Muddled thinking about probability

At the Hilton today, the guy on my right lost an all-in contest with pocket kings versus pocket aces. Can't blame him--it happens, and it's very difficult to get away unscathed if you're holding the cowboys. But after it was over, he rationalized his call of the all-in bet by saying, "I didn't think he had aces, because this guy [pointing to another player] just had aces on the previous hand."

Wow.

It's hard to know how to respond to such idiotic thinking. It's the classic gambler's fallacy--thinking that previous independent events influence the outcome of the current situation.

It's true that if at any particular time we ask, "How likely is it that pocket aces will be dealt on both of the next two hands at this table?" the answer is that it's pretty small. In fact, it's easy to quantify it. Each person has a 1/221 chance of being dealt aces on any hand.* There are ten players at the table, so on each deal there is a 10/221 chance that somebody will get pocket aces (about 4.5%). For that to happen twice in a row, the probability is 10/221 times 10/221, or about 0.2%.

But once somebody is dealt aces on the first hand that we're observing, the probability that a player will get aces on the next hand isn't 0.2%; it is, again, exactly what it was before, 10/221. Neither the cards nor the shuffler has any memory about what just happened. Even if aces are shown for ten hands in a row, the probability of somebody getting them on the next hand goes right back to precisely what it was (assuming the auto-shuffler is working properly, the dealer isn't cheating, etc.).

It's amazing how people cling to this fallacy, when just the tiniest amount of thought shows how wrong it is. Casinos often set up displays showing recent outcomes of the roulette wheel, and people idiotically consult it before making their bets. Some of them see, e.g., a run of odd numbers recently hitting, and decide that there's a trend underway, so they bet on "odd." Others look at the same accounting, decide that since so many odd numbers have hit it must be time for things to head back toward parity, and therefore bet on "even." Both conclusions are equally looney.** (For empirical evidence that gamblers really do act according to these irrational beliefs, see Sundali and Croson, "Biases in casino gambling: The hot hand and the gambler's fallacy," Judgment and Decision Making, vol. 1, #1, (July, 2006), pp. 1-12, available online at http://journal.sjdm.org/06001/jdm06001.htm.)

There's nothing particular shameful in getting felted*** because you had the second best possible hand up against the best possible hand. It happens to everybody from time to time. But if any part of your decision to commit your chips in that situation was because your judgment of the probability of the opponent having aces was skewed by a previous independent event, well, then you're an imbecile. It's just as ludicrous as saying you thought you'd win the showdown because your horoscope said that today was your lucky day, or because you're wearing your lucky socks.


*In case it isn't clear how this number is derived, here it is: To get pocket aces, first you have to be dealt one ace (duh!), then a second one (double duh!). The probability of the first one is 4/52, because there are 4 aces out of 52 cards in the deck. Now the probability of the second card being an ace is 3/51, because there are 3 aces and 51 cards left unaccounted for. Multiply 4/52 by 3/51, and you get 12/2652, which reduces to exactly 1/221.

**I'm aware that there's a kind of exotic exception to this. Roulette wheels, being mechanical devices, aren't quite perfect, and with tens of thousands of observations, one can, in theory, discover small biases in how the wheel is spinning, and thus gain a statistical edge over the house. Some people have actually managed to win some decent money this way, particularly in Europe, where, I understand, roulette wheels give a smaller built-in house profit margin because they have only a "0" spot, and not the added "00" spot that American roulette games usually have. There are two problems with trying to make money this way, however. The first is that it takes an incredible amount of work to gain what is usually a relatively small statistical advantage. The second is that casinos have ways of foiling your plans, such as switching the top ends of the roulette wheels, or moving the tables around, or doing maintenance on the mechanisms, all of which will tend to occur when you're not there to see it, thus making your hours of tedious observations worthless. Even moving the table to, e.g., vacuum under it may cause it to be put back down in a slightly different place on a slightly uneven floor, thus shifting the wheel's bias by a few degrees.

***Some may not have heard this term. It's another inventive Phil Laak-ism, meaning to lose all your chips: you're down to the felt on the table.

Addendum, September 13, 2007

In this week's installment of the 2007 World Series of Poker main event on ESPN, professional poker player Hevad Khan moves all in with Q-Q against an opponent's A-A. This opponent had apparently held K-K just a few hands earlier, though this hand wasn't shown on the broadcast. Khan gets unbelievably lucky and flops the other two queens. When the hand is over, he chats with the guy who had the aces and says, "Dude, you understand it's so hard to put you on aces and kings in one orbit." So, apparently Khan felt that his queens must be good because he couldn't believe that this opponent, having recently had kings, could have either aces or kings again so soon. This tells us that even some professional players can't get their minds around the fact that each hand is completely independent of previous ones.

Blue chip beggars

I wrote recently (http://pokergrump.blogspot.com/2007/07/dealers-who-are-stingy-with-their-1.html) about the "Blue chip Nazis" (a term suggested by one of the commenters)--dealers who are stingy about making change for red ($5) chips. Well, there's a parallel phenomenon: Players who are blue chip beggars.

These are the guys who, like the grasshopper in Aesop's fable (see http://www.bartleby.com/17/1/36.html), do not lay in a store of blue chips in anticipation of needing them to pay blinds, tip cocktail waitresses, etc. Then suddenly they discover that they need a blue chip or two, and they have none. Shame, shame!

I really don't care much about most in this group, because they just go to the dealer and get what they need. It's one sub-group that annoys me: the ones who instead turn to their fellow players.

Here I am with my carefully protected stack of 10 or so blue chips, and these idiots ask me for change for a $5 chip. I don't want to be the bad guy, so I do it, but then that means that I have to throw in a red chip for my blinds for a round or two, to get my stock of blues back up to where I want it. Hey, pal, how did your lack of planning become my problem to solve, eh? Get your own change!

I remember in junior high school and high school, there were always a couple of kids in the class who would show up without the standard supplies, then ask to borrow a piece of paper and pencil--as if it were beyond their feeble imaginations that such things might turn out to be useful in a classroom. It was always the same morons, day after day.

I have a theory that these same kids grow up to be blue-chip beggars. The ones who don't play poker very well end up on street corners holding up "Will work for food" signs.

Hey, that gives me an idea. Next time one of these imbeciles asks me for change for a red chip, I'm going to insist that he make and display a "Will work for blue chips" sign.

Another poker dump, and a study in contrasts

I am determined to eventually play at least once in every poker room Las Vegas has to offer. Today I scratched another one off the list: Arizona Charlie's-Decatur.

It will save a lot of time if I just say "ditto" to nearly everything I previously wrote about the El Cortez (see http://pokergrump.blogspot.com/2007/04/armpit-of-las-vegas-poker.html). The ACD poker "room" is just two tables roped off from the surrounding slot machines and blackjack tables. It doesn't even have a sign, so you can wander around the casino for quite a while before finding it. I had to ask a security guy, because I went around the entire perimeter and never saw a poker room or even a sign pointing to it--guess they wouldn't want to draw undue attention to it.

The main difference from the El Cortez is that ACD allows smoking right at the table. Today was only the second time I've sat in a game with smoking going on at the table. Fortunately, I was on an end seat, with a reasonably powerful desk fan right behind me, so I was upwind of all the smokers--which was literally everyone else in the game.

The other difference from the E.C. is that they were spreading a semi-normal hold'em game: $4-8 limit, with the unusual but not bizarro $1-2 blinds. I managed to lose only $26 in the course of an hour. I was actually ahead for most of the session, until losing one huge pot. After I flopped top two pairs, I pushed as hard as I could, but got called down and beaten by a backdoor flush. Now, this in itself wouldn't be too unusual, except that on the flop this guy had no pair and no draw, other than the runner-runner flush, yet he called a bet and a raise before the flop, and another bet and raise on the flop, with a suited K-5.* For readers who aren't obsessively into poker, I'll just say that this is seriously stupid play; in order to win, he had to get a club on the turn (about a 20% chance of that) and a club on the river (another roughly 20% chance), something like a 4% chance of both events happening. The point here isn't to complain about the beat or to garner sympathy, but to illustrate the level of play going on at ACD.

One elderly guy almost perpetually looked either asleep or dead. I kept wanting to suggest that the poker room manager call the paramedics to check his pulse and see if he was still alive. But then he'd stir a bit, scrape something out of his ear, look at it, and flick it on the floor. I thought that dead people probably wouldn't do this--at least not as often as he was doing it.

After my obligatory hour at ACD, I hopped over to the gorgeous Venetian, to put in a few more hours towards their 50-hours-in-August tournament promotion. What a contrast. It's hard to imagine two poker rooms farther apart on the scale of classiness. The Venetian is the top Editor's Choice on the only site (as far as I know) dedicated to comparing and commenting on Vegas poker rooms, http://www.allvegaspoker.com/. It's also usually one of the top two as voted by contributors to that site. See, e.g., this visitor's review, which I think is spot-on (except the part about Cassandra--sorry, but I just haven't paid much attention to the cocktail waitresses, other than to note that too many of them have gone way too big on the breast implants; it's kind of hard not to notice that): http://www.allvegaspoker.com/player_review_2382.html.

Arizona Charlie's-Decatur is like the anti-Venetian. If they were to come into physical contact, they would annihilate each other in a universe-shaking explosion. (Stephen Hawking, if you're reading this, please feel free to comment on the repercussions of such contact on the space-time continuum.)

I picked up my mandatory $1 and $5 chips for the souvenir collection, but when I look at them again some day, I expect that it will give me a shudder of disgust at the memory: "Ewwwwwwwwwww, I actually sat in that nasty room for an hour? What was I thinking?!"



*Probably the most thoughtful and refined guide to profitable selection of starting hands for a game like this is Lee Jones, "Winning Low Limit Hold'em" (3rd edition, 2005). K-5, suited or not, does not appear anywhere in his charts of recommended hands to play from any position. In other words, the consensus of studious, long-time winning players (yeah, it's a single-author book, but Jones has had a ton of input over the years his book has been in print, and he comments pretty profusely on how and why he has changed his mind on some points, after thoughtful comments from other pros he respects) is that K-5 (actually, any K with lower than a 9 kicker) is a dog in every situation, with a negative expected value over the long run (i.e., you'll lose more than you win playing it). So calling a raise before the flop was already pretty boneheaded, let alone calling another bet and raise after the flop when he had not improved.

Addendum, August 21, 2007: During a session at the Venetian today, I finally spotted the aforementioned "Cassandra." I can see why she caught the writer's attention. She is truly an exotic beauty--Hawaiian, maybe. Still, attractive women are about a dime a dozen in Vegas, and even if a place had a monopoly on gorgeous young women serving drinks, that would make for a pretty piss-poor reason to choose a place to play poker.

Something that is NOT against the rules, for a change

If you've played poker more than once or twice at the Las Vegas Hilton, you've probably run into Annie, a 59-year-old (which I know because it was her birthday last week, and she was telling everybody) Asian woman who practically lives there. She's too much of a character to fit into one blog post, so for now I'll limit my observations to this: If you beat her for a large pot because you played a superior hand well, she's a pretty good sport about losing. If you suck out on her with a worse hand, there will be hell to pay. And if you beat her by being tricky and non-straightforward, she will take it personally, and decide that you're scum unworthy of the game of poker. Aces and kings and queens deserve to win; if you win with junk hands, you're a disgrace, as Annie sees it.

I should note that Annie has actually been kind to me the vast majority of the time. She has some sort of clothing business, and once gave me a nice silk shirt and a silk sofa pillow, for no apparent reason. She engages in friendly chat, asks about my life, tells me about things going on in hers, etc. When there's an especially bad player at the table, sometimes we engage in quiet conspiratorial whisperings about that fact. As far as I can tell, Annie is a genuinely decent person--she just has this prominent flaw of being incredibly judgmental and rigid about how poker should be played.

I'm mostly a classic tight-aggressive player. Probably nine out of ten times that I put in a pre-flop raise, I have the predictable range of hole cards: big or medium pairs, big but unpaired cards, suited connectors. But about one time in ten I try to throw in an oddball, to keep people guessing about me: 4-2, 6-9, K-3, or whatever. When those hands hit, they are not only extremely difficult for opponents to guess that I have, but everybody watching me make such a play will remember the unorthodoxy of it, and as a result have less confidence in future situations that they can put me on a narrow, predictable range of hands.

For example, a few weeks ago I raised with an unsuited 8-9 and had one caller (Cy--a very nice guy, and better-than-average player). The flop was 5-6-7, giving me the nuts. I bet. Cy, who knows my style of play pretty well, raised. I never saw his hole cards, but he didn't need to have anything to raise me here. All he needs to know is that usually this kind of flop won't have helped my most likely holdings (such as A-K or A-J), and that therefore my bet is really a bluff and I'll have to fold to a raise. It was a smart play on his part, even if he had nothing at all. It would usually have worked. But in this instance, I had something he couldn't reasonably suspect. I re-raised and took the pot without further contest. (I think he knows me well enough to understand that I'm very unlikely to put in that re-raise without having the goods to back it up.)

Anyway, after playing for a while this afternoon, I thought too many of my raises had been with the expected hands, and it was time to throw a curve ball, so I put in a pre-flop raise with a 4-5 offsuit, from middle position. Annie was the only caller, in the big blind. As it turned out, she had a suited A-8.

The flop was 8-6-3, giving Annie top pair/top kicker, and giving me an open-ended straight draw. She checked, then called my bet. The turn card was a 2, completing my nut straight. Now I was confident that I had her, so my thoughts turned to how to get all of her chips. (I thought she probably had an overpair--9s or 10s or jacks--which was wrong, but wrong in a way that didn't matter.) She checked again, and I bet about half of her stack, figuring that if she called that bet, she'd be pot-committed and put in the rest on the river no matter what came. She called.

The river was probably the worst possible card for her: a third 8. She clearly thought this was Yahtzee for her, and moved all-in. I called and won.

It took her a few seconds to put the pieces together retrospectively and figure out what I had done, but then the torrent was unleashed: "You raised before the flop with 4-5???" That's where the tirade started. I won't try to quote the whole speech, but among other things she invoked a curse on my immortal soul. Let's just say that Annie most definitely does not approve of non-standard plays of that nature--especially when they end up with her losing her entire stack of chips.

Well, Annie, you sweet, crazy lady, I know there's not a chance in hell you'll read this, but I'll say it anyway: It's neither against the rules nor unethical to put in a pre-flop raise with a crappy little 4-5 offsuit. Yes, it's unorthodox, but that's the whole reason for it. Poker is a game of deception. Since I know you'll think I most likely have something like A-K or A-Q when I raise, doing it when actually holding 4-5 is deceptive. It's allowed. In fact, practicing occasional deception of exactly that sort is encouraged by every poker strategist I've ever read. And the reason, obviously, is precisely because of the kind of two-fold effect I achieved today. First, you were completely blindsided by the straight. Second, from this day forward, any time the board makes something like 4-5 a very strong potential holding, you'll have to wonder whether that's what I have again, when I actually have A-K and got no help from the community cards. It will make it easier to bluff successfully in such a spot, and/or you'll play your hand less aggressively than you should because you're afraid that I'm pulling another stunt like today's. That's exactly what I hoped to accomplish, on both counts.

So spare me your self-righteous moralizing about correct play, as if you are the Moses of poker, entrusted with stone tablets engraved by the finger of God, declaring which starting hands are worthy of a raise and which ones aren't. Nothing in poker is that cut and dried. (What a boring game it would be if it were!) It appeared to me--and I'm really not vain about this sort of thing so as to imagine it--that everybody at the table, except for you, admired my play and the outcome, and was laughing at you during your rant, because you weren't able to just say, "Nice hand," let it go, and move on. With your "there's only one right way to play" speech, you were the loser today, in ways more fundamental than who was stacking up the chips, and who had to re-buy.

I'm not dead yet! (A post with zero poker content, zero grumpy content)

That's me, as one of the French knights, peeking around the fortress door to get a look at the giant wooden rabbit left by the English ka-niggets. It's not Photoshop, it's one of those silly stick-your-face-in-the-hole photo ops. It's just outside the Spamalot theater at the Wynn.

As you might guess, I went to see that show tonight. It's great--even better than I expected. Maybe half the material is the best bits from the "Quest for the Holy Grail" movie, but half is new stuff, good stuff, stuff squarely in the Python tradition. The whole thing just works. I was delighted and forgot about my problems for a couple of hours of pure fun. Highly recommended.

Also had a celebrity sighting, because John O'Hurley (who is most famous for playing Elaine's nutty boss J. Peterman in "Seinfeld," and who plays King Arthur in "Spamalot") walked right by me while I was waiting for the doors to open.