Quote:
Originally Posted by Jon_locke
Having the best hand when neither player improved is worth too much. Ask yourself how often does A27 not improve, it's a significant enough number
If both improve, the exact solution is very complicated, because which wins depends on how much each improves. (From the perspective of the holder of A27, there are 14 ways A
2
7
can improve to a three-card six or better in one draw... and presumably if either A27 or 456 improves to a four-card badugi, you'd stop drawing. But if, for example,
A
2
7
improves to A
2
5
, then you'd draw again (hoping to catch a club or the 3
or 4
).
Meanwhile from the perspective of the holder of 4
5
6
, there are umpteen ways 456 can improve to a three-card five or better (and, again, presumably you'd stop drawing if you made a four-card badugi), but most of those umpteen ways to improve involve catching two or three favorable cards, low percentage plays compared to likely catching one favorable card). For example, drawing to 4
5
6
on the first draw you might catch the A
, and then, still having a three-card six, you might draw again and catch the 2
... and then, still having three-card six, you might draw again and catch the 3
.
To be exact, you'd have to compare the probability of each of the 14 ways of improving A27 to the probability of each of the umpteen ways of improving 456.
Let's approach the problem from the standpoint of not improving.
-------
With 14 outs and three draws, I believe
P of A27
not improving = C(34,3)/C(48,3)=~34.6%.
(Thus with three draws, A27 should improve to A23, A24, A25, A26, and/or a four card badugi about 65.4%).
With 13 outs and three draws, I believe
P of 456 not improving = C(35,3)/C(48,3)=~37.8%.
Thus the probability of neither hand improving is about
0.346*0.378=~13.1%.
In other words, neither A27 nor 456 will improve in three one card draws about 13.1%.
A27 will improve but 456 won't 0.654*0.378=~24.7%
456 will improve but A27 won't 0.622*0.346=~21.5%
And they'll both improve ~40.7%. Something like that.
If the above is reasonably correct, the question is when they both improve, which improves more. In order to make up the deficit, when they both improve, A27 has to improve more than about 9.9%.
- my math: 13.1+21.5-24.7=9.9
In other words, when they both improve, of the 40.7, I think A27 has to improve more than 456 by 25.3% to 15.4%.
- my math:
40.7=X+(X+9.9)
30.8=2X
15.4=X
And I don't think it does.
When they both make four-card badugis, the major way they both improve, and when that's the only improvement, then whichever gets the lower fourth badugi card wins. If they both get the same rank to make a badugi, then 456 wins. For the purpose of approximating (otherwise it's a morass) I'm ignoring various multiple draws where, for example, A27 makes A25 on the first draw and then makes a badugi on the second or third draw.
Anyhow, for heads up play when A27 is up against 456, I think I have to go with the 456. But if they're both against unknown hands, then I prefer the A27.
It's a bit like AAKQ vs. T987 in Omaha-8. AAKQ is better than T987 against unknown hands, but heads up against each other (T987 vs. AAKQ), T987 is better than AAKQ
Interesting comparison... at least I thought so.
Buzz