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Index > Poker games and rules > Odds and probabilities

What are my chances of sucking out on my opponent in Stud?

Author: Harry 026
Last updated: Oct 2003
Copyright © 2004 Harry 026
The official and up-to-date version of this answer is here.

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.