per shoe (again, another assumption that I have challenged) in the 66% penetration game the author specifies, where 1/3 of the cards are cut out of play, the player will be playing roughly 33 hands per shoe at the house edge. If he uses a spread of 100 to 1000:
$3000 x .01192 = $35.76 per shoe on his ace bets
$3300 x -.005 = -$16.50 per shoe on his waiting bets
35.76 – 16.50 = 19.26 player profit per shoe
19.26/6300 action = 0.3% win rate.
What can a player do to increase this win rate? For one thing, he has to address that 50/50 split on the aces with the dealer in this game. But in order to do this, he must either spread to multiple hands, and add in their costs, or play at a crowded table, and sharply reduce the number of aces he is able to bet per hour. In his p. 122 calculations of expected return, McDowell assumes 4 bets per hour at a full table. But to get this number of bets per hour, he assumes that the player will visually track 4 aces per shoe, and that none of these tracked aces will be cut out of play. Again, these are very unrealistic assumptions.
Assuming a 17.1% hit rate on our ace bets, and a full table where the dealer gets nothing beyond his accidental aces, the player will receive 9.4 additional aces beyond the 7.7 accidental aces he will receive.
9.4 x .51 = 4.794% expectation for player when he gets the keyed ace on his ace bet
To this you must add the expectation on the 90.6 hands per hundred where the 7.7 per 100 accidental or random aces will be appearing (for both dealer and player). This means 90.6 hands per 100 ace bets played at the house edge of roughly .5%.
90.6 x -.005 = -.453
4.794 – .453 = 4.341% edge on player bets for the keyed ace.
Again, this is assuming that the player never makes a visual tracking error, and never has a shoe in which the tracked ace does not make it to the expected post-shuffle deck.
But this is still not an overall win rate. Assuming the player is able to bet one ace per hour (again, a generous assumption given visual tracking at the crowded game with 66% penetration the author specifies), the player will be playing roughly 59 hands per hour at the house edge. If he uses a spread of 100 to 1000:
$1000 x .04341 = $43.41 per shoe on his ace bets
$5900 x -.005 = -$29.50 per hour on his waiting bets
43.41 – 29.50 = 13.91 player profit per hour
13.91/6900 action = 0.2% win rate.
And again, this makes no allowance whatsoever for visual tracking error.
Note that by correcting McDowell’s arithmetic on false keys and broken sequences, he is credited for a higher ace rate than he himself estimated. But this is still not a high-enough win rate expectation for consideration by professional gamblers, especially after McDowell’s false key probabilities are corrected.
The problem for real-life gamblers who want to locate aces for profit in casinos is that David McDowell’s single key, visual tracking methodology is not a good one even in the one pass riffle and restack shuffle, with two riffles, that he uses for calculating his win rate (p. 111).♠