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.