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Big Dave D wrote:
> I'm trying to work out a seven stud hi-lo eight or better
problem.
> You're heads up on 4th street, drawing to a busted low, say
a233.
> You are up against an obvious high, say KK with KK in the
hole.
> What are the odds that you make a low, any low. (which
would
> obviously get you half the pot back). My mental maths says
> about 2:1 against.
The odds against getting a low are about 1.67 to 1 against, so your "mental maths" estimate of 2 to 1 is pretty good.
Here is a chart (generated from a Markov chain) that might be helpful if another such question comes up. If you are hoping to get a low (in any seven-card game such as stud, Omaha-8, or hi-lo hold'em), a "good card" is any desirable low card of a rank you don't already hold, and a "bad card" is a card of any rank you already hold or of a rank higher than you want. For example, suppose you hold 24468, and you believe that you must get a 7-low or better to win the low half of the pot. At that point you hold 3 good cards and 2 bad cards. Find that row, and look below P7. The probability that you will make a 7-low is 0.0888. If you'd settle for an 8-low, then you hold 4 good and 1 bad, and the probability of making an 8-low is 0.5698.
This chart doesn't take into account any other cards, so in your case (where the opposition holds 4 kings) the probability of making your low is somewhat better than the 0.3238 from the table (it is 0.3745, for odds of 1.67 to 1 against).
G B P8 P7 P6 P5 0 0 0.1834 0.0849 0.0294 0.0058 1 2 0.0423 0.0181 0.0060 0.0012 2 1 0.1963 0.1072 0.0465 0.0125 2 2 0.0740 0.0370 0.0148 0.0037 3 0 0.4827 0.3325 0.1895 0.0715 3 1 0.3238 0.2091 0.1119 0.0398 3 2 0.1480 0.0888 0.0444 0.0148 4 0 0.7132 0.5872 0.4288 0.2343 4 1 0.5698 0.4496 0.3145 0.1647 4 2 0.3478 0.2609 0.1739 0.0870
Editors note, Oct 2003: for further reference, see the following.
Mike Caro has published an extensive set of probability tables for draw, stud, holdem, and lowball.