Poker News

Richard Burke

On a sunny day in early spring while waiting for a seat in a $4-8 Hold’Em game, I was trying to make an ordinary word from the letters, “G-N-AR- L-A,” when Hakim came into my local card room, signed up for a $4-8 Omaha/8 game, walked over, sat, and asked if I had figured why he was “always getting quartered, or worse” with the nut Low. I had. Howoften, he asked. Did Hakim know that the formula to calculate the number of ways to deal thirty-six cards among nine Omaha hands, C(35,3)*C(31,3)*…*C(11,3)*C(7,3), obtains 388,035,036,597,428 billion, a number about forty times larger than the number estimated for stars in the universe! Did he understand that the following results weren’t from a simulation, that they’re exact, from a mathematical proof never before published? Hakim wasn’t impressed.

There are six cards that could in tandem tie Hakim for Low. The table shows that 55% of the time at least one opponent will have two or more of the six nut-low cards. (45% of the time, the six other nut-low cards will be scattered among Hakim’s nine opponents or not even dealt, for an outright win.) The most likely pattern, 2-1-1-1-0, has five nut-low cards dealt among four opponents. The least likely pattern, 4-2-0-0-0, has four of the six nut-low cards in one hand and two in another.

I told Hakim that I examined each pattern to see whether it would tie once, twice, thrice, or not at all. For example, the pattern 2-1-1-1-0 will tie Hakim’s nut low 60% of the time. For another example, only the pattern, 2-2-2-0- 0, could possibly split the pot four ways, and the chance of that is 40%. For any pattern where there are four of the six nut-Low cards in someone else’s hand, Hakim must split the Low half with at least one other player. Hakim asked if I would please skip all these messy details, because he was getting drowsy. The next task was to multiply all the patterns’ probabilities by the chance that Hakim’s nut Low would win all, half, a third, or a fourth of the Low half of the pot, and then collect them. Using modern spreadsheet technology, those products and sums are straightforward. The table below showed Hakim the resultant probabilities by the number of Low winners. In a ten-handed Omaha/8 game, the nut Low will win outright over sixty-four percent of the time.

The dealer will split the Low half of the pot two, three or four ways, about 36% of the time. Hakim asked if I would get to the point pretty soon, because he had just been called to his Omaha/8 game. The point is that you’re NOT “getting quartered all the time”: your nut Low will win half the pot 64% of the time; a quarter 32% of the time; etc.; for an expectation of .4076, almost 41%. In the long run you’ll profit with the nut low with asfew as two opponents at the showdown. Thanks, Hakim said, leaving to go to his game. “No trouble at all,” I said.

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