Tuesday, August 28, 2007

Missed it by THAT much!

For those who have never heard some of the phrases made famous by Don Adams as Agent Maxwell Smart--including the title of this post--hear them at http://www.wouldyoubelieve.com/sounds.html.

There was another odd occurrence during my hour at Jokers Wild on Sunday. Its poker room, like many others, has high-hand jackpots. If you get four-of-a-kind or a straight flush (with certain conditions on the size of the pot, the number of people at the table, etc.), the casino pays you a jackpot, which grows over time when it isn't hit. There are, sadly, many players, especially at the low levels, for whom hitting such jackpots is the entire point of playing.

Anyway, an elderly woman won an unremarkable hand with pocket queens. I thought nothing of it.

On the next hand, the turn and river were both queens. When the second one hit, both this woman and the dealer let out loud groans, and then a couple of other players chimed in with sympathetic comments, like "Look at that," and "Oh, yeah--almost."

I remembered that this woman had shown queens on the previous hand, but didn't think it was possible that the occurrence of two queens on THIS deal was being fussed over. On the other hand, I couldn't see anything else remarkable going on that would cause the clucking.

When the dealer looked in my direction, I caught his eye, gave him a deliberately puzzled expression, and spread my hands, meaning, "What's going on?" He said, "She had pocket queens on the previous hand."

I asked him, "So what?"

He pointed out the high-hand jackpot. I told him that I knew about that, but it didn't qualify if two of the four cards came on one hand, and the other two on the next hand. He said, "Yeah, but it was so close."

I could hardly believe that these people were actually this stupid. I said, "Well, it's a whole lot easier for two queens to hit the board if the other two of them aren't in somebody's hand."

The dealer granted that, with a facial expression that seemed to indicate he had never considered this before.*

He replied, "Yes, but it seems that they always come on the next hand." Uh, right, if you consider 4% of the time (the probability that the next hand will have a board with two queens--see the math below) to be "always."

I pointed out that it makes no difference whether the other two queens came on the next hand or 100 hands later. He said, "I know, but it was so close."

To which I could again only think, "SO F'ING WHAT?????"

The only way these morons could have any sort of emotional response to this situation is if they're imagining that two queens hitting the board is completely independent of two queens being in somebody's hand, and you get the jackpot when these two events just happen to coincide. In that case, you could think of it being a "near miss," sort of like the roulette ball falling into the slot right next to the one on which you had bet your life savings--close, but not quite.

But that's not the situation here at all. Her having those queens in her hand radically affects the probability that we'll see two on the board. It's like reducing the width of the slot into which the roulette ball can fall to make you a winner to about one-fourth of the size of all the other slots, then trying to hit it.

(I suppose that I should mention the related poker phenomenon of "flop lag." This is a great term for when, e.g., you play a 7-8, have to fold to a large bet on a flop of A-K-K, then on the next hand see a flop of 4-5-6. This is "flop lag"--seeing the perfect flop for your hand, one shuffle too late. I have heard this discussed many times, and, of course, have experienced it. But it's just a fun observation, with no connection to reality. I have never heard anybody say anything that even remotely suggested seriousness about this experience being, somehow, "close" to a great thing happening. It's just a bit of silly amusement.)

In this case, it was also not only a completely different shuffle and deal, but a different deck of cards (because they alternate; one is shuffled while the other is in use). What's more, the queens that came on the board in the second hand were both black, whereas in the first hand her pair had been one red and one black. That second board could not have occurred in the first hand. None of that really matters in terms of the statistics, but the observations add to the lunacy of thinking that this was, in some meaningful way, a case in which Agent Smart would have had occasion to say, "Missed it by THAT much!"

No, you idiots, you missed it completely.

*Here's the exact computation of the effect.

Suppose we know that no players were dealt any queens (and, obviously, the same approach applies to any rank of cards you want to posit), so that all four are potentially available to be among the community cards. We don't care about the order they come in. The number of different boards of five cards that will contain exactly two queens is C(4,2)*C(48,3), where "C(x,y)" is the standard notation for the number of combinations of y objects you can pull from a group of x. Here we're specifying that two of the five cards on the board be queens. There are C(4,2)=6 different pairs of queens. With each of those pairs, there are C(48,3)=17,296 ways of pulling another three cards from the remaining 48 cards in the deck, so the total number of different boards with two queens is 6*17,296=103,776. There are 2,598,960 total possible boards--that's C(52,5)--so the fraction of all possible boards that contain exactly two queens is 103,776/2,598,960=0.0399. That is, it will happen 4% of the time, approximately.

But if we know that a player is already holding two of the four queens, now the number of boards containing the remaining two of them is given by C(2,2)*C(50,3)=1*19,600=19,600. The number of possible boards drops to C(50,5)=2,118,760. The probability of getting a set of five community cards containing two queens thus becomes 19,600/2,118,760=0.0093, or a little less than 1%. Putting two queens in a player's hand makes it more than four times less likely that a board with two queens will appear.

Isn't math fun?!

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