#5
10-30-2007, 11:52 PM
 MarkGritter Senior Member Join Date: Jan 2005 Location: Eagan, MN Posts: 1,376

So, if you pat he has about 8 outs (maybe 7) in 44 remaining cards, which is odds 4.5 to 1. 18% of the time he makes his hand. Let's say he gets an extra bet out of you every time. (He can't do quite that well if you play the game-theoretic optimal, but he definitely has the ex-showdown advantage when you are pat and he is in position.) So you earn 0.82 * p - 0.18 bets

Let's give him some crap like 762x. Again we'll vastly simplify stuff and assume 1 additional bet goes in on all the Badugi vs. Badugi cases and 0 bets otherwise. He has 10 outs to a badugi while you only have 9. (no Q!)

There are 1892 possible cases
In 61 of them you both make a Badugi but yours is better
In 29 of them his Badugi beats yours. (Total=90, good)
In 9*(43-10)=297 you make another Badugi but he doesn't
In 10*(43-9)=340 he makes a Badugi and you do not
The remaining 1165 cases you win with the best 3-card hand. (Check: does this make sense? (44-9)*(43-10) = 1155, close enough... there are 10 cases missing somewhere.)

So, you get 1165/1892 * p + 65/1892 * ( p + 1 ) + 29/1892 * (-1) + 297/1892 * p =
0.616 p + 0.034 p + 0.034 - 0.015 + 0.157 * p =
0.807p - 0.019

Given the assumptions above:
Patting =~ 0.82 * pot size - 0.18
Drawing =~ 0.81 * pot size - 0.02

This suggests drawing is slightly better unless the pot is very large. I think the 32A draw is actually worth more on the last round of betting than the above analysis suggests (even out of position) so I drawing is actually worth a bit more. But you'd need a game-theory solver to tell you how much it is really "worth".