Binions - There was a time when I wondered about that very question. At that time I had never played poker in a casino, although I did play in a weekly private, friendly, dealer's choice home game.
It was late at night. I was visiting an old fraternity brother on his boat, the "No Limit," which was moored in Cabo San Lucas. We had stayed friends through the years - several marriages for him and a couple for me. I was the only one awake on the boat and ran across my friend's copy of Ray Zee's book, part of my friend's poker library.
I had previously played Omaha-8 a few times in a friendly weekly game and vaguely wondered how often low was possible. I skimmed through my friend's copy of Ray Zee's book but didn't find the answer I sought.
When I got home, questions gnawed at me. How often was low possible? How would anyone figure out such a thing? Seemed like something anyone who played the game seriously would want to know. Why didn't Ray have the answer in his book?
Took me months to figure out how to solve the problem. My first complete solution took pages and pages and involved a cumbersome math system with notation and complex reasoning I have never seen anywhere else and have since forgotten. (But I got the correct answer). Now the solution would take maybe the better part of an hour and occupy maybe the back of an envelope.
I'll give you the answer, although you might get more enjoyment in the long run if you figured out how to do it yourself.
Probability is always considered from the perspective of the information you have. In poker, the more cards you have seen, the more accurate can be your estimate of probability.
Before you see your own cards, the probability of low is about 60%. (1,561,728/2,598,960)
After you have seen your own four cards, but before you have seen the flop, the probability of low ranges from 55% to 69%, depending on your own cards.
When you hold four low cards of four different ranks, such as A-2-3-4, the chance (probability) of low for anyone is about 55%, but your own chance of making low is only about 53%. If you held A-2-3-3, the probability of low would be yet a bit different. That's before the flop.
Then after the flop, you deal with different probabilities. If you hold Ac-2c-3d-4d, and the flop is 8h-9h-Th, unless there's an extraordinary amount of money in the pot from the first betting round and it will only cost you a single bet on the second betting round, you'll likely do best to fold to the single bet because you missed that flop.
The probability of low for you after that flop is only
(12*11/2 + 12*12 + 12*11/2)/990, or 276/990. Accordingly, the odds against your making low on the next two cards are about 2.6 to 1, and since with that flop you don't have much of a chance of winning for high, you're not even playing for the whole pot.
What's the chance of low? Depends on what cards you have seen. If none, then 60%, but as it turns out, that number actually is of little, if any, importance. I spent all that time seeking the answer to a question that doesn't even matter. (But I learned some other stuff in the bargain).
Buzz
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