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A common shuffle used by casinos is the "zone shuffle". Here, the cards are broken into piles, and then the shuffling is only performed between the piles. Thus, even with the uncertainty in pick sizes and riffs, a particular card has a zero percent probability of being in certain portions (most of) the shuffled pile, and a high probability of being in one or two particular portions of the shuffled pile. Casinos do not use more thorough shuffles, because more thorough shuffles take more time and reduce profits (and fortunately shuffling machines have not yet caught on.)
I am assuming that the reader has some knowledge of blackjack and card counting. A glossary of blackjack, card counting, and shuffle-tracking terms can be found in the appendix.
As an example of a simple tracking method, if the end-of-shoe count is -10, then you know that the count of the unplayed cards is +10. If the unplayed cards all get shuffled into the top half of the shuffled pile, where should you cut the cards?
The answer is right in the middle. The reason is that the unplayed cards had a count of +10 - that means there were 10 more low cards than high cards - you don't want to play those low cards which are now in the top of the pile, so you cut in the middle to put at least some of them out of play during the next shoe. You might also want to pat yourself on the back and raise your betting during the first half of the shoe, even though the count will probably start to go negative. The reason is that on average the first half of the shoe should now on average have a count of -10, which means there are ten more high cards than low cards. An important thing to remember during shuffle-tracking is that high count regions are bad and low count regions are good. This can be counter-intuitive (no pun intended).
There are two benefits to shuffle-tracking:
Suppose it is a four deck game, with three decks actually dealt. You record the running count for the first deck and call it A, the second B, and the third C. (The recorded running count is for each deck individually, so you must either take differences in the new and previous running counts.) The unplayed deck is D, and it is assigned the opposite of the final running count.
Suppose that shuffle starts by putting the unplayed cards on top of the played cards. Then the pile looks like this:
D C B AAnd now if the pile is cut in two it looks like this:
B D A CAnd if the top halves are shuffled together and then the bottom halves are shuffled together and placed on top, you wind up with this "profile":
A+C B+DThe profile shows how the tracking units are combined. The plus sign indicates that the estimated count in each two deck region is simply the sum of two tracking units. So if A=-4, B=+2, C=+1, D=+1, then the counts in each half of the shuffled shoe are estimated as follows:
-3 +3Here, you would cut as close to the bottom, trying to keep the -3 in front of the end-of-play card. You would bet more aggressively in the -3 region and more conservatively elsewhere.
In hand-done trials (which used a different casino shuffle), shuffle-tracking had a higher % advantage than card counting alone, statistically significant to the 90% confidence level. Also, while shuffle-tracking, I cut out more low cards than high cards, statistically significant to the 99.5% confidence level. The complete results of the hand-done trials are given in an appendix.
I also ran a full-blown computer simulation of shuffle-tracking an Atlantic City shuffle (the Random Pick Order Six Zone Shuffle) with AC rules plus late surrender, 75% penetration. A great deal of randomness was put into the shuffle, making it difficult to track, and the tracking and card counting was not done perfectly. The shuffle-tracker was given the cut card every time, however. With a 1-8 spread, never abandoning the table, it achieved a 1.0% advantage. In constrast, a simulated regular card counter did little better than break even in this game, if not permitted to abandon negative counts. Thus it would appear that shuffle-tracking provided a gain of nearly 1% here, but this is a tentative conclusion. By abandoning hopeless shoes, the shuffle-tracker's advantage could be increased - a regular card counter gains about 0.5% by abandoning true counts of -1 or worse on the AC game, and a shuffle-tracker has a much better of when a shoe is hopeless than a regular card counter.
A shuffle-tracker who is playing through the second shoe at a table will have four groups of chips: betting chips, count record chips, running count chips, and shuffle-track prediction chips. The betting chips are an unorganized mess from which all bets are placed and into which all winnings are placed. The count record chips are the counts of various regions in the current shoe. The running count chips denote the running count when the last count record chip was placed. And the shuffle-track prediction chips are the predictions of the counts in the current shoe.
The chip notation is used to record the numbers, such as counts in various regions of the shoe. For example, suppose that a four deck shoe is being tracked with four regions, A, B, C, and D (the latter being the unplayed cards). After the first deck (A), you place a chip to denote the running count. After the second deck (B), you subtract the current running count from the previous running count. You stack this chip on top of the previous and then record the current running count separately. For the next deck (C) you take the count difference and stack a chip representing this on the count record pile while also updating the record of the current running count. At the end of the played portion of the shoe, you take the opposite of the running count and assign this to D. (If there were more than one tracking unit that was unplayed, then this final count would be split among the unplayed tracking units as an estimate. If you are really sharp, you can split this count unevenly according to previous tracking information.)
Shuffle!
For casinos that use the same exact shuffle each time (with no randomness in the order of picks or plugging), you can analyze the shuffle away from the tables and come up with a "profile". This is just a precomputed diagram showing how to combine different portions of the shoe. A profile was listed in a previous section that look like this:
A+C B+DThis profile can be memorized and a small cheat sheet of it (perhaps on the back of a business card) brought to the casino in case you freeze under pressure.
For shuffles where the dealer has some randomization effect, like mixing up the order of picks, the tracking requires more of a brute force approach. Using brute force is simpler, but also less disguised. Here, you actually "shuffle" your chips the same way in which the dealer shuffles the cards. An example of this is given in a later section on the Random Pick Order Six Zone Shuffle.
As the shoe is played through, the shuffle-track prediction pile(s) shrink and the count record chip pile(s) grow. One should make a mental note of how close the newly recorded counts are to the estimates, and also to compare this to the prediction of the true count (i.e., the *opposite* of the true count.) Although you can't expect shuffle-tracking to be anywhere near 100% accurate in terms of sign much less magnitude of the count, you should be able to observe a correlation. If the shuffle-track predictions do not seem correlated to the observed counts, then you may be making mistakes or the shuffle may not be very trackable.
Deciding what tracking units to use is important. Generally, the tracking units relate to the dealer's pick sizes, otherwise the tracking predictions may be unnecessarily inaccurate. Also, if you choose too small tracking units, you will not be able to "eyeball" the discard tray to determine which tracking unit you're in, but if you choose too large tracking units, you may have insufficient information to give you much of an edge.
Shuffle-tracking teams can be effective. For shuffles that can be profiled, each team member can be responsible for generating a tracking prediction of some portion of the shoe, thus splitting the mental burden of shuffle-tracking. One team member can be responsible for just counting the number of cards that have been put into the discard tray and signaling the other members as the last few cards of each tracking unit get discarded. Teams can adjust the number of hands they play in order to make end of tracking units coincide with the end of a round (when lots of cards go into the discard tray at once.) Four or more team members can completely take over a seven spot table (each playing one or two hands), giving the team complete control over the cut card. This allows the team to build up a large clump of low cards that can be consistently cut out of play. However, four skilled shuffle-trackers might very well be better off each playing solo off of a pooled bankroll, so don't play at the same table unless you think there is enough of a benefit to outweigh the increased variance (the hands are correlated with each other at the same table, since all depend on the same dealer's hand.)
Intelligent cutting is one of the benefits of shuffle-tracking. Sitting at third base gives a shuffle-tracker an advantage, because it increases the likelihood of getting the cut card in casinos where the cut card is given to the third base player if it comes out while the dealer is resolving his own hand. (It's also nice to sit at third base, because there's usually lots of room there to spread one's chips out for shuffle-tracking purposes, plus third base has a slightly higher advantage for regular card counters anyway, due to more cards being seen before the player makes his plays.) A shuffle-tracker can also spread to two hands at the end of a shoe to boost the odds of getting dealt the cut card. Using both these techniques at a full 7-spot table would give one over a 3/7 chance of getting the cut card.
One can also often obtain the cut card simply by asking for it. Saying something like "I feel lucky - how about letting me cut for us?" usually does the trick. Players are usually either antipathetic or nervous about cutting the cards, so they will generally relinquish the cut card gladly. One should use some restraint in doing this while a pit critter is lurking nearby, however. You can attempt to tell people where to cut, but this is harder than it sounds, so it's better just to get the cut card yourself.
Often players leave during the shuffle, so be on the look-out for an abandoned cut card. If the player who had the cut card leaves, then pounce on it or ask another player to pass it to you.
Sometimes the shuffle-track predictions will not reveal any good place to cut the cards. At such times, obviously you don't need to fight to get the cut card.
Even if someone else cuts, you can still judge how good their cut is and decide whether or not to remain at the table. If there is an obvious good region that will be in play, it may be worth staying to bet big in that region even if another good region was cut out, yielding an overall bad shoe. However, always remember that if you leave to go to another table, on average an equal number of good cards and bad cards will have been cut out, whereas if you stay with a bad cut, then you are pretty sure that more good cards have been cut out than bad cards.
This is not to say that Atlantic City casinos do not care about card counters. They are very paranoid about them (which is not justified given the poor games) and they can and do take other countermeasures. Also, Nevada casinos will activate many other countermeasures before resorting to barring. The simplest and most effective countermeasure is the "shuffle up", when the cards are prematurely shuffled. If you place a large bet, they may shuffle up, or if the dealer is card counting too, he may shuffle away *any* favorable situation! This is very rare in Atlantic City, where the preferred countermeasure is to move the end-of-play card to perhaps the 50% point after the next shuffle. In Nevada, you will usually be kindly asked to play craps or any other game than blackjack before you are actually barred.
The casinos do not part easily with their money. In fact, I have seen many pit bosses get upset when someone (whom I can tell is about as intelligent as a squid) happens to get lucky and walks away from the table with a lot of money. The variance is so high in blackjack that that a very good counter can easily take a big loss for eight hours, while the squid keeps raking in the bucks, but this is something that card counters and casino employees do not in general appreciate fully. In the long run, of course, the counter grinds out a profit, while the squid will eventually lose everything unless it slinks away from the table first.
Since the average pit critter does not understand these facts of statistics, you must appear to lose and take little if any chips away from the table. This is difficult, since in order to shuffle-track you may need dozens of chips on the table at all times. Even if you buy in for the same amount that you leave with, pit bosses may get upset if you walk away from the table with a few hundred dollars in chips.
One method of hiding a win is to simply not let the dealer color you up. "Coloring up" or "coloring in" is where the dealer counts your chips and gives you a few higher denomination chips. If you don't let them color up your chips, then they will be uncertain as to how much you take away from the table. However, this itself will raise suspicions, especially in Atlantic City, where the casinos are quite insistent about coloring up a player's chips upon his leaving a table. Another method to disguise your winnings is to pocket chips *discretely*. The best way is to pick up a stack of chips and hold it in your hands to bet with for a few rounds - then when no one is looking and your hands are clasped around the chips, drop your hands off the table and shove the chips into your pocket while appearing to be interested in a cocktail waiter/waitress or something. Only do this if there are other players taking that denomination of chips away from the table, because the pit critters keep careful records of the chip trays.
Real gamblers often play with their chips. You should practice at home with real casino chips to learn to figit with your chips most of the time and to disguise your occasional moves to record information with some of your chips. It helps to be messy and careless with your chips. Leaning over your chips with your arms above them will help obscure the information in your chips from the eye in the sky and the pit boss' regular rounds. (Side note: gosh how I hate "leaners" when I'm trying to back-count - it makes it hard to see all the cards from behind!)
Remember that there is little risk of them mistaking you for a normal card counter, because you will sometimes be able to bet high off the shoe and high in the middle of the shoe in opposition to the true count. Thus, shuffle-tracking itself is a means of disguise, so long as you don't appear to be a little too interested in your chips.
There appear to be two shuffle techniques designed to frustrate shuffle-trackers. One is to introduce some dealer-driven randomness. This includes plugging unplayed cards into played cards at several random locations, and random orders of picks for zone shuffles. The other tracking countermeasure is the stutter shuffle, where two piles are shuffled by taking a pick from one pile and a pick from the already shuffled cards and putting the result on top of the already shuffled cards (the stutter pile), and then taking a pick from the other pile and a pick from the already shuffled cards and putting the shuffled result on top of the already shuffled cards, and so on.
It is interesting to note that I have never seen a casino shuffle that uses *both* of dealer-driven randomness and the stutter shuffle; it seems that the casinos feel that either one alone is sufficient to thwart shuffle-trackers. In truth, they are pretty much correct about the stutter shuffle,but sometimes the dealer dealer-driven randomness techniques can be conquered.
Each casino generally alters its shuffle every few months. Again, this is obviously intended to thwart long-term attacks from shuffle-trackers.
There are two main approaches one can take to analyze a shuffle. The most straightforward is to take eight decks of cards (or however many the casino uses), label the backs with letters denoting different tracking units, sort them according to tracking units, and then shuffle them as the casino does. You then count the numbers of cards from each tracking unit that wound up in each region.
The other way analyze a shuffle is to take a more symbolic approach. Start out with an equal number of copies of each tracking unit letter. (With the number of "copies" just being a convenient number that will avoid fractions of tracking units during the analysis.) For example, if you have four tracking units and two copies of each, then the cards might look like this before shuffling:
D D C C B B A AYou then manipulate this symbolically to arrive at the distribution of cards after the shuffle. This is the technique used previously in generating a "profile" of a simple shuffle, and this techique will also be used in the following sections on specific casino shuffles. The information here may be slightly dated - my last full survey of AC shuffles was around January 1991.
First, the Claridge plugs the unplayed cards into the discards (at three or four locations), while the Sands merely places the unplayed cards on top of the discards.
For the Sands/Castle version, take the top ~2 decks off the top of the discard pile (i.e., pretty much just the unplayed cards) to form what pile #3, and *then* cut the rest of the discard pile in half to the right to form piles #1 and #2. For the Claridge, on the other hand, cut initial pile in half to dealer's right, and call the piles #1 (bottom cards) and #2 (top cards). Pick .5 [variant: up to 1] decks off each of #1 and #2, and join the picks to form #3.
Cut #1 and #2 each in half, dropping the picked piles away [variant: toward] the dealer. Call the new piles #1a (bottom of #1), #1b (top of #1), #2a (bottom of #2), and #2b (top of #2). From now on, picks from #1a, #1b, #2a, and #2b will all be 1/3 [variant: 1/4, sometimes even less] of these piles, and picks from #3 will be 1/6 [variant: 1/8, sometimes even less] of this pile. Join picks from #2a, #1b, #3 (with #3 always on top) - riff, riff, put on done pile. Join #2b, #1a, #3 - riff, riff, put on done pile. Now back to #2a, #1b, #3, and continue alternating until all cards are in done pile. This results in 6 [variant: 8, sometimes even more] shuffled regions in the done pile.
This shuffle is the same for the four and six deckers as the eight deckers, except that picks are scaled down appropriately.
If you can find a dealer that is fairly consistent, then you can devise a profile of how this shuffle combines tracking units in the next shoe. Unfortunately, having that extra C pile leads to either a more complicated or less accurate (or both) profile.
Here's how to make such a profile using a symbolic approach. First, we start with, say, three copies of each of six tracking units (A-F). Then the initial cards look like this, assuming 1/3 of the cards (tracking units E and F) are undealt:
D D D C C C B E B E B E A F A F A FFor the Sands/Castle shuffle, the undealt cards are placed on top of the played cards and then taken right back off to form pile C. Then The played cards are broken into four piles:
(1b) (2b)
B D
B D (3)
B D E
E
(1a) (2a) E
A C F
A C F
A C F
The first part of the shuffle takes picks from #1b, #2a, and #3, the
second takes picks from #1a, #2b, and #3, and so on to yield:
A+D+F B+C+F A+D+F B+C+E A+D+E B+C+EThe tracking predictions should technically be divided by three. Also, tracking units E and F are the unplayed cards, and so they can be estimated to have the same count. Thus, a possible simplification might be to just ignore E and F since these counts are distributed evenly through the shoe.
The dealer plugs the unplayed cards into discards in three spots, usually one deck up from the bottom, the middle, and one deck down from the top. The pile is broken in two (to the right usually). Then each pile is broken in three, to create a line of six roughly equal piles. If the original pile after plugging looked like this:
F E D C B AThen the resulting six piles would look like from the dealer's perspective:
C B A D E FExcept that female dealers (with small hands) sometimes do this:
B C A D F EA 1/2 pile pick is made randomly from A, B, or C and riffed with a random pick from D, E, or F. The just riffed cards are then cut in two and riffed again. This is repeated until all the cards have been shuffled. Usually, the dealer does not take the second pick from a pile until all other piles have had their first pick, but this is not always true.
The result of this shuffle is quite nonrandom, and my shuffle-tracking simulator showed that it would be quite trackable, if only you can combine the appropriate tracking units. For example, suppose that the dealer always picked in the order A+D, C+F, B+E, A+D, C+F, B+E. Then the result would be this:
B+E C+F A+D B+E C+F A+DEasy, huh? Or if we break down the observed cards more finely, using A1 to stand for the bottom part of A and A2 for the top of A, and analogously for the rest, we start with this:
F2 F1 E2 E1 D2 D1 C2 C1 B2 B1 A2 A1And after being shuffled this becomes...
B1+E1 C1+F1 A1+D1 B2+E2 C2+F2 A2+D2So, if the count for B1 were -5 and the count for E1 were -2, then we can estimate that the first sixth of the shoe shuffled as above would have a count of -7.
Now the above letters sort of assumed no plugging. A plugged pile might start like this, if the unplayed cards are units E2, F1, and F2:
E1 D2 F2 <- PLUG D1 C2 C1 F1 <- PLUG B2 B1 E2 <- PLUG A2 A1And the above would then wind up like this:
E1+F1 D2+B2 F2+B1 D1+E2 C2+A2 C1+A1In practice, the individual counts of the plugged cards (tracking units E2, F1, and F2) are not generally known, so the end-of-shoe count can be split between them as an estimate (possibly biased by previous tracking information.) Also, the above profile cannot be used in general, since the order of the picks and placement of the plugs varies.
But that's okay. We can use brute force.
If penetration is 75%, then you've got a stack of nine chips representing the played cards and a stack of three chips representing the unplayed cards. This is convenient for the above shuffle, because the unplayed cards are split into three and plugged into the played cards. A brute force approach is required to track this. Watch the dealer carefully, and just plug your chips wherever he plugs his cards! Cut the chips in half the same direction (mirrored, so usually you cut to the left) that the dealer cuts the cards. Break the chips into six piles of two chips in the same pattern as the dealer breaks the cards six piles. You can be a little less conspicuous by not doing these activities exactly when the dealer does them. For example, I usually cut my chips 5-4 before the dealer and I do the plugging, and I break mine into the smaller piles before he does. I just have to keep one eye on the dealer to make sure he is doing the normal routine. Then, as the dealer takes a pick from each of two piles, I take a chip from each of my corresponding piles (mirrored). I put these into two separate piles. The next picks are stacked on top of these piles.
The end result is two piles of six chips. At each level, the sum of the two chips is the estimate of the count in that region. As I put down the chips, I note where excessively positive (bad) and negative (good) regions are. If I get the cut card, I then attempt to cut just below the worst regions in order to cut them out of play. If someone else is cutting, I note where they cut. I then discretely (no hurry) cut my chips and restack them. The top chips on the two piles added together are then an estimate of the first 1/6'th of the shoe. I can adjust my betting and playing appropriately. I don't consider the information to be very reliable unless both chips are pointing strongly in the same direction - then chances are very good the shuffle-tracking has at least the sign of the count in that region correct.
Place unplayed cards on top of played cards. Split the eight decks into two ~four deck piles, call them #1 (bottom) and #2 (top). Picks are taken from each pile, shuffled once, and placed in the "stutter pile", #3. Then a pick is taken from one of the piles (usually #1) and shuffled with a pick from #3. The result is placed *under* #3. A pick is taken from the other pile and shuffled with a pick from #3, and the result is again placed *under* #3. After this point, it continues to alternate between #1 and #2 (both with picks from #3), but the results are placed on *top* of #3; the shuffle proceeds until all the cards are in #3.
The dealers at Bally's Grand tend to use very large pick sizes, but in the following profile, I will assume that the pick sizes are just 1/2 deck, so that this will be compatible with the following section on the Stutter Plus Shuffle. Here is how the profile looks for this stutter with sixteen regions from A (bottom) to P (top), where (x y) is defined as the average of the counts of regions x and y:
1. (I (A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))) 2. (I (A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))) 3. (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))) 4. (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))) 5. (B (K (C (L (D (M (E (N (F (G (H P))))))))))) 6. (K (C (L (D (M (E (N (F (G (H P)))))))))) 7. (C (L (D (M (E (N (F (G (H P))))))))) 8. (L (D (M (E (N (F (G (H P)))))))) 9. (D (M (E (N (F (G (H P))))))) 10. (M (E (N (F (G (H P)))))) 11. (E (N (F (G (H P))))) 12. (N (F (G (H P)))) 13. (F (G (H P))) 14. (G (H P)) 15. (O (H P)) 16. (O (H P))For example, the bottom deck (regions #15 and #16) is composed of 1/2 region O and 1/4 each of regions H and P. Obviously, this is not a practical prediction scheme, especially with respect to the top deck. This could be simplified by using larger tracking units; for example, there could be just four tracking units: A/B/C/D, E/F/G/H, I/J/K/L, and M/N/O/P. But even this would be complex. Other simplifications could be made, but only at the cost of much accuracy. So long as there exist zone shuffles, the shuffle-trackers should avoid stutter shuffles like the plague!
1. ((L (D (M (E (N (F (G (H P))))))))
(O (H P)))
2. ((C (L (D (M (E (N (F (G (H P)))))))))
(O (H P)))
3.((K (C (L (D (M (E (N (F (G (H P))))))))))
(G (H P)))
4. ((B (K (C (L (D (M (E (N (F (G (H P)))))))))))
(F (G (H P))))
5. ((J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))
(N (F (G (H P)))))
6. ((A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))
(E (N (F (G (H P))))))
7. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))
(M (E (N (F (G (H P)))))))
8. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))
(D (M (E (N (F (G (H P)))))))
9. ((L (D (M (E (N (F (G (H P))))))))
(O (H P)))
10. ((C (L (D (M (E (N (F (G (H P)))))))))
(O (H P)))
11. ((K (C (L (D (M (E (N (F (G (H P))))))))))
(G (H P)))
12. ((B (K (C (L (D (M (E (N (F (G (H P)))))))))))
(F (G (H P))))
13. ((J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))
(N (F (G (H P)))))
14. ((A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))
(E (N (F (G (H P))))))
15. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))
(M (E (N (F (G (H P)))))))
16. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))
(D (M (E (N (F (G (H P)))))))
Note that the pattern for 1-8 is the same as the one for 9-16.
What the above mess means is that most cards have a chance of being
*almost* anywhere in the resulting shoe. It is effectively not
trackable, especially considering that the randomness of the
dealer's pick size and riff will considerably alter the distribution
of cards. Avoid such thorough shuffles if at all possible.
In "Break the Dealer", Patterson and Olsen published the first book describing shuffle-tracking. It is still the only book whose primary focus is on shuffle-tracking. This is unfortunate, since the book serves more as an advertising tease for their TARGET system than as a treatise on shuffle-tracking. (TARGET is a non-counting blackjack system that has been criticized by a number of blackjack experts as having no scientific basis or empirical proof of its effectiveness.) Patterson and Olsen describe shuffle-tracking in nice graphical terms, but don't go into much detail. In general they believe that certain shuffles bias the cards for or against the player, and there just isn't any evidence to support this claim (and quite a bit to refute it.) In particular, they fear the strip shuffle (where the order of the cards is reversed with a rapid series of pick), and there is no reason to fear this shuffle, unless you are trying to shuffle track and the strip mixes up your tracking units. Also, they fear like-card clumping (the natural tendency of similar cards to cluster given the order they are discarded), and this is not something to be worried about. In sum, be very skeptical about anything these authors say, though they are not *always* wrong.
Mason Malmuth has the only other publication speaking of shuffle-tracking in any depth, "Blackjack Essays" . (He calls shuffle-tracking "card domination".) He is very enthusiastic about it - perhaps overly so, since he estimates an expected win rate of 4.5%, which is very unlikely. Mason added a note in which he backs down from this estimate on the basis that the *actual* advantage in blackjack is rarely 4.5% and shuffle-tracking will not even identify all of these situations. Mason recommends that the shuffle-tracker not try to keep very detailed information about the deck composition; this is not my philosophy, and perhaps Mason is wrong about this too.
Shuffle-tracking is explained briefly in Zender's "Card Counting for the Casino Executive", and it is noted that dealers can employ shuffle-tracking in reverse, to shuffle the good cards to where they will be cut out of play.
Blackjack magazines occasionally have information on shuffle-tracking. Snyder's "Blackjack Forum" and Olsen's "Blackjack Confidential" have both mentioned shuffle-tracking in certain articles, though I have not seen any in-depth analyses of shuffle-tracking in these magazines.
+8 +9 +5 -2 +8 0 +3 +3 +7 +8 -11 +12 +19 +3 +9 -5 -10 +11 -5 +3 +3 -8 -5 -10 +1 +8 +9 +2 +17 +6 +9 +5 +7Each of the above numbers is the number of extra low cards versus high cards left *unplayed*. (This is not quite the same as the cards behind the cut card, since the cards after the cut card are used to finish the round in which the cut card appears.) It is the additive inverse of the count at the end of the shoe (though I counted the values of the remaining cards at the end to reduce the chance of errors in the statistics.) The trials are broken into three lines to note when I switched shuffle tracking strategies. The second line was intended to be an improvement over the first, and the third was intended to be an improvement over the second - each improvement involved switching to a finer grain of tracking units.
Let u0 = 0 be the mean of the null hypothesis distribution Let u be the mean of the alternative hypothesis distribution Let n be the number of trials (shoes) Let x be the average count of the unplayed cards let s^2 be the variance in the count of unplayed cards H0: u = u0 H1: u > u0 s^2 = 53.45 n = 33 x = 3.606 We should reject H0 if in LISPish notation... (/ (/ (- x u0) (/ s (sqrt n)))) > t(alpha, n-1) Plugging in we get, (/ (- 3.606 0) (/ (sqrt 53.45) (sqrt 33))) = 2.8334 Looking in a table, we find, t(.005, 30) = 2.750 and t(.0025,30) = 3.030Thus, there is significant difference, up to the 99.5% confidence level, assuming I did all the computations correctly. Therefore, we should reject the null hypothesis that shuffle tracking does not work, and hence we must believe that it does work, at least for me in my own home. By "work", I mean that you can certainly cut out, on average, more low cards than high cards. This effectively raises your running count!
Now, if we assume that the above average of 3.6 more low cards than high cards is a realistic average (warning: the 99.5% confidence does not apply to this assumption), then that implies if I shuffle track at the Claridge and cut the cards myself, then the running count is *effectively* 3.6 points higher than the actual running count. The commonly quoted number in blackjack books is that one true count point of High-Low is worth .5% advantage. Thus, in the played portion of the shoe, my gain from shuffle tracking is (* (/ 3.6 6) .5%) = 0.3%. Note that this benefits everyone at the table; stupid players and basic strategy players will get this percentage increase; card counters will win an extra amount with reduced risk; and I will argue that shuffle tracking counters will win much more at even lower risk, because of the benefits of having information about the distribution of high and low cards in the shoe.
I argue that my gains from shuffle tracking could potentially be much higher than 0.225%-0.9%, because not only do I have the benefit of cutting out the low cards, but also shuffle tracking provides a rough profile of the clumps of high cards (and low cards) in the played part of the shoe; this indicator coupled with the true count is very powerful. (How many times have you raised your bet on a high count, only to have the count go still higher while you lose? This doesn't happen often with shuffle tracking.) This local information I believe is worth more than cutting out the low cards, so even when I don't get the cut card or when I accidentally cut out more high cards than low cards, I can still benefit greatly from shuffle tracking.
Shuffle tracking doesn't always work. The inherent randomness sometimes makes the technique backfire. But it works on average to cut out the low cards as the above statistics have shown.
See my previous post for the explanations of playing conditions. Minimum bets are always $5, and the maximums used within a particular shoe are listed below by the bankroll. When the max bets are on two hands, the bets are listed with a plus sign in between. Note that sometimes the bets on two hands are not equal, since I was trying for a bet size in between (and besides it's good cover in an actual casino.) The recorded bankrolls are *after* a given shoe has been played. I estimate that 25 rounds were played per shoe, and I averaged maybe 1.25 hands at a time and a total bet average of maybe $10. By the way, I believe the relatively low observed variance is due largely to late surrender (this is a benefit that few people mention about this rule.)
WITHOUT SHUFFLE TRACK WITH SHUFFLE TRACK ###################### ########################## SHOE BANKROLL MAX BET BANKROLL MAX BET CUT-OUT ---- -------- ------- -------- ------- ------- - 300.00 - 300.00 - - 1 300.00 5 347.50 5+5 +8 STARTED TRACKING METHOD I 2 340.00 5+5 315.00 5 +9 3 320.00 5 320.00 5+5 +5 4 347.50 5 300.00 5 -2 5 377.50 5 375.00 10+10 +8 6 382.50 10+10 342.50 5 0 7 390.00 10+10 395.00 5+5 +3 8 387.50 10+10 407.50 5 +3 9 372.50 5 415.00 5 +7 10 402.50 5 427.50 5 +8 11 402.50 5+5 425.00 5 -11 12 407.50 5 422.50 15+15 +12 13 412.50 5 432.50 5+10 +19 14 410.00 10+10 395.00 5 +3 15 412.50 5+5 345.00 5+5 +9 16 367.50 5+5 327.50 5 -5 17 342.50 5 322.50 5+10 -10 18 327.50 15+15 412.50 10+10 +11 19 335.00 5+5 405.00 5+5 -5 20 322.50 5 375.00 5 +3 21 315.00 5+10 375.00 5+5 +3 22 337.50 5 382.50 5+5 -8 23 330.00 5 465.00 10+10 -5 STARTED TRACKING METHOD II 24 335.00 5+5 565.00 15+15 -10 25 327.50 5 615.00 5+5 +1 26 337.50 5 595.00 5 +8 27 282.50 10+10 597.50 5 +9 28 290.00 5+5 607.50 5 +2 29 297.50 5 590.00 5 +17 30 570.00 5 +6 STARTED TRACKING METHOD III 31 572.50 5+5 +9 32 732.50 10+10 +5 33 720.00 5+5 +7As you can see there is an impressive bottom line difference ($2.50 loss versus $420.00 gain) between the "without tracking" and then "with tracking" play records. Note that the earnings in the shuffle tracking column don't really take off until the shoes where tracking methods II and III are used. This is probably because "method I" did not provide a detailed profile for the shoe - it only indicated where to cut - whereas the other methods gave me a reasonable idea where to find the clumps of high versus low cards. Unfortunately, there were relatively few trials involvings these advanced tracking schemes.
Is the difference statistically significant?
I played about thousand hands for each series (about 906 for without "column", and 1031 for the "with" column). 1000 trials is "starting" to be a signicant number, though often millions or billions of trials are required to get above the inherent "noise" in blackjack, at least when trying to detect *small* gains in excepted value.
What is the probability of being (720-300)/5 = 64 units ahead after 1031 bets? Expected value is about zero for the null hypothesis that we are no better off than using the count with a 1-4 spread. Variance per hand is about 2.0 units squared, according to my simulations of a 1-4 spread on an 8 deck game with AC rules. Normalizing this we get z=(64-0)/sqrt(1031*2.0)=1.41, and looking this up in a statistical table for the normal distribution, we get about 8%. There is an 8% chance of results as good as mine even with no advantage. This is not low enough that I can disregard the possibility that my winnings could have just been luck. In addition, the variance was probably a bit higher than 2.0 units squared for the shuffle-tracking betting.
We will assume that the variance is unknown. At first I thought that the variances should be equal (though unknown) for the "with" and "without" trials, since our null hypothesis is going to be that shuffle tracking doesn't do squat for our winnings, in which case one might think it shouldn't do squat to the variance either. However, the shuffle tracking did allow me to increase my bet size more often; therefore the variance should be higher in the "with" trials. Given that the null hypothesis is that the means (but not necessarily variances) are equal, the alternative hypothesis is that shuffle tracking gives us a higher average win per shoe.
Here is the data for per shoe wins without tracking:
0.00 +40.00 -20.00 +27.50 +30.00 +5.00 +7.50 -2.50 -15.00 +30.00 0.00 +5.00 +5.00 -2.50 +2.50 -45.00 -25.00 -15.00 +7.50 -12.50 -7.50 +22.50 -7.50 +5.00 -7.50 +10.00 -55.00 +7.50 +7.50Where is the data for per shoe wins with tracking:
+47.50 -32.50 +5.00 -20.00 +75.00 -32.50 +52.50 +12.50 +7.50
+12.50 -2.50 -2.50 +10.00 -37.50 -50.00 -17.50 -5.00 +90.00
-7.50 -30.00 0.00 +7.50 +82.50 +100.00 +50.00 -20.00 +2.50
+10.00 -17.50 -20.00 +2.50 +160.00 -12.50
Let u1 be the actual average shoe win "without tracking"
Let u2 be the actual average shoe win "with tracking"
Let x1 be the mean of the observed shoe wins "without tracking"
Let x2 be the mean of the observed shoe wins "with tracking"
Let n1 be the number of observed shoes "without tracking"
Let n2 be the number of observed shoes "with tracking"
Let S1^2 be the sample variance of the observed shoe wins "without tracking"
Let S2^2 be the sample variance of the observed shoe wins "with tracking"
Then
H0: u1 = u2
H1: u2 > u1
u1 = ?
u2 = ?
x1 = -0.086207
x2 = +12.727
n1 = 29
n2 = 33
S1^2 = 412.277
S2^2 = 2058.85
We should reject the null hypothesis if |t0*| > t(alpha, v) and t0* negative
where in LISPish notation,
t0* is (/ (- x1 x2) (sqrt (+ (/ S1^2 n1) (/ S2^2 n2)))
and v is (- (/ (sqr (+ (/ S1^2 n1) (/ S2^2 n2)))
(+ (/ (sqr (/ S1^2 n1)) (+ n1 1))
(/ (sqr (/ S2^2 n2)) (+ n2 1))))
2)
(Hairy formulas courtesy of "Probability and Statistics in Engineering
and Management Science" by Hines and Montgomery)
Plugging in we get...
t0* = -1.46395 v = 46.41Looking in the t distribution table, t(.1, 40)=1.3 while t(.05,40)=1.7 so we can reject the null hypothesis, but with only 90% confidence. So again there is that ~10% that I cannot ignore.
The results are astounding and clear: my shuffle-tracking procedure is clearly better for sizing one's bet than using true count given the Random Pick Order Six Zone Shuffle. This contradicts Patterson's wisdom, expressed in "Break the Dealer", that one should use shuffle-tracking for cutting the cards, but not for sizing one's bets; I should perhaps state that I've never had much faith in Patterson's assertions, and what I know of his TARGET system doesn't impress me - sorry Jerry. My shuffle-tracking procedure allows one, given the cut card, to remove an average count of over +6 from the played cards (the first 5 1/3 decks). This in itself neutralizes the base casino advantage with Atlantic City rules, even for basic strategy players unknowingly at the same table as the tracker. However, in addition, the shuffle-tracking estimates of advantage are more accurate than the true count.
I used shuffle-tracking units of 2/3's decks, and my shuffle-tracking procedure calls for just 6 additions of 2 numbers during the shuffle. This is an approximation for simplicity, but it is fairly accurate. If I had the computer use the precisely correct shuffle-tracking procedure and had the simulated dealer perform perfect pick sizes, then the computer would predict exactly the count of the next region; however, this would not have been very informative. Instead I had the computer simulate my human-manageable shuffle-tracking procedure and a nonperfect casino shuffle. I tried to make the shuffle as realistic as possible; it is for the Claridge's eight deck "zone" shuffle. All the simulation's picks were subject to a +-5% error, and the simulation's riffle shuffles were imperfect. This simulation does not actually play blackjack; it just flips over the cards and places them in the discard tray one at a time, but this allows us to see how accurate tracking is in predicting when the high cards are going to come out.
I ran the simulation for 100,000 shoes (starting with a totally pseudorandomly shuffled shoe), so the results are very accurate (to maybe +-0.02.) The "regions" mentioned in the data summary are two tracking units, or 4/3's decks. Penetration was set at 2/3's, so only 4 of the 6 regions are dealt out, which is lousy but realistic. "ACTUAL_COUNT" is the count for the cards in that region of the shoe. "-TRUE_COUNT" is the negative of the running count divided by the remaining regions, which is the true count's prediction of the next cards about to come out; obviously the count predicts "0" for region 1 before any cards have been dealt. "SHUFFLE_TRACK" is the shuffle tracking prediction, obtained by those 6 additions performed over the actual counts for the tracking units from the previously observed shoe. "Count cut out" is the count of the unplayed cards (the ones in regions 5 and 6).
Here are the first five of 100,000 shoes run with random cutting:
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -3 0 5
2 -1 1 0
3 -5 1 -13
4 11 3 -6
Count cut out = -2
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 1 0 14
2 2 0 2
3 -11 -1 -9
4 5 3 5
Count cut out = 3
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -7 0 1
2 6 1 0
3 14 0 7
4 -5 -4 -8
Count cut out = -8
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -17 0 -11
2 8 3 6
3 1 2 -4
4 10 3 12
Count cut out = -2
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -6 0 1
2 -10 1 -2
3 -1 4 -5
4 8 6 0
Count cut out = 9
If you compare "ACTUAL_COUNT" to each of "-TRUE_COUNT" and
"SHUFFLE_TRACK", then I think you can see that "SHUFFLE_TRACK" is
better correlated to "ACTUAL_COUNT" than "-TRUE_COUNT" is, though it
is still only a rough estimate. Since it is cutting randomly, it
cuts out an average count of 0 in the limit.
Here are the first 5 of 100,000 shoes run with intelligent cutting:
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 7 0 0
2 -3 -1 -13
3 5 -1 -6
4 9 -3 4
Count cut out = -18
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -21 0 -8
2 1 4 0
3 -6 5 -10
4 -7 9 -3
Count cut out = 33
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -14 0 -6
2 6 3 7
3 -5 2 2
4 -9 4 -9
Count cut out = 22
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -5 0 -5
2 -6 1 -10
3 2 3 10
4 4 3 1
Count cut out = 5
PREDICTIONS
REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK
1 -4 0 1
2 0 1 2
3 7 1 2
4 -9 -1 -5
Count cut out = 6
Note that it usually cuts out a positive count with the intelligent
cutting, though it messed up big time on the first shoe (I would
guess because of dealer randomness.)
If you don't believe these few trials, then here is a summary for 100,000 trials, with random cutting:
Number of decks: 8 Number of tracking units: 12 Number of cards per track unit: 34.666666 Number of track units dealt: 8 Number of track units per statistics region: 2 Number of trials (i.e., shoes examined): 100000 Counting system (A, 2, 3, 4, 5, 6, 7, 8, 9, 10): -1 1 1 1 1 1 0 0 0 -1 Conservative factor for shuffle tracking: 1 Type of cutting (random or intelligent): Random +-% error in player cutting cards: 0 +-% error in dealer pick sizes: 5 % chance dealer drops 1 card in riff: 66 % chance dealer drops 2 cards in riff: 26 % chance dealer drops 3 cards in riff: 5 % chance dealer drops 4 cards in riff: 2 % chance dealer drops 5 cards in riff: 1
Average count cut out: -0.010
IDENTIFYING FAVORABLE REGIONS
-------------------------------
CORRELATION ABSOLUTE ERROR % FALSE POS. % FALSE NEG.
--------------- -------------- -------------- --------------
REGION COUNT TRACK COUNT TRACK COUNT TRACK COUNT TRACK
------ ----- ----- ----- ----- ----- ----- ----- -----
1 N/A 0.59 5.36 4.41 N/A 33.97 41.27 25.75
2 0.19 0.59 5.21 4.36 46.53 34.01 39.16 25.52
3 0.31 0.59 5.04 4.36 42.54 33.93 35.48 25.66
4 0.45 0.60 4.79 4.39 37.55 33.55 31.95 25.77
5 (not dealt)
6 (not dealt)
OVERALL 0.29 0.59 5.10 4.38 41.10 33.87 37.46 25.67
Overall % error in identifying favorable/unfavorable regions...
COUNT: 38.10
TRACK: 28.82
Note: a region is `favorable' if and only if its count is <= -2 per region.
Okay, I've got lots of things to explain and interpret.
First, if you look up there, you'll see that the average count cut out was close to 0, which is what we'd expect for random cutting.
Next, look at the correlation columns. The count subcolumn is the correlation coefficient of the true count to the actual count for the region noted to the left. The track subcolumn is the same, but for shuffle-tracking. For the statistically underprivileged, the correlation coefficient is a measure of the predictivity of one statistic related to another. Correlations are always between -1.0 and +1.0. A correlation of +1.0 means that they relate to each other perfectly. A correlation of -1.0 means that they are exactly the opposite of each other. A correlation of 0 means that they are unrelated.
I can't compute the correlation for the true count for region 1, because true count is always 0 at the start of the shoe, and this causes the correlation equation to divide by zero. However, as one would expect, the correlation of true count to actual count increases as you move deeper into the shoe. In fact, for predicting the count of the last region (6 in this case), the true count would have a correlation of +1.0, because then you *know* the count of the remaining region. Unfortunately, casinos realize this and thus they don't deal anywhere near all the cards. In this experiment, the count correlation reaches a maximum of +0.45 during the fourth region. Compare this to tracking. With tracking, you have equal knowledge about every part of the shoe. Thus, the tracking correlations are all statistically indistinguishable from each other, at about +0.59. This is considerably better than the best count correlation, which was only obtained during the fourth region! Overall, counting scores a +.29, while tracking scores a +.59. Hence, the true count is only weakly correlated with the actual count, while the tracking estimate is fairly strongly correlated with the actual count. Now, look at the absolute error columns. These are the averages of the differences of the actual count with the true count and tracking predictions. A similar pattern emerges. True count is very inaccurate at the start of the shoe (being no help at all by guessing 0 all the time and winding up with an error of 5.36), while deeper in the shoe it gets better to reach a minimum error of 4.79, which is still worse than the average error of 4.38 for the shuffle-tracking estimates. Note that this error of 4.38 is not a huge improvement over the base absolute error of 5.36; the shuffle-tracking technique never gets extremely accurate in terms of predicting the absolute value of the actual count, as a result of the simplifications in the tracking technique and the randomness introduced into the shuffle.
You may be saying this is all well and good, but what about what counters really care about: predicting favorable situations in order to size one's bet. Examine the four columns on identifying favorable regions. I defined a region as "favorable" if and only if it contains a count of *less* than -2. For the statistically underprivileged, false positives are when the prediction was for a favorable region, but in actuality, the region was unfavorable; false negatives are when the prediction was for an unfavorable region, but in actuality, the region was favorable. Percent false positives is the percent of the time that when the prediction is positive it is wrong, and analogously for percent false negatives.
Since true count never predicts a favorable first region prior to seeing it, there are no false positives, but no correct positives either (so I cannot compute the percent false positives.) The percent false negatives for true count in the first region therefore happens to be the percent of actually favorable regions: about 41%. If you look back to those sample shoes, you'll see that quite a few shoes are "favorable" in terms of the actual count being -2 or lower. This may be surprising, especially when you think about all the literature that says that true count indicates an advantage less than 20% of the time on these eight deckers, but that's just it - much of the time you have an advantage, and much of the time that you have an advantage you didn't predict it! However, like the previous statistics, shuffle-tracking again does better at identifying favorable regions in any part of the shoe than the count for the last played part of the shoe. Overall, the true count is wrong about the favorability of regions 38% of the time, while tracking is wrong only 29% of the time. Just by guessing "unfavorable" all the time, you get get an error rate of 41%, so the 38% error of true count is pretty bad. By the way, either counting or tracking estimates can be made more conservative in order to reduce the percent false positives, but usually at the cost of increasing the percent false negatives and the overall percent error.
Here's the summary of 100,000 trials with intelligent cutting.
Number of decks: 8 Number of tracking units: 12 Number of cards per track unit: 34.666666 Number of track units dealt: 8 Number of track units per statistics region: 2 Number of trials (i.e., shoes examined): 100000 Counting system (A, 2, 3, 4, 5, 6, 7, 8, 9, 10): -1 1 1 1 1 1 0 0 0 -1 Conservative factor for shuffle tracking: 1 Type of cutting (random or intelligent): Intelligent +-% error in player cutting cards: 0 +-% error in dealer pick sizes: 5 % chance dealer drops 1 card in riff: 66 % chance dealer drops 2 cards in riff: 26 % chance dealer drops 3 cards in riff: 5 % chance dealer drops 4 cards in riff: 2 % chance dealer drops 5 cards in riff: 1
Average count cut out: 6.45
IDENTIFYING FAVORABLE REGIONS
-------------------------------
CORRELATION ABSOLUTE ERROR % FALSE POS. % FALSE NEG.
--------------- -------------- -------------- --------------
REGION COUNT TRACK COUNT TRACK COUNT TRACK COUNT TRACK
------ ----- ----- ----- ----- ----- ----- ----- -----
1 N/A 0.55 5.40 4.17 N/A 26.89 58.31 37.43
2 0.21 0.56 5.26 4.47 34.48 32.57 44.06 27.48
3 0.34 0.56 5.03 4.48 34.29 34.96 39.70 25.87
4 0.51 0.54 5.31 4.23 14.00 28.44 51.88 36.11
5 (not dealt)
6 (not dealt)
OVERALL 0.32 0.57 5.25 4.34 24.66 30.27 48.71 30.93
Overall % error in identifying favorable/unfavorable regions...
COUNT: 46.66
TRACK: 30.59
Note: a region is `favorable' if and only if its count is <= -2 per region.
Okay, as you can see up there, the average count cut out was +6.45. This in itself is reason enough to shuffle track. The other statistics for this run are somewhat bizarre, owing partially to the effect of this +6.45, and partially to the fact that the computer shuffle-tracker generally thinks it has cut out more positive cards than it actually has. From the percent false negatives for region 1, you can see that about 58% of the regions are now favorable given this intelligent cutting, up from 41% with random cutting. The result: any stupid gamblers at a shuffle tracker's table may think they are riding an incredible lucky streak, though their stupidity may make them still lose. The count percent false positives drops from 41% with random cutting to 25% with intelligent cutting, while the percent false negatives zooms up from 37% to 49%, pushing the overall error from 38% to 47%, which is *worse* than the 42% error you could get from just guessing "favorable" all the time. This is because the count is being much too conservative in the intelligent cutting case. Since +6.45 count is being cut out from the first 5 1/3 decks, +1.2 should be added to the true count for betting purposes given an intelligent cut! If the true count were adjusted in this manner, then its error in identifying the favorability of regions would drop back down. Interestingly, the overall percent error of shuffle-tracking stays at about 30% (though there is a statistically significant but pragmatically insubstantial increase in its error with the intelligent cutting.)
In summary, I assert that shuffle-tracking can kick butt over true count in terms of sizing one's bet (and deviating intelligently from basic strategy, as well). If you think of yourself as an expert blackjack counter on multi-deck games but you do not shuffle-track, then think again... you are not an expert at multi-deck blackjack unless you shuffle-track. The above statistics have hinted at the tremendous benefit of shuffle-tracking.
Empirical results from a full-blown blackjack simulation integrated with the shuffle-tracking and realistic shuffles produced what seemed to be close to a 1% boost in advantage over the non-shuffle-tracking case. This was explained in the section "Empirical Results in Support of Shuffle-Tracking".
In order to have 100% control of the cut card, you must take over a table with a shuffle-tracking team, but contrary to popular belief, a whole team is not necessary to perform the actual tracking operations, except for complicated shuffles. Also, while having control of the cut card is nice, one can still profit from shuffle-tracking without having the cut card.
I would like to leave you with a word of caution. It can be difficult to devise and implement a good shuffle-tracking scheme. Also, some casino shuffles are much harder to track than others. If you attack a tough shuffle or use a suboptimal shuffle-tracking scheme, you could easily get creamed. That's why these simulation results are so nice. I am now reasonably sure that I, armed with my shuffle-tracking scheme, can cream this casino, rather than the other way around (at least when the casino provides a fairly easily tracked shuffle.) We're not talking about some measly 1.5% advantage that is the common wisdom for the maximum advantage of counting; as Synder wrote in the April 1990 issue of Blackjack Forum, "It is worth noting here that a good shuffle-tracker could absolutely murder the grossly shuffled games." Oops, I was going to leave you on a cautionary note... how about "your mileage may vary"?!