pat J & pat 9 calculations

[euri10]

I know that a pat J is a favorite against 1 opponent who draws 1 card on the last draw and that a pat 9 is a favorite against 2 players who each draw 1 on the last draw.
My question may be dumb, but can I conclude this ?
Is it a "simple" enumeration exercise ? If yes I would be very interested in reading the demonstration, I was quite good at school on probabilities and enumaration problems, but I can't figure out how to start with these 2 problems.

[MarkGritter]

[ QUOTE ]
I know that a pat J is a favorite against 1 opponent who draws 1 card on the last draw and that a pat 9 is a favorite against 2 players who each draw 1 on the last draw.
My question may be dumb, but can I conclude this ?
Is it a "simple" enumeration exercise ? If yes I would be very interested in reading the demonstration, I was quite good at school on probabilities and enumaration problems, but I can't figure out how to start with these 2 problems.

[/ QUOTE ]

There was an earlier thread where somebody (randomstumbl?) worked out several cases of this, and actually came up with a couple situations where the J was tied or worse, I think.

I like to approach problems like this from a worst-case/best-case point of view. The best case for your opponent (if you hold the J) is the holding that maximizes his outs.

Suppose you have JT432 and your opponent has 7432x. Then you hold only two of his outs. (If you held J7432 you would hold the equivalent of 3.5 of his outs, because the remaining J's would only chop.)

Any J, T, 9, 8, 6, or 5 wins it for your opponent. This is 22 cards. Based only on the current holdings there are 42 cards left (because x is presumably not one of these cards) so the 7's chance of winning is 22/42, or 0.524.

But, we usually have seen other cards previously. The more non-out cards the players have seen, the better the drawing chances. Suppose Villian has already had two 7s in his hand, and holds the 4th 7 right now. Then there are still 22 outs but only 40 unknown cards, so Villian's equity has improved to 22/40 = 0.55.

If Hero saw four of Villian's outs previously before sticking with his JT (say, TT in his hand originally, J9 after the first draw, then got his JT) then there are 4 less outs and 4 fewer unknown cards, so Villian's equity is now 18/36 = 0.50. If Hero had completely bricked with four non-out cards before drawing JT then Villian's equity is 22/36 = 0.61!

So, it is certainly possible to come up with a worst-case situation where the pat J is not a favorite. (I believe there are cases where your equity is improved by breaking a J as well.)

Working out the 9 case is only slightly more complicated; but you can use the same reasoning to work out the probability that at least one Villian outdraws hero. I wouldn't be surprised if the worst case put the 9 as a dog here, either.

Something like 98432 vs. 7432x vs 7432x--- each villian has 14 outs out of 37 cards remaining, or 0.378. The chance of them both bricking is thus (23/37) * (22/36) = 0.380, which is Hero's equity, so Hero is a significant dog here before adding in any previous discards.

However, these are atypical situations. Something as bad as JT986 holds 5 of Villian's outs (with 7432x) so the draw is 19/42 = 0.452 to come in. Ironically, the smoother your pat hand the more likely that a smooth drawing opponent will beat you. (But the less likely your opponent is also drawing smoothly, etc.)