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Systems 4, 6 and 8, which are also normalized, have the first new idea. They assign a point count of zero to the ace for strategy purposes. This is consistent with the evidence: in most instances that have been examined, the optimal strategy seems to be relatively unaffected by changes in the fraction of aces in the pack. However, the player's expectation is generally affected by aces more than by any other card (Table 22). Therefore these systems keep a separate ace count. Then the deviation of the fraction of aces from the normal 1/13 is incorporated for calculating the player's expectation for betting purposes*
The (c) column in Table 21 still remains to be explained. It is a numerical assessment of a particular system's closeness to an ideal system based on the change in expectation values contained in Table 22. The calculation of the (c) value eliminates the necessity of simulating a large number of hands (say a million) to evaluate a strategy. The computation of these numbers requires some advanced mathematical background, so its explanation is left to the appendix.
Cheating: Dealing Seconds
Various card counting systems give the blackjack player an advantage, provided that the cards are well shuffled and that the game is honest. But many methods may be used to cheat the player. I have been victimized by most of the more common techniques and have catalogued them in Beat the Dealer.
One of the simplest and most effective ways for a dealer to cheat is to peek at the top card and then deal either that card or the one under it, called the second. A good peek can be invisible to the player. A good second deal, though visible to the player, can be done so quickly and smoothly that the eye generally will not detect it. Although the deal of the second card may sound different from the deal of the first one, the background noise of the casinos usually covers this completely. Peeking and second dealing leave no evidence. Because these methods are widespread, it is worth knowing how powerful they are.
Does even a top professional blackjack counter have a chance
against a dealer who peeks and deals seconds? Consider first the simple case of one player versus a dealer with one deck. This is an extreme example, but it will illustrate the important ideas.
I shuffle the deck and hold it face up in order to deal practice hands. Because I can see the top card at all times, dealing from a faceup deck is equivalent to peeking on each and every card. I will deal either the first or second card, depending on which gives the dealer the greatest chance to win. I will think out loud as an imaginary dealer might, and the principles I use will be listed as they occur. The results for a pass through one deck are listed in Table 23 (pp. 2021). There were nine hands and the dealer won them all.
On hands one, two, four, six, eight and nine, the dealer wins by busting the player. Because there is only one player, it does not matter what cards the dealer draws after the player busts.
When there are two or more players, the dealer may choose a different strategy. If, for example, die dealer wishes to beat all the players but doesn't want to peek very often, an efficient approach is simply to peek when he can on each round of cards until he finds a good card for himself on top. He then retains this card by dealing seconds until he comes to his own hand, at which time he deals the top card to himself. That strategy would lead to the dealer having unusually good hands at the expense of the collective player hands; because some good hands have been shifted from the players, the player hands would be somewhat poorer than average.
A player could detect such cheating by tallying the number of good cards (such as aces and 10s) which are dealt to the dealer as his first two cards and comparing that total with the number of aces and 10s predicted by theory. In Peter Griffin's book, The Theory of Blackjack, he describes how he became suspicious after losing against consistently good dealer hands. Griffin writes that he .. embarked on a lengthy observation of the frequency of dealer up cards in the casinos I had suffered most in. The result of my sample, that the dealers had 770 aces or 10s out of 1,820 hands played, was a statistically significant indication of some sort of legerdemain." Griffin's tally is overwhelming evidence that something was peculiar. The odds against such an excess of tenvalue cards and aces going to the dealer in a sample this size are about four in ten thousand.
Another approach the dealer might select is to beat one player at the table while giving everybody else normal cards. To do this, the dealer peeks frequently enough to give himself the option of dealing a first or second to the unfortunate player each time that player's turn to draw a card comes up. Dealing stiffs to a player so that he is likely to bust is, as we see from the chart in Table 23, so easy to do that the player has little chance.
IJF all dealers peeked and dealt seconds according to the cheating strategy indicated in Table 23,1 estimate that with one player versus the dealer, the dealer would generally win at least 95 percent of the time. With one dealer against several players, the dealer would win approximately 90 percent of the time. Anyone who is interested can get a good indication of what the actual numbers are by dealing a large number of hands and recording the results.
The deadliest way a dealer can cheat is to win just a few extra hands an hour from the players. This approach is effective because it is not extreme enough to attract attention, or to be statistically significant and therefore detectable over a normal playing time of a few hours. For example, the odds in blackjack are fairly close to even for either the dealer or the player to win a typical hand. Suppose that by cheating the dealer shifts the advantage not to 100 percent but to just 50 percent in favor of the house. What effect does this have on the game?
If we assume that the player plays 100 hands, a typical total for an hour's playing time, and we also assume that the player bets an average of two units per hand, then being cheated once per 100 hands reduces the player's win by one unit on the average. A professional player varying his bet from one to five units would probably win between five and 15 units per hour. The actual rate would depend upon casino rules, the player's level of skill, and the power and variety of winning methods that he employed. Let's take a typical professional playing under good conditions and
Table 23

Top 
Card 

Plr. 
Plr. 
Dir. 
Dir. 

Hand 
Card 
Dealt 
Comment 
Gets 
Total 
Gets 
Total 
Result 
1 
2 
First 

2 
2 




5 
Second 4) 
Dir. trys for good card 


4 
4 


5 
Second 110) 
Dir. will have 9: Plr. gets stiff 
10 
12 




5 
First 


5 
9 


4 
First 
Worsens Plr. stiff 
4 
16 




5 
Second (81 
Prevent Plr. 21 
8 
24 bust 


Dir. wins 

5 
First 
Doesn't matter 


5 
14 


9 
First 
Doesn't matter 


9 
23 bust 

2 
4 
First 

4 
4 




K 
First 



10 
10 


Q 
First 
Give Plr. stiff 
10 
14 




J 
Second (J) 
J will bust Plr. ISecond turns out to be J, too!) 
J 
24 bust 


Dir. wins 

J 
First 
Doesn't matter 


J 
20 

3 
A 
Second (3) 

T " 





A 
First 



A 
1. 11 


3 
First 
Bldg. potential stiff 
3 
6 




J 
First 



J 
BJ 
Dir. wins 
4  
■ z 
First 

2 
2 




9 
First 
Would make Plr. 11. so Dir. takes 


9 
9 


K 
First 
Give Plr. stiff 
K 
12 




Q 
Second (3) 
Guarantees Dir. win 


3 
12 


Q 
First 

Q 
22 bust 


Dir. wins 

8 
First 



8 
20 

5 
5 
First 

5 
5 




J 
First 



J 
10 


A 
Second (Al 
Guarantees Dir. win (Second was A, too!) 
A 
6,16 




A 
First 



A 
BJ 
Dir. wins 
Table 23 (continued)
6 
8' ' 
hirst 

8 
8 




Q 
First 



Q 
10 


6 
First 
Give Plr. stiff 
6 
14 




2 
Second (31 



3 
13 


2 
First 
Give Plr. worse stiff 
2 
16 




6 
First 

6 
22 bust 


Dir. wins 

4 
First 



4 
17 

7 
7 
First 

7 
7 




6 
Second (A) 



A 
1. U 


6 
First 

6 
13 




6 
Second <KI 



K 
BJ 
Dir. wins 
8 
'6 ~ 
First 

6 
6 




10 
First 



10 
10 


K 
First 
Give Plr. bad stiff 
K 
16 




9 
Second (21 
Dir. must win 


2 
12 


9 
First 

9 
25 bust 


Dir. wins 

10 
First 
Doesn't matter 
10 

10 
22 bust 

—r 
■' 8 
First 

8 " 
8 




Q 
First 



Q 
10 


7 
First 

7 
15 




10 
Second 15) 
Dir. must win 


5 
15 


10 
First 

10 
25 bust 


Dir. wins 

7 
First 



7 
22 

assume that his win rate is ten units per hour and his average bet size is two units. Given those assumptions, being cheated ten times per hour or onetenth of the time would cancel his advantage. Being cheated more than ten percent of the time would probably turn him into a loser.
Cheating in the real world is probably more effective than in the hypothetical example just cited, because the calculations for that example assume cheating is equally likely for small bets and big bets. In my experience, the bettor is much more likely to be cheated on large bets than on small ones. Therefore, the dealer who cheats with maximum efficiency will wait until a player makes his top bet. Suppose that bet totals five units. If the cheat shifts the odds to 50 percent in favor of the house, the expected loss is 21/2 units, and j ust four cheating efforts per 100 hands will cancel a professional player's advantage. A cheating rate of five or ten hands per 100 will put this player at a severe disadvantage.
We can see from this that a comparatively small amount of cheating applied to the larger hands can have a significant impact on the game's outcome. This gives you an idea of what to look for when you are in the casinos and think that something may be amiss.
Missing Cards: The Short Shoe
I have heard complaints that cards have been missing from the pack in some casino blackjack games. We'll discuss how you might spot this cheating method.
In 1962,1 wrote on page 51 of Beat the Dealer, "Counting the... cards... is an invaluable asset in the detection of cheating because a common device is to remove one or more cards from the deck." Lance Humble discuss« cheating methods for fourdeck games dealt from a shoe in his International Blackjack Club newsletter. He says, "The house can take certain cards such as tens and aces out of the shoe. This is usually done after several rounds have been dealt and after the decks have been shuffled several times. It is done by palming the cards while they are being shuffled and by hiding them on the dealer's person. The dealer then disposes of the cards when he goes on his break." But cheating this way is not limited to the casino. Players have been known to remove "small" cards from the pack to tilt the edge their way. The casino can spot this simply by taking the pack and counting it; the player usually has to use statistical methods.
In the cheating trade, the method is known as the short shoe. Let's say the dealer is dealing from a shoe containing four decks of 52 cards each. In 52 cards, there should be 16 tenvalue cards: the tens, jacks, queens and kings. Logically, in four decks of 208 cards, there should be 64 tenvalue cards. I'll call all of these "tens" from now on. Casinos rarely remove the aces—even novice players sometimes count these.
Suppose the shift boss or pit boss takes out a total often tens; some of each kind, of course, not all kings or queens. The shoe is shortened from 64 tens to 54 tens, and the four decks from 208 cards to 198 cards.
The loss of these ten tens shifts the advantage from the player to the dealer or house. The ratio of others/tens changes from the normal 144/64 = 2.25 to 144/54 =2.67, and this gains a little over one percent for the house. How can you discover the lack of tens without the dealer knowing it?
Here is one method that is used. If you're playing at the blackjack table, sit in the last chair on the dealer's right. Betasmall fixed amount throughout a whole pack of four decks. After the dealer puts the cut card back only, let's say, ten percent of the way into the four shuffled decks and returns the decks into the shoe, then ready yourself to count the cards. Play your hand mechanically, only pretending interest in your good or bad fortunes. What you're interested in finding out is the number of tens in the whole fourdeck shoe.
Let's say the shift boss has removed ten tens. (Reports are that they seem to love removing exactly ten from a fourdeck shoe.) When the white cut card shows at the face of the shoe, let's say that the running count of tens has reached 52. That means mathematically that if all 64 tenvalue cards were in the shoe, then, of the remaining 15 cards behind the cut card, as many as 12 of them would be tens, which mathematically is very unlikely. This is how one detects the missing ten tens because the dealer never shows their faces but just places them face down on top of the stack of discarded cards to his right, which he then proceeds to shuffle face down in the usual manner preparatory to another fourdeck shoe session.
Although at first the running count is not easy to keep in a real casino situation, a secondary difficulty is estimating the approximate number of cards left behind the cut card after all the shoe has been dealt. To practice this, take any deck of 52 cards and cut off what you think are ten, 15 or 20 cards, commit yourself to some definite number, and then count the cards to confirm the closeness of your estimate. After a while, you can look at a bunch of cards cut off and come quite close to their actual number.
In summary, count the number of tens seen from the beginning of a freshly shuffled and allegedly complete shoe. When the last card is seen and it is time to reshuffle the shoe, subtract the number of tens seen from the number that are supposed to be in the shoe—64 for a fourdeck shoe—to get the number of unseen tenvalue cards which should remain. If 54 tenvalue cards were seen, there should be ten tens among the unused cards. Then estimate the number of unseen cards. You have to be sure to add to the estimated residual stack any cards which you did not see during the course of play, such as burned cards. Step four is to ask whether the number of unseen tenvalue cards is remarkably large for the number of residual cards. If so, consider seriously the possibility that the shoe may be short. For instance, suppose there are 15 unseen cards, ten of which are supposed to be tenvalues. A computation shows that the probability that the last 15 cards of a wellshuffled fourdeck shoe will have at least ten tenvalue cards is 0.003247 or about one chance in 308.
Thus the evidence against the casino on the basis of this one shoe alone is not overwhelming. But if we were to count down the same shoe several times and each time were to find the remaining cards suspiciously tenrich, then the evidence would become very strong. Suppose that we counted down the shoe four times and that each time there were exactly 15 unseen cards. Suppose that the number of unseen tens, assuming a full four decks, was nine, 11, ten, and 13 respectively. Then referring to Table 24, the probabilities to six decimal places are H(9) = .014651 to have nine or more unseen tens, and for at least 11, ten, and 13 respectively, the chances are H(U) = .000539, H(10) = .003247, and H(13) = .000005. These correspond to odds of about 1/68,1/1,855,1/308 and 1/200,000 respectively. The odds against all these events happening together is much greater still. In this example, the evidence strongly suggests that up to nine tenvalue cards are missing. There can't be more than nine missing, of course, because we saw all but nine on one countdown.
If the casino shuffles after only 104 cards are seen, it is not so easy to tell if ten tenvalue cards were removed. A mathematical proof of this is contained in the appendix. *
This discussion should make it clear that the method suggested is generally not able to easily spot the removal of tenvalue cards unless the shoe is counted several times or is dealt down close to the end.
One of the interesting ironies of the short shoe method of cheating players is that neither the shift boss nor the pitboss—the latter bringing the decks of cards to the dealer's table—need tell the dealer that his shoe is short. Thus, the dealer doesn't necessarily have to know that he's cheating. After all, he's just dealing. It's an open question how many dealers know that they're dealing from a short shoe.
Reports are that the short shoe is a frequent method that casinos use in cheating at blackjack using more than one deck. The tables with higher minimums (say $25) are more tempting candidates for short shoes than those with the lower minimums.
An experienced card counter can improve the method by counting both tens and nontens. Then he'll know exactiy how many unseen cards there are, as well as unseen tens. Table 24 can then be used with greater confidence.
In practice, you don't need to count through a shoe while bet
Table 24
K Number Of Unseen TenValue Cards 
P(K) Probability Of Exsctlv K Unseen Ten« 
K(K) Probability At least This Manv Unseen Tens 
0 
.003171 
1.000000 
1 
.023413 
.996829 
2 
•07S8IB 
.973416 
i 
.160423 
.994598 
4 
.220732 
.734176 
$ 
.217437 
.513443 
S 
. 15S3SO 
.296006 
7 
.036431 
.137626 
s 
.036132 
.050712 
9 
.011404 
.014651 
10 
.002707 
.003247 
11 
.000475 
.000539 
12 
.000053 
.000065 
13 
.000005 
.000005 
14 
.000000 
.000000 
IS 
.000000 
.000000 
ting (and thus losing money in the process) to find out that the casino is cheating. If you suspect foul play, count while standing behind the player to the dealer's right.
You might easily catch a short shoe by simply counting all the cards that are used, whether or not you see what they are. Then if the remaining cards, at the reshuffle, are few enough so you can accurately estimate their number, you can check the total count. For instance, you count 165 cards used and you estimate that 31 ± 3 cards remain. Then there were 196 ± 3 cards rather than the 208 expected, so the shoe is short.
A casino countermeasure is to put back a 4,5 or 6 for each ace or tenvalue card removed. Then the total number of cards remains 208, and the casino gets an even greater advantage than it would from a short shoe.
Cards do get added to the deck, and there's a spooky coincidence to illustrate this. On page 51 of Beat the Dealer, I wrote in 1962,' 'One might wonder at this point whether casinos have also tried adding cards to the deck. I have only seen it done once. It is very risky. Imagine the shock and fury of a player who picks up his hand and sees that not only are both his cards 5s, but they are also both spades." And then 15 years later in 1977, a player in a onedeck game did get a hand with two of the same card—the 5 of spades. Walter Tyminski's casino gaming newsletter, Rogue et NoirNews, reported on page 3 of the June 15,1977 issue, "What would you do if the player at your right in a single blackjack game had two 5 of spades? Nicholas Zaika, a bail bondsman from Detroit, had that experience at the Sahara in Las Vegas on May 24 at a $5 minimum table.
"Zaika wasn't in the best of humor because he had reportedly lost $594,000 at other Sahara tables, by far the largest loss he has ever experienced. Zaika had the blackjack supervisor check the cards and there were 53 cards in the deck, the duplicate bein the 5 of spades.. .The gamer has engaged the services of Las Vegas attorney George Grazadei to pursue claims he feels that he has against the casino...
"The Sahara denies any wrongdoing and says that it is cooperating fully with the investigation... Players aren 't likely to introduce an extra 5 because the presence of the extra 5 favors the house and not the player."
Suppose instead of just counting tens used and total cards used, you kept track of how many aces, 2s, 3s, queens, kings, and so on were used. This extra information should give the player a better chance of detecting the short shoe. The ultimate proof would be to count the number of each of the 52 types of cards which have been used. Mathematical readers might wish to investigate effective statistical or other ways of using information for detecting shoes in which the numbers of some of the cards have been changed.

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