Poker News

Richard Burke

It cost you $200 to play in the satellite for a seat in the $10,000 buy-in, No Limit Hold’Em tournament at the World Series of Poker (WSOP). Fifteen tables were packed with ten players each eagerly waiting for the dealers to “put ‘em in the air.” The tournament started. A player in middle position opened for three times the Big Blind. Everyone between folded. On the button, you peeked at your hand and saw Ka- Kd. “Hot damn,” you thought, “Maybe I can double up on the very first hand!” “All-in,” you announced. Everyone else folded but the raiser who almost beat you into the pot with the rest of his chips. As soon as he went all-in, you just knew he had pocket Aces. He did. His Aces held up and you were first out.

Returning to your usual card room the next day, you an-swered your friends’ questions by saying that you were knocked out early by a “bad beat.” And you sighed and la-mented your wretched plight in running up against pocket Aces when you had pocket Kings. But just how bad a beat was it, really? That’s an easy question to ask: obtaining the correct answer is somewhat complicated. Here goes.

Given that you hold pocket Kings, there are fifty un-known cards pre-Flop, the thirty cards remaining in the deck and the twenty cards dealt to the ten other players. The four Aces have to be somewhere, so there are five cases to con-sider: no Aces were dealt; one Ace was dealt; two Aces were dealt; three Aces were dealt; and, all four Aces were dealt out to your ten opponents. The first two cases don’t concern us because no one could have pocket Aces. The probability for exactly two Aces being dealt in your opponents’ twenty cards is C(4,2)*C(46,18)/ C(50,20), or 0.3289. Similarly, the probability of exactly three Aces being dealt is 0.1485. The chance that all four Aces are in enemy hands is 0.0210. The combined probability that two or more Aces are somewhere in those ten enemy hands is 0.5284, about a 53-47 chance. “But,” you wail, “pocket Kings beaten by pocket Aces is a ‘bad beat,’ not a fifty-fifty deal at all!”

We’re not done yet. There are 19!! ways, called semi-deals, to have dealt twenty cards into ten, two-card hands. For all three cases that concern us, the Aces could be scat-tered among your opponents such that no one has a pair, that someone does have a pair, or, that two enemies each have pocket Aces. For two Aces having been dealt out, the chance that someone has them both is 17!!/19!!, or .0526. For three aces having been dealt out, the chance that anyone has two of them is given by the expression, C(3,2)*17!!/19!!. For all four aces out, the chance that one opponent has two of them in the pocket is C(4,2)*16*15!!/19!!, or .2972. The chance that two opponents each hold pocket Aces is .0093.

After multiplying, collecting, and summing terms, the probability that one opponent will have pocket Aces is 0.0486. (The probability that two will each have pocket Aces is .0002.) Your pocket Kings will run up against one or more pocket Aceswith a probability of .0488, about one time in 20.5. That’s not a “bad beat,” that’s just bad luck.

Only if it were to happen to me would it be a bad beat.

Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold ‘Em, 2nd ed., available from amazon.com, gamblersbook.com, and www.kokopellipress.com.

Filed under: Poker News