Hypothetical Situation

Okay,


Here's a question for you poker math gurus? Let me preface this by saying that I don't know the answer, so this isn't a quiz. I'm just curious as to whether anyone can work it out.

Here's the scenario:


You're playing heads-up against someone. A situation arises where you move him all-in. He calls and flips over his cards: aces. Now, you have a special power that allows you to change your two cards to any that you want, except the other two aces. Which hand do you choose that gives you the maximum chance of winning?


Or put more simply, which hand has the most outs against aces in a heads-up situation?


The Bear

65s not in his suit.

76s would also be as good.

Just to pick nits, 76s wins slightly more often than 65s, but does not tie as often. I think this makes 65s "better". It has better straight EV, for example.


As Ac vs. 5h 6h


aces win 1314307

65 wins 391582


tie 6415


As Ac vs. 6h 7h


aces win 1315168

76 wins 391637


tie 5499


Sure, they (and 87s) are as close as dammit. 98s is only about 7000 hands worse -- 87s wins when the board is QJT9x, but 98s loses when the board is KQJTx. T9s, JTs, etc. are progressively worse because it's easier to make the second-best straight.

Let's turn the scenario around .. the big blind is all-in, and you call with the two black aces and get heads-up with him. He closes his eyes to pray to the Poker Gods for good cards and you close your eyes and pray that he has the


{insert two cards here}


to maximize your chance of winning.


--JMike

Kc 2s, or Ks 2c. I'm not sure, though.

Any combination of another A and a black K.

K2o I think, with both black.


Pat

I am confused as to how you got these numbers- I could be wrong but I still don't see how 65s is any better then 76s.

What a mental block I had .. yeah, ace-and-a-black-six looks like the worst:


bash-2.02$ cmp2.exe as ac ks 2c

1538197 165043 (ties = 9064)


bash-2.02$ cmp2.exe as ac ah ks

1588961 100271 (ties = 23072)


bash-2.02$ cmp2.exe as ac ah 6s

1598778 96808 (ties = 16718)


----------------------------------------------


Goat -- there's a fast poker hand evaluation program called "fish" available out on the net. (go to your favorite search engine and type in "fish poker hand evaluator" and maybe "ftp" as well and you'll find it). cmp2 is a program that takes the two given hands, runs all C(48,5) boards and evaluates which hand wins with each.


So I just look at the numbers and see


bash-2.02$ cmp2.exe as ac 6h 5h

1314307 391582 (ties = 6415)

bash-2.02$ cmp2.exe as ac 7h 6h

1315168 391637 (ties = 5499)

bash-2.02$ cmp2.exe as ac 8h 7h

1315602 391672 (ties = 5030)


and, well, I believe them because I believe the guys that wrote 'fish'.


The only difference I can come up with off the top of my head is that there is a little more room to make a high heart flush on the board -- but I think the difference is only about 30 hands. I'm not sure where the other extra ties come from.

I ran my situations through a freeware program touted by Andy Glazer. You can get it at


http://www.brecware.com/Software/software.html

Turns out that the guts are the same -- when they talk about "Cliff's evaluator", they are talking about the same underlying program.