Texas Holdem: Odds a given hand will win (not *your* hand)

I'm having a hard time finding anything on this one.

The question is: given a number of players, what is the probability that a high card will win, a pair will win, etc.

This is an extremely difficult question, it seems. Let's eliminate actual played hands and just suppose hypothetically that no one ever folds, to weed out uncertainty. Clearly, this has a definitive answer. You just need to "list out" all possible dealings and what the winning hand is for each. But how hard is this to do?

On a similar note, is there real-world data on how frequently each poker hand wins? Is there a simulator I could get/use that would show this?

Thanks in advance

Here is a couple of simulators:

link

link

The equity numbers you get are the number of times the hand will win if no one folds,percentage wise.

I don't think there's an "easy" way of doing this - at least, not without some thought, or without doing a "Monte Carlo" calculation (where you simulate millions of hands and determine the results from that).

There are 2,118,760 different possible sets of community cards for any particular pair of your hole cards. This can be reduced, in the same way that the 2,598,960 different possible 5-card hands in Video Poker are actually only 134,459 unique hands (e.g. because any particular flush in each of the four suits is really the same hand counted four times), but, like I said, that would take some thought, and even then, the final number would be overwhelming.

For each of the 2,118,760 sets of community cards, there are (45)C(2) = 990 pairs of cards that "player #2" (assuming you are "player #1"); that's a total of 2,097,572,400 possible results, and that's just for heads-up. If you add a third player, it's (I think) almost a trillion; add a fourth and it's now over 25 trillion.

I think a Monte Carlo is the best way to go. All the sims I've come across so far including the 2 mentioned above don't appear to run the sim I'm thinking of.

Ideally, I'd like a sim where I could enter a number of players and a run time (or hand count) and just have it keep a count or relative frequency of each winning hand. It sounds like the wizard uses a sim like this?

Here's what I get (each number is the percent chance of winning given a particular pair of hole cards and a number of players):

Sweet table. This is the kind of table I'd love to see only in this case instead of all hole card hands on the the left column you have all the possible winning 5-card hands (treating a a a a k = a a a a q and suits equal and so on),
and the percentages are the % of time those hands won over the long haul.

So for example, if you were curious how often in an 8 player game (simulating after the river, no folding) the hand 2 2 2 A J won, it would tell you that percentage. Granted, it would be better to simulate folding to some extent - the ideal monte carlo sim could have sliders for tightness of play and things like that.

ThatDonGuy - are those numbers from a sim you built?

Quote: Zodd

ThatDonGuy - are those numbers from a sim you built?

My numbers? Yes...and DO NOT PM ME ABOUT DOING OTHER SIMS. I only even consider "requests" posted to the public forum.