Counting combinations in Texas Hold'em Ultimate

Good morning,
I am a mathematician and aspiring casino game developer, recently dealing with this very interesting poker variant "Texas Hold'em Ultimate".

Needless to say, google sent me here.

With reference to the Wizard analysis (/games/ultimate-texas-hold-em/), I would like to understand how the combinations of different scenarios are computed.
To be precise, I am referring to the “Analysis” table, which lists the probabilities of a certain event to occur, together with the relative contribution to the total House Edge (under the assumption of optimal strategy). We have 27,813,810,024,000 of possible outcomes, and I understand that this number is given by C(52,2)*C(50,3)*C(47,2)*C(45,2), where C is the binomial coefficient.

However, I am having a hard time trying to figure out how to compute theoretically the number of combinations for any specific event, i.e. why the number of combinations for the event: “the dealer qualifies, the poker hand is less than a straight, and the player wins, in the assumption of a 2X raise” is exactly 3,671,050,165,880 (first item in the Analysis table on the Wizard of Odds website)?

Are those numbers of combinations been derived analytically or are they the result of numerical simulations?

Thank you
m

Welcome to the forum.

Link fixed, for convenience: https://wizardofodds.com/games/ultimate-texas-hold-em/
(Don't worry, it's not your fault.)


I'm reasonably sure that those are calculated numbers, not simulated.

Thank you for the link fix, and for your reply.
I see, indeed they are too precise to be simulated.. But how would you compute them?
My understanding is that to compute, for example, the number of combinations giving a Royal Flush, I should add all the combinations of getting it by the flop, the turn and the river, and for any given suitable starting hand, then adding the assumption that the dealer qualifies or not.. and then add this possible combinations together? It looks too convoluted, but maybe this is the only way to get an exact result?

Quote: Matteo

Thank you for the link fix, and for your reply.
I see, indeed they are too precise to be simulated.. But how would you compute them?
My understanding is that to compute, for example, the number of combinations giving a Royal Flush, I should add all the combinations of getting it by the flop, the turn and the river, and for any given suitable starting hand, then adding the assumption that the dealer qualifies or not.. and then add this possible combinations together? It looks too convoluted, but maybe this is the only way to get an exact result?
link to original post

Yes, this is how it is done. But it is even more complicated than you describe. First, you need to find out which choice is optimum for every single one of the 21 billion possible river raise/fold decisions for every starting hand consisting of your two hole cards. Without doing this, you cannot possibly know what the EV is for getting to the river and therefore you do not know the precise EV of any of the earlier playing decisions. Then you need to go back to the flop raise/check decision, and then the initial raise/check decision.

You can make the task easier by realizing that the EV for pocket aces is the same no matter what suits the aces are, etc. You only need to calculate the EV for 169 unique starting hands.

Modern computers are very fast, but it helps if you are programming in a language that allows you to manipulate bits and do floating point math efficiently. C++ is a good choice. This is not a particularly difficult programming problem. If you tackle it, start by looking at the river decision. There are only two dealer cards to consider, and there are only 45 unseen cards.

Casino Holdem is a simpler game than UTH, with only one decision point. But the number of possible combinations is the same as for UTH. I can calculate the EV for Casino Holdem in 158 minutes (exactly, no simulations).
https://wizardofvegas.com/forum/gambling/tables/38937-casino-holdem-poker-revisited/

Thanks Mental,
your explanation clarified my first idea on how to compute all that. Also, I think that the trick of using equivalence classes will save me a lot of time!

btw, is it possible to retrieve the detailed procedure and calculation to compute all the combinations above?
maybe there is a book were I can look into ?
cheers
m

Quote: Matteo

btw, is it possible to retrieve the detailed procedure and calculation to compute all the combinations above?
maybe there is a book were I can look into ?
cheers
m
link to original post

What is your interest in the problem? If you just want to know the EV and basic strategy, multiple people have published it including our own Wizard. There are online calculators that will give you the correct EV for each set of cards.

If you want to learn how to do the calculations yourself, then the best way to learn is by doing. I might be able to help, but I don't even know your level of programming skills or what you are trying to get out of the project.

I am fairly certain that the Wizard's calculations were done with a "looping code" that he writes that lists every single possible combination of cards for player, dealer, 3-card flop and 2 card river and evaluates the poker hands for each combination and remembers the frequency of outcomes. I'm not sure exactly how decisions in Wizard's looping code are made/optimized at the three decision points.

I believe that Wizard does not use "suit-folding" to reduce the runtime in this type of calculation.

thanks mental for your question. indeed I don't care about the EV themselves, I am not a gambler and as you said they are online already.. but the math behind this poker variant looks so challenging - yet doable given my expertise in math and coding - that I cannot help but trying to do it myself. The process itself which led the Wizard to obtain his results is somehow cryptic to me though, this is why I came here to ask about the method for deriving the aforementioned combinations

BTW there is one thing that I am not sure I have fully understood.
what does exactly mean that this game has the following returns:

Total main bet: 99.47%
Initial main bet: 97.82%
Best Five bonus bet: 96.50%.

In particular, could you help me understand what is the difference between the total main bet and the main bet? is the main bet just the Ante + Blind assuming no raise at all during the game?

Also, I promise this is my last question, how it is known that the maximum RTP is actually 99,47%? I cannot find anywhere a justification of this fact..
thanks in advance!