UNUSUAL CASINO POKER SITUATION

I have been having a running and often loud discussion with a friend about the following. NOBODY can figure it out.

Two casinos. Both offer high hand bonuses. One is 300 every 20 minutes, must use two cards, any full house qualifies. The other is 500 every 30 minutes, can use only one card, aces full or better. He says the 300/20 is better for two reasons: Must use two cards means less likely to get beat; any full house qualifies. I say, 1 or 2 cards makes no difference in the long run because everybody has the same distributionof luck, and the 500 vs. 300 more than makes up for ANY other factor. He says the HUGE advantage is ANY full house. I say it's so rare for ANY full house to hold up that the advantage is negligible. Now, I am not looking for an opinion. I’m sure everybody has one. But this can DEFINITELY be solved mathematically, and that’s what I’m looking for, with proof. There are too many variables for any “opinion” to be valid. Thanks in advance. I don't think it's relevant, but if you need the number of tables, say ten.

On the surface 500/30 is better because more cash is given out. But 300/20 is better because there are more winners.

The requirement of using one card vs two cards is irrelevant. While you're correct that one card means more qualifiers, and therefore more chance for your big hand to get beaten, it also means more chance for you to beat someone else's current qualifier. So that part doesn't matter.


But the big variable you didn't mention is, is the part you didn't think was relevant. How many players does each poker room typically have? That's the real deciding factor.

Quote: DJTeddyBear

But the big variable you didn't mention is, is the part you didn't think was relevant. How many players does each poker room typically have? That's the real deciding factor.

Yep! And if there are differences about the action requirements for a hand to qualify, too. Sometimes 4 or 5 people have to see the flop, or the pot has to be over $50, or some other hurdle before the hand qualifies regardless of the cards involved. If that's different at each place, it'd have an impact as well.

So far I got two responses, and I appreciate them both. However, while you may be right, you may also be wrong. I asked for mathematical proof. I think nobody short of a mathematician with a computer or maybe the Wizard himself can solve this WITH PROOF, NOT OPINION. I see no math whatsoever in either response. If you asked me which product is higher 53785*26938 or 43789x25699 you could make an eyeball guess, but you could also do the math and prove it. ANYTHING less is an opinion.

Quote: mansherman

So far I got two responses, and I appreciate them both. However, while you may be right, you may also be wrong. I asked for mathematical proof. I think nobody short of a mathematician with a computer or maybe the Wizard himself can solve this WITH PROOF, NOT OPINION. I see no math whatsoever in either response. If you asked me which product is higher 53785*26938 or 43789x25699 you could make an eyeball guess, but you could also do the math and prove it. ANYTHING less is an opinion.

43789<53785 AND 25699<26938. So, 53785*26938 must be larger than 43789*25699. No need to multiply for proof:-)

What if I told you that 53785 x 26938 was because both factors were higher without actually doing the math? Is that enough proof?

Sorry POTUS, I didnt see you typed basically the same thing as me while I was typing mine. You win! LoL

Quote: DJTeddyBear

But the big variable you didn't mention is, is the part you didn't think was relevant. How many players does each poker room typically have? That's the real deciding factor.

He said in his initial post to assume both casinos have ten poker tables. You may assume the tables are generally full, or specify an average number of players per tables.

Q: Is this a bad beat bonus, in which the full house must be beaten? Or is just being dealt such a full house enough to qualify for the bonus?

This is not a hard situation to analyze mathematically. I would do it myself but I am busy tonight with family stuff. Hopefully one of the usual suspects will step up and do this calc.

As described, it is not a high hand bonus the frequency of which would be determined by the qualifications for "high hand." The frequency and amount are specified. It is the highest hand of the specified time period. Only the time period and amounts matter. It makes absolutely no difference whatsoever what the qualifying hand is set at, given that they will be paying the highest hand among those that qualify. That one, and only that one. Unless these rooms are virtually empty, it should be intuitively obvious to anyone who plays poker at all that it is not plausible that there would be no hands that qualify in a poker room in any of the specified time intervals.

Who gets the highest of all the hands that qualify in any specified time period under either set of qualifications is equally random. It has the same effect as a random drawing of tickets or seat numbers. High hand of the hour, or of the half hour, or of the third of an hour will be paid to some random player. Raising or lowering the means by which the high hand is counted does nothing at all to change that fact.

One room randomly distributes $1000 per hour, and the other does so at the rate of $900 per hour. The only other variable is the number of players among whom those payouts will be distributed. Nothing else. The other stuff is just meaningless noise of no consequence at all, given that this kind of promo WILL pay someone the specified amount in each specified time period. Divide the hourly rate of payouts (in this case either $1000 or $900) by the number of players to determine EV of the promo payouts to each player. Then subtract the average hourly amount of the promo drop that pays for it. That is all. Nothing else. And once you do so, you will most likely discover that the net of the promo payouts minus the promo drop is something very close to zero. As it must be for this or any promo to continue, as overages in the fund will have to be paid out and shortfalls would have to be made good.

It is not unusual. It is a common way for a poker room to distribute the promo fund which accumulates from the extra promo drop that many rooms take on top of the house rake. I'd estimate there are about 15 or 20 poker rooms around Las Vegas doing some version of it right now as I type this.

I think, mathematically, it does matter that the 300 requires both cards to be part of the high hand, and the 500 can use one, the other, or both to complete it. (Assuming that's what the word "can" means in the OP conditions.) Even with all the FH's in play, there may be a larger frequency of hands that qualify on the 500 because at the AceFull point and above, there will be less hands made using both cards vs. only 1 card plus all using both cards.

I could be wrong. But I think the formula has to account for that.

I'd also like to know whether the bonuses roll over if they're not awarded, since there's a qualifying threshold for both.

No. One is paid per time period. Always one. Exactly one. No more. No less. Qualifications don't matter. That's how it was described, quite clearly, and no doubt correctly because that's how they work. They are not paying every full house or better. Nobody does. Nobody ever could or would. Almost none of the full houses, including aces full, will end up getting paid. And not due to the minimum qualification for bothering to record the "high hand." Due to the fact that it will rarely if ever be high enough to be the once per time period highest hand.

Quote: mansherman

One is 300 every 20 minutes, must use two cards, any full house qualifies. The other is 500 every 30 minutes, can use only one card, aces full or better. He says the 300/20 is better for two reasons: Must use two cards means less likely to get beat; any full house qualifies. I say, 1 or 2 cards makes no difference in the long run because everybody has the same distributionof luck, and the 500 vs. 300 more than makes up for ANY other factor. He says the HUGE advantage is ANY full house. I say it's so rare for ANY full house to hold up that the advantage is negligible. Now, I am not looking for an opinion. I’m sure everybody has one. But this can DEFINITELY be solved mathematically, and that’s what I’m looking for, with proof. There are too many variables for any “opinion” to be valid. Thanks in advance. I don't think it's relevant, but if you need the number of tables, say ten.

Quote: mansherman

So far I got two responses, and I appreciate them both. However, while you may be right, you may also be wrong. I asked for mathematical proof. I think nobody short of a mathematician with a computer or maybe the Wizard himself can solve this WITH PROOF, NOT OPINION. I see no math whatsoever in either response. If you asked me which product is higher 53785*26938 or 43789x25699 you could make an eyeball guess, but you could also do the math and prove it. ANYTHING less is an opinion.

You gave horrifically incomplete information. Therefore, nobody can do anything but guess.

He gave precise and fully complete information to determine the answer. Along with additional information that doesn't matter in any way at all, except as a procedural convenience for dealers in the room not bothering to note ridiculously not so "high" hands such as straights and flushes and smaller ranked full houses that have no chance whatsoever of being the 1 per 20min or 1 per 30min high hand. There is nothing complex about it.

Quote: DrawingDead

No. One is paid per time period. Always one. Exactly one. No more. No less. Qualifications don't matter. That's how it was described, quite clearly, and no doubt correctly because that's how they work. They are not paying every full house or better. Nobody does. Nobody ever could or would. Almost none of the full houses, including aces full, will end up getting paid. And not due to the minimum qualification for bothering to record the "high hand." Due to the fact that it will rarely if ever be high enough to be the once per time period highest hand.

Quote: mansherman

One is 300 every 20 minutes, must use two cards, any full house qualifies. The other is 500 every 30 minutes, can use only one card, aces full or better. He says the 300/20 is better for two reasons: Must use two cards means less likely to get beat; any full house qualifies. I say, 1 or 2 cards makes no difference in the long run because everybody has the same distributionof luck, and the 500 vs. 300 more than makes up for ANY other factor. He says the HUGE advantage is ANY full house. I say it's so rare for ANY full house to hold up that the advantage is negligible. Now, I am not looking for an opinion. I’m sure everybody has one. But this can DEFINITELY be solved mathematically, and that’s what I’m looking for, with proof. There are too many variables for any “opinion” to be valid. Thanks in advance. I don't think it's relevant, but if you need the number of tables, say ten.

You're kind of assuming, not without justification, that there will be a qualifying hand every time. I do think it happens occasionally, especially in a slow house, that no one would get a high enough hand during a 20 or 30 minute period. But I can see where they simply wouldn't pay it if that happened.

Quote: beachbumbabs

You're kind of assuming, not without justification, that there will be a qualifying hand every time. I do think it happens occasionally, especially in a slow house, that no one would get a high enough hand during a 20 or 30 minute period. But I can see where they simply wouldn't pay it if that happened.

A very slow house. One that is teetering on the edge of closing due to lack of business, with < about three tables running. And one that is not likely ever setting the promo payouts at these rates, unless it is deliberately donking off their fund balance before closing. They don't set promo payouts at rates that don't get hit sufficiently to pay out the promo drop. That isn't very "promotional" and can't continue for long, as the promo fund balance will have to get distributed. They can't keep it. It isn't a choice.

ADDENDUM: But yes, if it is such a slow business that there's actually doubt about whether there will be a full house or aces full in the room, say at place that's a very small room even at peak times that was somehow also foolish enough to institute and continue this promo at six-thirty on a Tuesday morning with something like one short handed table, then yes, in that peculiar unlikely circumstance (with a room manager who obviously has never been in a poker room before last week and has no supervision or accounting department) the minimum qualification could actually exceed the highest hand. And you'd best get it while the getting is good, before anyone running the place and responsible for keeping their gaming license shortly notices there's a problem rapidly brewing with the promo fund.

Given the terms of this (and many similar type promos) that use a time period to determine the frequency of a set amount to be paid, the EV per player is highest at off-peak times when there are the fewest players, if you make the large assumption that everything else is equal about the desirability of the games or that you don't lose more from tight games in off-peak hours than you gain from gaming the promo (usually promos aren't very important compared to other factors at a poker table). But it is also likely that the promo is time specific for that very reason. Poker room managers are usually not that clueless, and even when they are, their fund balance will hit them over the head with the clue.

What games are allowed is one variable. I would guess only holdem is played/allowed, but OP left out some stuff.

Are these promos run at the same time causing you to choose where to play? If yes, tell your friend he is right (and anyone else willing to listen), and then play in the other room with fewer players.

My "Spidey sense" is tingling after reading this thread....

Quote: Keeneone

What games are allowed is one variable.

Good point. I assumed hold 'em, hold 'em, and more hold 'em. Usually a room that also commonly has something like omaha running keeps a separate fund with entirely different promos, but if someone doesn't.... ahem. Sit at the omaha table and rape the hold 'em players' fund all night. There would tend to be a fierce sucking sound at the HE tables depositing most of their promo drop on the heads of the omaha folk.

Mathematically, there would be an ambiguity in the calculation of FH frequency at a Texas Hold'em table.

Consider a table where the 5 common cards are 2 2 7 3 3. A FH would be made by any player holding a 2, a 3 or a pair of 7s. But how many starting hands holding a 2 would stay in to see the flop? How many starting hands with a 3 would stay in to see both the flop and the turn?

The frequency of full houses doesn't matter unless there aren't many. In a room that supports a promo payout of $300, and specifies this high frequency rather than hourly or per two hours, they know without doubt that there are plenty. The difference in the hourly payout rate in the room if there are twenty full houses or higher vs. only ten or five or three or just one of them = zero. No difference. None. At all. Ever. Only the difference between 0 and 1 of them matters.

The minimum qualifying hand for a "highest hand of the [insert time period]" is not going to be set at a level where it isn't going to be hit. I have never seen that as a common condition in many thousands of hours, including many dozens of rooms running such promos at one time or another, and I'm pretty sure I never will, since it isn't a sustainable condition as a practical matter of managing a poker room with a promo fund. It is going to be set at a level commensurate with the volume of play, whether through rough estimate or calculation or actual experience, that results in many impulsive and intellectually lazy gambling-oriented recreational players (who are the ones that respond to promos) 'qualifying' and excitedly hoping that their hand holds up as the one that's highest, and then foolishly imagining "awww, I almost got it that time." As many as possible to feel like a lucky "(almost)-winner" for a few minutes. The more the better, without excessively slowing the games by burdening the staff with promo-crap from calling the floor over constantly for even more piddly stuff that even the most clueless newbie wandering in from the bingo parlor will understand isn't going to be getting the cheese of the hour, or kewpie doll of the half-hour, or third of an hour, or whatever.

The hands set as the theoretical minimum aren't going to realistically have any relevance at all to the final outcome of the time period. That isn't what they are for. The payout in such a room with this kind of promo with the conditions described in the OP is routinely going to be won by straight-flushes, perhaps occasionally something as low as quads if things are a little slow at the time, and not very likely something as low as aces full.

But apart from the reality, by all means carry-on with the theory if it is pleasing to do so for its own sake. But good luck with the conundrum of estimating the frequency of player behavior such as seeing a flop with 'x' and staying in the pot to the river with 'y' without reference to the experience of actual high-hand frequency per table per hour of people running high volume poker rooms with their specific player population.

Try this with your friend/antagonist, Mansherman: So let's say the qualifying hand is raised to something a little harder, let's call it 'aces full of tens or better' and so there are only a dozen or so of them among the sixteen tables in half an hour, and so only eleven of those are beat for the bonus. What is the result then, Leroy? The result is that one guy or gal gets $300. Now let's say with exactly the same players at the same tables with the same cards being dealt just as they were before, the qualifying hand is 'two cards - which may or may not make a pair or sumpin' and so there are 2720 of them in half an hour, 2,719 of them getting beat by higher 'high' hands for the bonus. What is the result now, Leroy? One guy or gal gets $300. Pay me, and shut up."

[And I officially give up. If folks still feel the 'qualifying-hand' information must matter, then by golly, it matters to ya.]

DrawingDead,

I don't see it written, but doesn't the length of both promotions play a role, along with the amount of time the player plans on staying? Wouldn't also your bank role and table limits have an effect. Although I'm sure the latter portion is not as important. Or am I just adding unnecessary details?

Quote: BoulderDamIt

DrawingDead,

I don't see it written, but doesn't the length of both promotions play a role, along with the amount of time the player plans on staying? Wouldn't also your bank role and table limits have an effect. Although I'm sure the latter portion is not as important. Or am I just adding unnecessary details?

"Just when I thought I was out, they pull me back in..."

No, I sure don't think so. The length of the promo is either 20 minutes, or else 30, in these mentioned in the OP (or may be something like hourly in some other rooms doing the same). And may, or may not, be followed by another, without any effect on those preceding or following. 20 minutes. Or 30. That's the whole enchilada. The end. And then the new beginning. And then the end, again.

Each of those intervals is independent of the others that do or don't follow. I don't follow if what is meant is the length of playing time changing the relative EV. Same value within each of the specified intervals, each independently, unaffected by one's presence or absence in others. It is a promo paying a set amount within each short time frame, with random effect. There is neither an effect per hand or per unit of time or a relative difference among the two slightly different room promos which differs when playing six hours vs. playing one hour.

With amounts and player numbers held constant, that difference varying the value over time would only potentially occur in the case of a promo with a progressive increase from an accumulation of unpaid funds over time that it was not hit, such as an automatic payout for a specific high hand, as with a progressive royal jackpot for example, or a bad beat bonus with a gradually escalating amount. Those are common too, but this is the opposite of that. The probability of any one hand being that time period's highest doesn't vary over time, and most importantly for this question neither do the payouts made in each time segment. I think people are confusing this "high hand of the [x-time period]" with something also common, but very different: those progressive jackpot high-hands.

If what is meant is that room 'A' is full of 10/20 tables playing tight, and by contrast room 'B' is exclusively 1/2 games playing loose, then there is no difference among the two when they both pay per highest hand in that room per time segment in same amount just as often. It only matters to the extent your play varies from others in the room.

[Sidebar: Tighter play could in theory result in somewhat lower promo drop paid per player, if it resulted in such tight play that hands commonly ended without seeing a flop. But that is not typically a consideration worth bothering with. Speed of play (hourly hand count) is generally much more significant than anything else for amount of drop measured per table per hour, both from the house rake and the extra promo fund chip. That's often determined by the extent to which the room has the sort of imaginary pokah-savant kiddies who must pretend they are soul reading before making the fold everybody rolling their eyes knows they are going to do after the snot impresses his girlfriend and himself with his antics. And which is graciously tolerated by sensible players because the posturing 'psychological soul readers' at poker tables are profit centers with a charming habit or making rebuys that compensate for the juvenile annoyance. But faster or slower play won't matter for one's chance of collecting on this promo, unless it is different from the other tables in the same room in the same time period.]

The only time someone's bankroll could matter to the value of this promo would be if you were so short you had it all on the table to begin a hand. Then the consequence could be that you might get all-in with a potential bonus winner without having to decide to call subsequent bets as action continued in the hand. But that effect would be the same if someone was short-stacking deliberately, regardless of their bankroll. Some do choose to do that for several reasons, but this alone wouldn't be a very good reason for anyone doing so. You might prefer low-variance games & promos over others if you are playing with little bankroll, or maybe you choose to seek high variance if you are hoping for a miracle score to get well, but that doesn't change the EV. It is not a progressive slot machine that you can potentially play until it eventually must pay out something related to the accumulation of what you individually stuffed into it. It has no more or less EV in the 200th 30 minute promo period you play with your last twenny-dollah from pawning grandma's silver, than the 1st one when sitting down days earlier with $10k on the felt. No difference.

The stakes of the game would not intrinsically change the $/hr expected value of the promo, unless indirectly through your play at your particular table becoming tighter or looser than the average of others in the room as a result of the stakes. It would represent a lesser proportion of someone's chips in play at a 10/20 game than at a 1/2 game, but only change your chance of being paid the bonus to the extent your individual play differed from others throughout the tables in the room at the time. (And if your play is affected at all by this kind of bonus, or almost any kind, you aren't usually one who is going to be leaving poker rooms with chips all that often.)

They could run a time specific promo intended to encourage play in slack periods, then change it at times the room is always overflowing anyway regardless of any promos, such as Tuesday mornings vs. Saturday nights. That is common. And they may be doing that in these two examples. But that does nothing to change the relative value among these two *IF* the two promos operating as specified are distributing it among the same number of players in room 'A' as in the promo time of room 'B.'

With any definte payout per time period, whether for high hand of the 'x' or a seat drawing or any other of that same type:

->-> (amount of set payout per hour / number of players) - (average promo drop) = value per player per hour. <-<-

He gave all that, except for the promo drop which is pretty standard. That is all that there is that determines the player value from the promo structure. He also mentioned other stuff no doubt printed on the room promo info sheets that may as well have been: "the first room is painted blue with chandeliers, and the other one is beige and has hanging ferns." The rest is just fog folks are determined to get lost in for no reason, apparently most especially his pal (who would likely be a profitable addition to a poker table), with details taken from the room's procedures for running it, without relevant effect on EV for a time specific and defined fixed-level payout promo.

I think he's getting his money's worth out of posting the thread, anyways.

I got an accurate estimate of the number of tables, so amend the question this way: Assume the $500 casino has 20 tables, ten players. Assume the 300 casino has 8 tables, ten players.

Thank you all for taking up this thread.

As I stated originally, I believe the 1 or 2 card issue is a wash. However, I can be wrong. My friend says it makes a HUGE difference. That's why I posted this question. To a much smaller extent, in my opinion, is the ANY FULL vs. ACES OVER TENS issue. I also now think, which I did not consider originally, that the 20 vs 8 tables is a huge issue. With only eight tables, if you do manage to get on the board, you have way less competition to knock you off, and your odds of getting on in the first place are the same no matter how many tables there are, WITH ONE MAJOR EXCEPTION: That is with more tables it is more likely somebody will already be on with a higher high hand than you have. Another variable is that in the 20 minute bonus it is more likely for a high hand to roll over, thus doubling the payoff. Of course, there are other variables, as stated. Originally I thought the main question is does the extra money at the one card casino outweigh the disadvantage of being more easily knocked off the board because you only need one card. Now I'm starting to think that a mathematical solution might not even be possible. Somebody else was confused about what qualifies. As stated in the original question, in the one card $500 bonus room, it is ACES FULL that qualifies. In the two card $300 room it is ANY FULL HOUSE. In neither case is it ANY HIGH HAND..

Here is the original question (before changing the number of tables to 20 and 8:

Two casinos. Both offer high hand bonuses. One is 300 every 20 minutes, must use two cards, any full house qualifies. The other is 500 every 30 minutes, can use only one card, aces full or better. He says the 300/20 is better for two reasons: Must use two cards means less likely to get beat; any full house qualifies. I say, 1 or 2 cards makes no difference in the long run because everybody has the same distribution of luck, and the 500 vs. 300 more than makes up for ANY other factor. He says the HUGE advantage is ANY full house. I say it's so rare for ANY full house to hold up that the advantage is negligible. Now, I am not looking for an opinion. I’m sure everybody has one. But this can DEFINITELY be solved mathematically, and that’s what I’m looking for, with proof. There are too many variables for any “opinion” to be valid. Thanks in advance. I don't think it's relevant, but if you need the number of tables, say ten.

Quote: mansherman

I got an accurate estimate of the number of tables, so amend the question this way: Assume the $500 casino has 20 tables, ten players. Assume the 300 casino has 8 tables, ten players.

Thank you all for taking up this thread.

As I stated originally, I believe the 1 or 2 card issue is a wash. However, I can be wrong. My friend says it makes a HUGE difference. That's why I posted this question. To a much smaller extent, in my opinion, is the ANY FULL vs. ACES OVER TENS issue. I also now think, which I did not consider originally, that the 20 vs 8 tables is a huge issue. With only eight tables, if you do manage to get on the board, you have way less competition to knock you off, and your odds of getting on in the first place are the same no matter how many tables there are, WITH ONE MAJOR EXCEPTION: That is with more tables it is more likely somebody will already be on with a higher high hand than you have. Another variable is that in the 20 minute bonus it is more likely for a high hand to roll over, thus doubling the payoff. Of course, there are other variables, as stated. Originally I thought the main question is does the extra money at the one card casino outweigh the disadvantage of being more easily knocked off the board because you only need one card. Now I'm starting to think that a mathematical solution might not even be possible. Somebody else was confused about what qualifies. As stated in the original question, in the one card $500 bonus room, it is ACES FULL that qualifies. In the two card $300 room it is ANY FULL HOUSE. In neither case is it ANY HIGH HAND..

Here is the original question (before changing the number of tables to 20 and 8:

Two casinos. Both offer high hand bonuses. One is 300 every 20 minutes, must use two cards, any full house qualifies. The other is 500 every 30 minutes, can use only one card, aces full or better. He says the 300/20 is better for two reasons: Must use two cards means less likely to get beat; any full house qualifies. I say, 1 or 2 cards makes no difference in the long run because everybody has the same distribution of luck, and the 500 vs. 300 more than makes up for ANY other factor. He says the HUGE advantage is ANY full house. I say it's so rare for ANY full house to hold up that the advantage is negligible. Now, I am not looking for an opinion. I’m sure everybody has one. But this can DEFINITELY be solved mathematically, and that’s what I’m looking for, with proof. There are too many variables for any “opinion” to be valid. Thanks in advance. I don't think it's relevant, but if you need the number of tables, say ten.

Lost my (longer) reply. Ugh. Recap:

Nobody is arguing for no qualifier; DD is saying it's irrelevant because it's virtually never a factor. I'm saying it's probably a minute one, depending on the amount of players, happens occasionally. It may be so rare as to be mathematically insignificant. But if the money rolls over, it probably cancels out on profitability over time (new info from you, above). If it DOESN'T roll over, then it lessens the profitability over time by some fraction.

Difference of opinion on 1 card vs. must use 2 cards. DD says not relevant; I say it is, though I don't know how much without the math.

First calculation: using these WoO tables, there are, in 7 cards:

All possible hands: 133784560

Full house and above: 3473184
Full house Aces up and above: 537824

Percent of hands containing FH and up: 0.0279525, or about 2.8% (1 in 35.77)
Percent of hands containing FH Aces up: 0.0040201, or about .4% (1 in 248.75)

What I don't know how to calculate is being forced to use 2 + any 3 of 5, vs. forced to use 1 or 2 + any 4 or 3 of 5. Both of those modify the numbers above by making all hands rarer, since they both eliminate hands that could fully fall on the board, and in the case of forced 2, eliminate hands that use 1 + any 4 board cards. And the 1+4 would have to be considered separately for EACH of the 1, less duplications like pairs where only 1 fits, or both ends of a SF in hand.

It would be those totals, minus the subsets of card orders that don't qualify (since the original numbers don't discount for order received). In the case of MUST use 2, I would think the percentage drop would be significant, and where you can use either 1, or 2, it would still matter, but not nearly as much.

However, the higher qualifier is the one that can use 1 card or 2 in hand. The lower must use both. So that's a balancing mechanism that may wind up close to cancelling out the threshold difference.

Once you had those figured out, you could use DD's suggested formula for the larger calculation, modified by these qualifiers.

Found your sidebar particularly amusing, DD:

Quote: DD

Sidebar: Tighter play could in theory result in somewhat lower promo drop paid per player, if it resulted in such tight play that hands commonly ended without seeing a flop. But that is not typically a consideration worth bothering with. Speed of play (hourly hand count) is generally much more significant than anything else for amount of drop measured per table per hour, both from the house rake and the extra promo fund chip. That's often determined by the extent to which the room has the sort of imaginary pokah-savant kiddies who must pretend they are soul reading before making the fold everybody rolling their eyes knows they are going to do after the snot impresses his girlfriend and himself with his antics. And which is graciously tolerated by sensible players because the posturing 'psychological soul readers' at poker tables are profit centers with a charming habit or making rebuys that compensate for the juvenile annoyance. But faster or slower play won't matter for one's chance of collecting on this promo, unless it is different from the other tables in the same room in the same time period.

Saw a lot of these guys in WSOP events last year. Hysterical commentary from the box (David Tuchman et al) during many of them.

Here's the schedule for 2015 WSOP coverage, beginning May 27.

This post will be somewhat off-topic from what I take to be the main questions you intended to pose. So by all means ignore it as a side comment.

From the information you added about the size of the rooms, one thing I can immediately see from very simple arithmetic is the likely promo drop in each room vs. the scheduled promo payout amounts. And if everything is paid out as scheduled, and these rooms have a standard single $1 chip per hand promo rake, this promo is not self-funding. More is scheduled to be paid out per hour than a standard promo fund drop would collect from the tables. That's unusual.

The $500 per half hour room is stipulated to have 20 tables. If one assumes all tables always have active games running, and that they are getting out 35 hands an hour (which is a pretty high hand count, but within the high end of what is plausible in live poker hold 'em games with experienced players and highly skilled staff), and that all hands dealt take a $1 promo drop, then with these optimistic assumptions $35/hr. * 20 tables = $700 per hour, while the promo distributes $500 per half hour & $1,000 per hour creating a shortfall of $300/hr, or at least a buck and a half per hour per player if all seats are always filled in that room.

The $300 per 20 min room is stipulated to have only 8 tables, and has a bigger shortfall if funding the promo from the usual standard $1/hand promotional drop taken in most rooms with such promos. Staying with the deliberately high assumptions made above on hand count and all tables always remaining active, we have 8 tables * 35 = $280/hr going into the promo fund box, while distributing $300 per 20 minutes = $900/hr, which is obviously not sustainable if intended to be self-funding from the players under the most common structure of funding these things.

And the amount scheduled to be paid out obviously exceeds the amount players pay in promo drop by the largest total amount per hour ($620) and per player (at the astounding rate of $7.75/hr per player) in the second room, creating a huge potential overlay benefiting the players at the expense of depleting the fund or costing the casino to subsidize it from their own house rake, assuming there isn't something highly unusual about the promo funding mechanism used in these rooms.

But now back to what you wanted to get responses about. I think you'll find it difficult to get very real about it through calculation of qualifying hand results, because as actually played it is heavily dependent on player behavior in betting, calling, raising, and folding their hands. Much more so than what any theoretical mathematical modeling construct can tell you that it could be. But good luck with seeking that.

Quote: beachbumbabs

First calculation: using these WoO tables, there are, in 7 cards:

All possible hands: 133784560

Full house and above: 3473184
Full house Aces up and above: 537824

Percent of hands containing FH and up: 0.0279525, or about 2.8% (1 in 35.77)
Percent of hands containing FH Aces up: 0.0040201, or about .4% (1 in 248.75)

Babs, I started out this way as well but then realized that this is mis-leading. The possibility that n hold'em players will not get a FH in a short time period must take into account the fact that on each deal at each table the outcome of the players hands are coupled by the fact that they all share the flop, turn and river cards.

Let's look at the 'Aces-Over FH' calculation (it is easier) and start like this:

Five common cards
- do not contain an Ace. Prob= 65.9% Zero probability any player has an Aces Over FH.

- contain one Ace and no pair Prob =19.5% Zero prob any player has Aces Over FH

- contains one ace and one pair Prob= 3.25% Player has Aces Over FH only when he/she has a pair of Aces
How often does a pair of aces (pre-flop) get to the flop and then to the showdown?

- contains two aces and no pair Prob = 3.25% Player with an Ace and another card that pairs the board will make an Aces-Over FH. But keep in mind that players with a weak Ace will often fold in early position. And if the flop comes with 2 Aces and the player with A-x bets, then the other players may fold - thus no showdown (and no turn and river card which may be the ones that pair the player's other card, forming the AAA-xx FH.)

and so on. One needs to go through all of the scenarios and consider each one separately. The problem with an AAA-xx FH is that it is such a strong hand that it often does not go to showdown. Indeed, once a player has either trip aces or two pair AA-xx, there is a chance they will bet out and try to end the hand in order to prevent an opponent making a straight or flush. Remember also that a significant fraction of hands end with a continuation bet after the flop and some hands end with a pre-flop raise. DD may indeed be correct that 8-20 poker tables will almost always produce an AAA-xx FH in a 20 or 30 minute window -but I would like to see an orderly analysis that supports that assertion. Again, I'm not disagreeing with DD -I would just like to see the analysis.

Quote: gordonm888

I'm not disagreeing with DD -I would just like to hear the analysis.

I might disagree with DrawingDead. At only eight tables, and using the short time interval of only 20 minutes rather than something like the common hourly hand, from my experience my assertion that it won't be an issue could start to get dicey.

Quote: gordonm888

Quote: beachbumbabs

First calculation: using these WoO tables, there are, in 7 cards:

All possible hands: 133784560

Full house and above: 3473184
Full house Aces up and above: 537824

Percent of hands containing FH and up: 0.0279525, or about 2.8% (1 in 35.77)
Percent of hands containing FH Aces up: 0.0040201, or about .4% (1 in 248.75)

Babs, I started out this way as well but then realized that this is mis-leading. The possibility that n hold'em players will not get a FH in a short time period must take into account the fact that on each deal at each table the outcome of the players hands are coupled by the fact that they all share the flop, turn and river cards.

Let's look at the 'Aces-Over FH' calculation (it is easier) and start like this:

Five common cards
- do not contain an Ace. Prob= 65.9% Zero probability any player has an Aces Over FH.

- contain one Ace and no pair Prob =19.5% Zero prob any player has Aces Over FH

- contains one ace and one pair Prob= 3.25% Player has Aces Over FH only when he/she has a pair of Aces
How often does a pair of aces (pre-flop) get to the flop and then to the showdown?

- contains two aces and no pair Prob = 3.25% Player with an Ace and another card that pairs the board will make an Aces-Over FH. But keep in mind that players with a weak Ace will often fold in early position. And if the flop comes with 2 Aces and the player with A-x bets, then the other players may fold - thus no showdown (and no turn and river card which may be the ones that pair the player's other card, forming the AAA-xx FH.)

and so on. One needs to go through all of the scenarios and consider each one separately. The problem with an AAA-xx FH is that it is such a strong hand that it often does not go to showdown. Indeed, once a player has either trip aces or two pair AA-xx, there is a chance they will bet out and try to end the hand in order to prevent an opponent making a straight or flush. Remember also that a significant fraction of hands end with a continuation bet after the flop and some hands end with a pre-flop raise. DD may indeed be correct that 8-20 poker tables will almost always produce an AAA-xx FH in a 20 or 30 minute window -but I would like to see an orderly analysis that supports that assertion. Again, I'm not disagreeing with DD -I would just like to see the analysis.

I agree with what you say, Gordon; I mentioned part of it in the further discussion you quoted from, but your point is even more pertinent, in that hands have to make it to showdown to even be considered. No way of knowing how many of those happen, beyond DD's bringing up loose/tight tendencies earlier in the thread.

The threshold math question of estimating whether there will always or almost always be a payout in each time period is interesting and may become relevant depending on factors like the volume of business in the room, and the precise calculation of the theoretical expected number of hands meeting that threshold would be a useful number to have to then get at estimating the lower number of hands as actually played. But, apparently even after that threshold is assumed to be crossed the OP and/or the other party in the dispute seems to be stuck on thinking somehow in some way it could possibly matter what hands the poker room notes as potentially qualified before paying the one and only one that is highest, or that how they go about doing so makes it 'easier' or 'harder' in a way that changes the expected value of it even after stipulating that it is being paid to one.

Quote: mansherman

Originally I thought the main question is does the extra money at the one card casino outweigh the disadvantage of being more easily knocked off the board because you only need one card. Now I'm starting to think that a mathematical solution might not even be possible. Somebody else was confused about what qualifies. As stated in the original question, in the one card $500 bonus room, it is ACES FULL that qualifies. In the two card $300 room it is ANY FULL HOUSE. In neither case is it ANY HIGH HAND..

I don't think this will do any good, but I'll give it one last shot. There is no such "disadvantage" provided that the qualifier is not set unrealistically high and therefore there is actually *a* hand that is counted as the one highest. So long as the qualifier is such that the bonus is paid, it makes no difference how many lower high hands there are. This is not an opinion. This is a problem of basic fundamental logic which is either intuitively obvious or it is not. This fact is not a math problem.

Making it easier or harder to have what they choose to note as a possible 'high hand' for this purpose changes nothing about the expected value per player, provided that there is one that is paid. There is no calculation to help with that. There is no confusion involved about what qualifies. It doesn't matter what qualifies, or how many, so long as there is one. Only one gets paid. The highest one of that time period. Doesn't matter if it is made 'easier' or 'harder' so long as there is one. The resulting number is the same. The number is one. Exactly one. Only one. The highest one.

The value of the random one does not change as the means of determining it results in it being of higher or lower rank, among more or fewer others that are noted. The distribution of cards is equally random, and does not become more or less so depending on whether two hole cards must play, only one, or if the player to receive the one payout is determined by the number of sequins on the cocktail waitress uniform. Provided that there is one per specified time period it remains random, and continues to be one payout randomly distributed among those playing in the room. There is no math to be done to further reveal that fundamental aspect of it.

If, on the other hand, there is often no qualifying hand at all in some time periods, than everything I just wrote above is wrong. Then, and only then, the method of qualifying matters a lot for the expected value of the promo. Good luck.