Hello everybody. I am a casino executive and I am considering starting a new promotion in our poker room. I have an idea for the following promotion, but I need help in figuring out how often it will hit.

I would like to pay out a jackpot whenever somebody loses with Aces full.
Could anybody tell me what the probability is of getting Aces full beaten in a hold 'em game? The parameters are that both hole cards must play in the LOSING hand only and the hand must go to the river. Any aces full will do- even AAA22.

I really appreciate any help!

This would likely require a computer simulation; I don't think it's feasible to come up with an analytical solution, at least not for a full-ring game.

A simulation would be fairly easy, but... as a casino executive, shouldn't you hire someone to do it? I know someone you could hire -- heck, I wouldn't even cost that much! (And my credentials are unimpeachable!)

BTW, it's going to be roughly in the neighborhood of 1 out of every 200,000 hands (for a table, not a given player). Something like that.

I have done this type of problem analytically for a cardroom before, and it is feasible. I even had a model that attempted to take into account the way people play. It would be easy to adapt that to the present problem. Problems should always be done with a manual calculation and then verified with a simulation whenever possible. When you get the same answer both ways you can be confident that it's right. Never blindly trust a simulation.


The wizard of odds has extensive simulation data here. He has about 1 in 21,700 for a 10 player table if both hole cards must play for both players, and 1 in 5900 if either player can use any 5 cards. Again, I strongly suggest that these be verified by manual calculation and independent simulation.

I'm talking about real-world solution that takes into account how people play, not a no-foldem analysis. You're saying you have an analytical solution to that problem? (And when I say "analytical solution", I mean an exact calculation, not an approximate model.) With all due respect, I don't think that's possible. But feel free to prove me wrong.

That's a no-foldem scenario. Obviously jackpots are going to be much rarer when you don't have all ten players going to the river every hand.

Turbo Texas Holdem counts BBJ's, and IIRC (I will re-run some this afternoon if I have time), it's closer to 1 in 200,000 for their simulated players (2-4 players to a flop, typically).

OK, I ran some more TTH simulations. It suggests an Aces-Full BBJ (where both hole cards must play, and no kickers allowed) happens about once every 38,000 deals. (My recollection of 200,000 was for quads beat.) If you allow kickers in the quads, it becomes much more likely, so that's an important detail of the rules to define.

This is based on a limit holdem simulation where players are probably somewhat looser than in a low or midstakes NL game. If you have a NL game where typically 2-3 players see a flop, and there's often no showdown, it's going to be comparatively rarer than 1 in 38,000.

It was an exact calculation with some reasonable stated assumptions about how often people would continue with certain hands. When people are going to make say quads by the river, there are only certain hole cards and flops that they could have had, and many of these will be continued almost automatically. The cardroom manager didn't even ask for that analysis, but I provided it anyway. He just wanted the odds for a no-fold'em scenario as a conservative estimate because it seemed to be happening too often. It was enlightening to demonstrate that they will happen more often than he would think even once you know the probability just because of the character of the exponential distribution. People are often surprised when another one happens again just a day after the last one, but of course that's the most likely day that it will happen again.

How do you handle the scenario where two players hit straight flushes with A2s and 67s or whatever? Or where a straight flush holding 67s beats quads with 22?

I think you'd have to make an awful lot of assumptions about how people play these kinds of hands.

That's proprietary.

Fair enough!

But I would think that at some point, you'd end up having to make so many assumptions that you might as well make it a simulation -- which is, essentially, a very elaborate set of assumptions about how people play. But it can at least take into account how players might react to raises, re-raises, check-raises, flop textures, bluffing, etc. etc. etc.

I just don't see how you can deal with all that stuff analytically while also being fairly realistic.

If your answer is affected by all those things, then you should certainly NOT be trying to take them into account. A Turbo Texas Hold'em simulation requires you enter a huge number of parameters for each player. That's great if the casino exec is only interested in offering his promotion to your simulated Turbo Texas Hold'em players. The parameters you mention will vary widely depending on the location of this cardroom, and the limits that are being played. We have no idea what those parameters should be correctly averaged over all of the players in his cardroom.

It isn't the goal of a mathematical model to try and account for everything in the world that can happen. I don't make assumptions about how every type of hand will be played. I make only a handful of assumptions about how often certain hands will be continued. These assumptions could be varied by the client by changing numbers in a spreadsheet. The goal isn't to provide a super accurate number that will match his exact situation because that's impossible. The goal is to give some idea about how much the answer can change from a no-fold'em scenario based on reasonable assumptions about things that we KNOW are going to happen. And chances are the OP would be happy with even a no-fold'em scenario because it provides a conservative estimate.

You handle this with sensitivity analysis. I suspect that many of those parameters would have a negligible impact on the results. But for the ones that do impact the results, then it seems to me that you want to know about them. Those are precisely the kinds of things your assumption/model-based approach is likely to miss.

Why not? If the exec is really interested in accuracy, we could spend some time gathering data. It wouldn't be that hard to do -- just spend a day or two looking at the cameras.

For starters, you'd want to know how many players go to the river on average, and what kinds of hands they're winning and losing with. If you could get your simulated players to approximate something close to that, you would likely have something in the right ballpark. (BTW, TTH gathers all these stats automatically.)

OK, but the question is whether the results from your model would be any better than the simulation. If it matters that the Turbo players are not like real players, then it matters that your assumed players are not like real players.

Of course, I obviously can't argue with your assumptions without knowing what they are.

Sure, if that data is available, or the client is willing to pay to fly you to his location and for your time to get it. It's completely unnecessary if a simple model gives him an answer that satisfies his needs. You know it won't happen any MORE often than what the no-fold'em model says. The only reason to get a more refined estimate is if he wanted to change the qualifying hands for the BBJ because it didn't happen often enough and people are becoming bored. That's not very likely for a BBJ jackpot, and we have no reason to think that the OP wants to do this.

I would spend my time making sure that my results agree with the sim for whatever assumptions that the OP wants to make. That isn't trivial. I don't know how much time you've spent thinking about this problem, but I have over 100 man hours into it. I value accuracy. I know based on your past statements that you think producing an answer that is off by 28% is no big deal even when the answer can be just as easily computed almost exactly, but I find that sort of thing unacceptable.


The results from the model are better because the model is chosen to match the needs of the client.

Dude... JFC... it was a response to a student on the internet about an exam question, from an exam he had already taken nonetheless. I even told him the answer I was giving him was an approximation, and I told him that because I was reasonably sure (and correctly so) that the full, exact answer (which I also gave him) was over his head. And your "28%" number was a ratio -- not the absolute error -- which makes a big difference when talking about small percentages.

But if you want to make it personal, I can assure you Bruce, I care about accuracy. I have done statistical simulations for several large federal agencies, and I taught our simulation techniques to some of the most sophisticated actuaries around. When someone gives me a grant or a contract to do work for them, I assure you sir, it is accurate.

Now I'm going to bow out of this discussion, because I don't appreciate it when people launch into personal attacks. Sad to see it coming from a moderator.

So teaching a student starting out in life who comes here trying to learn is somehow less important??? I would invoke the saying "Watch how a man handles the smallest of things, and that will tell you volumes about how he will handle everything else", except that this isn't a small thing. Mentoring the young is one of the most important and lasting impacts that we make on this world.


First of all, approximations generally come with an idea of how accurate they are, especially when they are crappy non-standard approximations that no one should ever make. Second of all, if you are going to use an independence approximation, you should at least choose variables that are as close to independent as possible, which means that you must include draws. To do otherwise just makes the already bad approximation twice as bad for no good reason. Third of all, even if he wasn't expected to know what a covariance was, it was certainly not over his head to understand this very simple concept. Fourth of all, the answer was off by 28%. It's the difference between 1 in 29.4 and 1 in 21.3. If the answer was 1 in a thousand, and you said it was 1 in a million, that's an absolute error of less than 0.001, but more importantly, it's an error of 3 orders of magnitude no matter what units you use to measure it.


Just make sure you teach them to use that multinomial covariance to determine how many simulation iterations they need for a certain confidence whenever they're simulating an EV of a game where there can be draws.


So does that mean that you never trust unproven software that you didn't write yourself? Like normal distribution programs off the web that give wrong answers? After I told them their normal program was wrong, they sent me a form letter that said "Thank you for your interest in our site. The problem is fixed." But it wasn't fixed. All they did was average 2 values in their table some of the time, and because of a bug they didn't even do that right, and it still produced wrong answers. So I rewrote their code for them to do a proper linear interpolation. I never heard back from them again, so I looked at their site, and they had implemented a series solution using Horner's method that looked like it came out of a numerical recipes book. I haven't checked it carefully, but it is now correct for the few cases that I have checked. They just didn't bother to tell me. I don't think they like me. I don't know why as I was perfectly sweet and kind to them as I am to everyone. A mathematical site with that kind of process and those types of errors is completely worthless and should never be trusted by anyone. That goes for just about every website that does calculations that I see nowadays, except perhaps for something like wolframalpha.com.

I should probably be careful though. I heard that Ted Kaczynski was correcting a bunch of stuff like this just before he lost it and became the Unabomber.


That wasn't a personal attack. It was a statement of fact. If I wanted to act like a moderator ITT, I'd throw you an infraction for solicitation of services. That is a clear violation of 2+2 policy. We have a marketplace forum for that where you must pay to place an ad. Don't do it again.

I suppose it would be, except, Newsflash: OP is not actually a "casino exec". LOL.

I'm just gonna leave that there.