View Full Version : Heads up in Hold'em

05-08-2003, 07:00 PM
An interesting problem came up last week when talking to a friend about Heads-up all-in Hold'em play.

Let's call a 'power value' the exact probability for any given hand to win a pot in a heads-up situation.

At the risk of being too verbose, I'll go into detail about the problem.

There are 52!/(50!*2!) = 1326 possible Hold'em hands. In a heads up situation, your opponent can get 50!/48!*2! = 1225 possible hands.

We know that there are 48!/(43!*5!) = 1,712,304 possible boards that can be dealt.

Using these three numbers we can see that a heads-up Hold'em situation can occur in 1326 * 1225 * 1,712,304 = 2,781,381,002,400 possible ways. Yup, that's nearly 3 trillion!

Hopefully, I've got your interest by now, cuz this is where it becomes clear that we have entirely too much time on our hands. /forums/images/icons/wink.gif

AA has an advantage over every hand, so you would have no problem going all-in preflop when your heads-up opponent raised enough that puts you in that situation.

It's also clear that 32 off suit is trash versus every other hand.

The question is, what is the 'power value' of all 1326 hands in a heads-up all-in Hold'em situation?

If we know the 'power value' of all hands, we can then rank them and know the 'power ranking' of each hand.

We've approached solving the problem in the following manner.

We have run a simulation that ran through all hands that include at least 1 spade. We have output that includes the number of wins and losses for each hand tested. Now, we can rank these hands.

To re-iterate, we have tested AsKs versus all 1225 hands it can play against with the board flopping all 1,712,304 possible ways and have a win/loss record for AsKs. Now, we know the probability of winning with AsKs in a heads-up all-in situation. We continued and crunched AsQs, AsJs, all the way down to 2s2c. We want to know the same information for ALL 1325 other possible hands.

By the way, it took 5 days of non-stop computing to get the results for all hands that include a spade. ( FYI, it was on a 500MHz Pentium with 512MB of RAM ).

If we just consider "suited" and "un-suited", it doesn't matter that the suit we tested was spades, right? Obviously, we should get the same results if we had tested any suit. Therefore, we should expect the same results if we tested hearts, diamonds and clubs in the future.

Is it true that we have covered all possible situations already?

05-08-2003, 07:49 PM
Is this what you're looking for?


05-10-2003, 07:32 AM
Yup! Exactly! Thanks!