Between hands at his Omaha eight-or-better-to-qualify, high-low split (Omaha/8, for short) table, Jeff hied over to my low-limit hold 'em table to ask about his chances of getting a hand with ace-deuce. He had played Omaha/8 for several orbits of the dealer button without receiving an ace-deuce, and couldn't believe his bad luck. "How often should I expect an ace-deuce," he asked.
Including suits, there are 270,725 different Omaha starting hands. There are 16 ace-deuce doubletons and 946 doubletons without an ace or a deuce, so there are 15,136 hands with exactly one ace and one deuce. The probability of receiving an Omaha hand with exactly one ace and one deuce is their quotient, .05591, about one time in 18 hands.
In a nine-handed game, that's once in two orbits of the dealer button on the average. "Another way to look at it," I said, "is that you have a 50-50 chance to receive a hand with exactly one ace and one deuce within a dozen hands. It's 99-to-1 that you'll pick up a hand with an ace and a deuce within the next 80 hands."
Jeff's question led me to thinking about Omaha/8's very desirable, very rare starting hand, double-suited A-A-3-2. There are six ways possible for the suits: spades hearts; spades-diamonds; spades-clubs; hearts-diamonds; hearts-clubs; and, diamonds-clubs. The doubletons are A-2 or A-3 in one suit and its complement in the other suit. Multiplying yields a total of 12 hands out of the 270,725 possible. The probability of receiving A-A-3-2 double suited is the quotient, 12/270725, which equals 0.00004433. Now that's a rare starting hand! A-A-3-2 DS is rarer than pocket quads. Think about that. There are 13 ranks so there are 13 hands of pocket quads, versus 12 hands for A-A-3-2 DS. Pocket quads are rare, but not quite so rare as A-A-3-2 DS. If you were to play Omaha eight hours per day, five days per week, fifty weeks per year, then you would receive 60,000 hands per year at 30 hands per hour. Omaha/8 takes more time because of splitting the pots among the winners: at 25 hands per hour you would receive 50,000 hands in a year.
Multiplying the probability times 50,000 equals 2.2. In other words, you should expect to see A-A-3-2 double suited about twice a year. You don't play 50,000 hands a year? You don't have the brass-plated, cast iron bottom typical of a diehard Omaholic? The table shows another way to look at it. You have a 50-50 chance to pick up A-A-3-2 double suited in 15,637 hands. If you play only four hours a day, five days per week, fifty weeks per year, the chances are 50-50 that you'll receive that rare hand within 71/2 months. The table shows that the odds are 9-to-1 that you'll receive that hand in 52 thousand deals. The odds are 99-to-1 that you'll receive that hand in 104,000 deals.
The odds in the table are a difficult for most to grasp. For example, if you don't receive A-A-3-2 double suited within 15,637 hands, it doesn't mean that you're due: it means that there's a 50-50 chance you'll pick up that hand within the next 15,637 hands. Similarly, the odds are 99-to-1 that you'll pick up A-A-3-2 double suited within 104 thousand hands. But if you don't, you still aren't due: it remains 99-to-1 that you'll pick up that hand within the next 104 thousand hands.
Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net
