“If you get 8 heads out of 10 flips, then on your next 10 flips, you are very likely to get less than 8 heads.” The more amazing your first 10 flips, the harder it will be for your second 10 flips to match that.

Exceptionally lucky or exceptionally unlucky performances are unlikely to repeat.

No. I’m quite familiar with the Gambler’s Fallacy. Note I said “usually”. If they suspect you of counting, they won’t do this; in fact probably the opposite. But if you’re not counting, the house has the edge, so the law of large numbers says that your overall winnings (or losings) will most likely approximate the expected mean (which is a loss for you), given enough trials. Particularly if you’re given both alcohol and other inducements to make you feel “good” and “lucky” (read: “foolish” and/or “less competent”).

Cheers,

Hyman Rosen: Pintler, you are incorrect. There is no regression to the mean when outcomes are based on independent trials, which is the case in a sequence of fair coin flips. Regardless of previous outcomes, the next coin toss still has exactly the same chance of landing heads as tails.

This is incorrect. While it’s true that the next coin toss still has exactly the same chance of landing heads as tails, they will not have an equal affect on the regression to the mean. If the past has been predominantly heads, one more head will not take us as much further from the mean as one more tail will take us closer to it.

For example, suppose we have flipped 100 coins, and by luck we’ve had 80 heads. There are several different ways to measure how exceptional this is, but by any measure, if we flip 100 more coins, and we have, say, 50 more heads, it will be a lot less exceptional. Even with independent trials, outcomes still regress towards the mean.

In other words, given two slot players, one having incredible good luck and one having incredible bad luck, it’s a safe bet that the one having incredibly good luck will see worse luck and the one having incredibly bad luck will have better luck. The more exceptional the luck so far, the more likely it is that it will seem less exceptional with more trials. (Think about it if this is not obvious to you.)

Or, as an example of reductio ad absurdum, with a single flip of a fair coin, whatever outcome you get is exceptional — all one and none the other. With 50 flips, all one and none the other is much less likely.

Not quite correct I think. The central limit theorem speaks to the approaching normality of the distribution of a sample mean as sample size increases. It is the much simpler and accessible law of large numbers that speaks to the smaller variance of a sample mean as sample size increases.

The big payoff for the card counter in facing fewer decks comes as a result of: (1) the greater variations in the “true count”, that is in the density of high and low cards remaining in the cards to be played; (2) the effect of this on the player’s expected probability of winning the hand; and (3) the player’s ability to ratchet up his bets.

For example, whether using 1 deck or 8 decks, if one could magically remove all the fives from the deck the player’s expected return would move from about -1/2% to +3%. The likelihood of all the fives coming out in the first hand of a deal with one deck is enormously greater than the likelihood that they will come out in the first eight hands dealt from an eight deck shoe. This is illustrative of the fact that the number of truly favorable betting opportunities becomes trivially few as the number of decks increase.

traveler496: zuch,
If you read Steve’s “first card” as “first card you receive in this hand,” his analysis is correct regardless where you are in the deck.Completing his analysis, the probability of being dealt a blackjack at an arbitrary point in a 1-deck game, averaged over a large number of random shuffles, is(4/52)*(16/51) + (16/52)*(4/51), or 32/663,where the first term covers A-first and the second term covers 10-first….

I don’t know the rules of black jack, so I was wondering if this proof is sufficient to show that multi-deck shoes are worse for players. I looked up and saw that when the player gets a blackjack, they are supposed to win 3:2 odds. I assume the player only loses their wager after 2 cards if the dealer gets a black jack and the player does not.

Could you tell me what happens if both the dealer and the player draw a blackjack? Does the player still get the payout? Or does the player just keep his money but lose nothing? Or does the player still lose his money.

I know that if neither gets the blackjack, the round continues. I just want to be clear on the math related to the winning status after only 2 cards are dealt. However, I can’t unless I know the rules!

normal human being: Whats the big deal, they have the right to add more decks, shuffle up each hand, change the limits at the table, why do they need the right to ban a card counter when they can effectively (just with the shuffle up right) ruin their odds? I always thought this was stupid anyway. The casino should advertise when card counters show up. Even if they lost $1 mil to a card counter, the PR from it would make all the 99% of the people who think they can count, but can’t, go to the casino and lose tons.

Exactly right. Indeed that reflects the history. When card counting began in the 1950s the casinos were initially welcoming. Thinking that this was another stupid losing strategy. Then they became frightened that this would bankrupt them in the same way a single cheating dealer could. Eventually most learned that there were very few really disciplined and skilled counters with large enough bankrolls and even they had only at best a 2% advantage. Even the best card counters will have horrendous runs and hemorrhage tens of thousands of dollars in a day. On the other hand the average player operates at a 6% disadvantage. And the numbers of those players have multiplied many times over by knowledge that blackjack is a beatable game.

byomtov: David Schwartz,In one sense you are right, but only because you are defining “rules” to make it so.To my mind, the rules describe the mechanics of the game: the bets the players may make: how the cards are dealt, how bets are settled, how various possible irregularities are dealt with, etc. They say nothing about what decision rules players may use to determine their bets in the context of those mechanics (which may include table minimums and limits). If you prohibit doubling down you are changing the rules. You are eliminating a betting option. If you prohibit counting you are barring a player from using a particular decision procedure which has nothing to do with the mechanics.

I disagree. Prohibiting card counting changes the bets a player may make by prohibiting particular betting and playing patterns. Casinos don’t read people’s minds to determine that they’re counting cards, they infer it from the bets they make.

In fact, the reason rules say nothing about what decision rules players may use is because they don’t restrict them. A rule need not specifically state that a player may choose to double down using any set of rules he or she pleases, all it need do is not prohibit particular rules or strategies.

The casinos, by prohibiting a particular playing strategy, are effectively adding a rule that prohibits that strategy.

You are absolutely correct that the odds on the next flip of the coin aren’t affected by a prior run, but if the previous ten flips have by chance had many heads, say, the odds are that the next ten will have fewer heads. The term is ‘regression to the mean’.

This can lead to interesting results – some recent book (‘Predictably Irrational’??) discusses a study of pilot instructors. When a student makes a worse than average landing, the instructor can either respond positively (helpful hints, encouragement, etc) or negatively (derision, threats of washing out, etc). Some instructors adopted one strategy, some the other. Both feelings were reinforced because if the student had just made a much worse than normal landing, the next landing was likely to be better than the previous, regardless of what the instructor did.

Second, computers and the Internet have really changed the counting world. Today people can practice their card playing, both blackjack and video poker. And deals get burned out fast. In 1993 Stanford Wong sent his premium customers a notice that the Grand Casino in Biloxi had made a mistake on their Sic Bo felts and was paying 80-1 on a roll of 4 or 17. The actual odds are 72-1. This deal lasted almost a month after this mailed notice. Today, with the Internet, this deal wouldn’t last a week.

Third, I think there are some state and federal statutes about discriminating against passengers on riverboats. Not just racial, I’m talking about statutes going back 100+ years. This ought to include gamblers on boats as well.

Finally, it is not really hard to count a big shoe. The problem is that a single-deck will swing somewhat positive or negative after the removal of a single card. For a six-deck shoe generally two decks have to be dealt out before you will have a single advantageous play (if any).

In one sense you are right, but only because you are defining “rules” to make it so.

To my mind, the rules describe the mechanics of the game: the bets the players may make: how the cards are dealt, how bets are settled, how various possible irregularities are dealt with, etc. They say nothing about what decision rules players may use to determine their bets in the context of those mechanics (which may include table minimums and limits).

If you prohibit doubling down you are changing the rules. You are eliminating a betting option. If you prohibit counting you are barring a player from using a particular decision procedure which has nothing to do with the mechanics.

Another argument is that, whatever a rule is, it should apply equally to all players. If you don’t allow doubling down no one can double down, period. But the casinos don’t want to bar counters. They only want to bar good counters. That’s like saying Babe Ruth’s HR’s don’t count, but Willie Keeler can swing for the fences all he wants.

Anon21:
From my vague knowledge of blackjack, this strategy probably doesn’t work for experienced and skillful players of that particular game; as a commenter upthread noted, the “basic strategy” is a learnable collection of actions which will always minimize the house’s advantage and keep the player consistently winning. (Obviously, results from any one night may be bad, but over time the player will turn a nice profit.)

In most cases this is not true. There is a “perfect storm” of game variations that can lead to a player’s edge, but at your average table playing perfect basic strategy merely narrows the house edge to roughly 0.5%. There’s a reason most places will let you openly use a basic strategy reference right there at the table.

How is saying “we will eject you if you use strategy X” not the same as saying “our rules prohibit strategy X”?

By ejecting anyone who plays using a certain strategy, the casinos are in effect imposing a new set of rules about what strategic options are available to the players.

No. The rules are the same. Counting doesn’t create any new bets unavailable to non-counters.

Babe Ruth didn’t change the rules of baseball by hitting lots of home runs. He just was very good at exploiting a tactic which was always available.

Waste93: Card counting is routinely used in Poker in a fashion.

Yes, but card counting in poker does not decrease the amount of rake the house takes on a hand. In fact, if a good card counter is up against a bad card counter (or a bad player in general), the good card counter can sucker a lot of money out of the bad player, which increases the rake the house gets.

@Neurodoc – there is no fallacy there. The odds clearly show that the house will “probably” win on the next hand regardless of how many times in a row the player wins. It’s not a coin flip between the house and the player.

zuch: Anderson:Usually, they try to make winners welcome, for more than one reason. They give them comps, drinks, free hotel rooms, etc.. This is not only because it’s good customer relations, but also because they know that if the person continues to gamble, they’ll probably lose their winnings back again, and the casinos will recoup the losses.Cheers,

There may be a bit of a fallacy implicit here, the same one that imagines that if a coin lands tails a number of times in a row, it is more likely that it will come up heads the next time than when a coin hasn’t land too many times heads or tails in a row. Let someone give you a coin from their own pocket, so they won’t question whether it is a “fair” coin, then invite them to bet on the next flip. Most people will be more eager to bet or bet more that they can call the next one after a run of heads or tails, demonstrating their mistaken notion that their chances have improved to greater than 50-50. Casinos want to encourage as much betting on their games as they can, since their take is a function of how much is wagered. It matters not whether the player has been on a “winning” or a “losing” streak, so long as they continue to gamble. It’s that “law of large numbers” thing taking all comers together, not looking at them individually.

“Smart” gamblers may favor those games that give the house less of an edge or even let the gambler get the edge from time to time (card counting in blackjack) rather than games like roulette or pumping the slots. But so long as the gambler isn’t going to have an edge by some means, casinos will be happy to have “smart” gamblers as well as “stupid” ones, their degree of happiness depending only on how much the gambler wages. When people are winning, they are more inclined to keep on playing than when they are losing, and that’s why the casinos want to give them an extra nudge to stay longer, wager more, not because the casino will win it back from those individuals who are ahead. The casino would be no better or worse off, and hence no happier or less happy, to have losers go on playing as opposed to winners, so long as they gambled as much, and more is always better from the casino’s perspective.

Also, when others see those who appear to be winning in their midst, they are encouraged to gamble more, so another reason for casino’s to want winners on display, the flashier the better.

Yes and no. Basic strategy reduces the house’s advantage to less than one-half of 1% under most sets of rules, but does not eliminate it or give the player a slight advantage like skillful card counting can do. You are guaranteed to “win” only if you play perfect basic strategy for small stakes while collecting free drinks.

Nor am I sure that this is indeed the best interpretation of the statutory text; I haven’t looked closely enough at the statutes to form a reliable opinion, but the statutes that the court cites don’t strike me as persuasive on this — that the statutes control the “rules of the game” doesn’t mean, I think, that they control the casino’s choices about whom to allow to play the game.

I don’t know anything about the statutes either, but the general line of reasoning makes perfect sense to me. By ejecting anyone who plays using a certain strategy, the casinos are in effect imposing a new set of rules about what strategic options are available to the players.

There is a perfect strategy for any given set of rules/displayed cards.

Which is why blackjack is so hideously boring compared to poker.