Help me solve a poker math problem

Hello! New to this forum. I've done some searching on the internet for the answer to this but can't find anything. I started trying to solve it myself but it looks like it's going to be very time consuming the way i'm doing it so I'm looking for a better method or if anyone knows a site where the probabilities are already listed.

1) What are the odds of making AAAxx (aces full of anything) or better using both hole cards in texas holdem? (so, AAAxx, quads, and straight flushes).

2) What are the odds for each hand individually (using both hole cards)?
I.e.
the odds of making AAA22 are a (you can have A2 in your hand on AA2xx or AA in your hand on A22xx or 22 in your hand on AAAxx)
the odds of making AAA33 are b
....
the odds of making 7777x are c (not necessarily using pocket pairs. So, the board can be 77xxx and you can hold 77 in your hand, or the board can be 777xx and you hold A7 in your hand)
the odds of making 8888x are d
...
etc.

In south florida most of the poker rooms have a 'high hand' promo where they give away $300-$1000 for making aces full or better in a 30 minute period. But if someone beats your high hand in that 30 minute period, they win. Lots of people will get a high hand with 5-30 minutes left in the high hand period and sell it to someone else. i.e. I have 7777 with 15 minutes left, high hand is $500 and I sell it to someone for 250 meaning that if I win it, they get the 500 and I keep the 250. If I lose it, I still keep the 250.
I'm trying to calculate the EV of selling a high hand for the different ranks of hands with different times left. I've already done it for using 0 hole cards, but i'm having trouble doing it using both hole cards.

Hi Sunchips, and welcome to the forums.

You should not have sold your $500 high hand of quad 7's with 15 minutes left. Math aside, I'll tell you this right now.

The questions you have would be slightly time consuming, but I can point you in the right direction... You can start with pai gow. 7 cards (just like hold'em 2 in the hand and 5 community). From there you can see the odds of getting a full house/etc... Then you can multiply by the odds of it being a full house with aces, etc. That could be your base.

Normally, high hands (in most the poker rooms I've played in) do not require you to use both your hole cards but simply take your best 5 card hand in to account. Having to use both hole cards would add a bit of complexity, and definitely lessen the likelihood of people hitting high hands (again making your selling point of $250 at 15 minutes left a poor choice).

I pretty much wouldn't sell my high hand if I made quads. I'd consider selling an Aces-Full hand, if there's still 20 minutes left or more. 30 minutes, while yes people will hit qualifying hands, is not a big window. If you indeed must use both hole cards this would defintiely lessen the amount of 'high hands.' Another thing to take in to account is the size of the poker room and the number of tables running / active players. If you're playing a smaller room I'd rarely sell the hand. If you're playing a medium sized room (20-30 tables) then I would consider selling low quads/etc with a good chunk of time left (and I'd sell for more like 75%). If you're in a large poker room (50+ tables) then I'd look to sell anything less than quad 10's with a majority amount of time left (but again, sell for more like 75%).


Simple... P(four 7's) = 4/52 * 3/51 * 2/50 * 1/49 = .077 * .059 * .04 * .02 = .0000036, or about 1 in 275,000....

I hope you can see why this depends on the size of the poker room. If you have 10 tables playing 1 hand a minute then you're playing 10 hands per minute, or 300 hands per half hour. The odds of someone beating you is dismal if you have quad 7's.

However, if you're playing a large poker room, with 100 tables, playing 1 hand per minute, then this would be 3,000 hands per half hour. While still quite the dismal odds of beating your quad 7's, this is still 10x more likely than in the small poker room example.

EDIT - I also simply googled "odds of quads in poker" and got a slue of sites that could probably give you some numbers you could find of use. FYI.

The 7777 hand was just an example... I normally don't sell high hands unless I know I am getting a very good deal. Also, your math for getting 7777 is only correct if you are playing 4 card stud. The actual odds for getting 7777 out of the 7 cards you can use in holdem is .000129 (I took this from the wizard of odds site)

I've already done the math for it with 0 hole cards. I actually have an excel sheet that takes in to account # of tables being played, # of different types of game (more players to a flop in low stakes limit games than no limit games), time left on clock, and strength of high hand. So I can calculate the amount you have to sell the high hand for in order to at least break even under different conditions.

For instance, these are the values of 2222x using 0-2 hole cards in a room with 3 low stakes limit tables, 7 low stakes nl tables, and 1 midstakes nl table with 10-15 min left for a $500 high hand (assuming 25 hands per hour per table).

10 $245
11 $228
12 $212
13 $197
14 $184
15 $171


Most high hands in south florida require two hole cards, not just best 5 card hand. That is why i'm looking for these numbers specifically.

Hence why I suggested a simple starting point such as Pai Gow... with 7 cards, where the odds of quads = 0.00199472. A lot of posters don't know how to calculate basic odds, and this is just a direction I was pointing in due to seeing no math in your opening post / first post on the site. Overall the info could be good to know, but if you're attempting to make a chart of when to sell a hand and when not to, I think you'll find it to be a waste of your time as most of the answers are fairly obvious. If you play poker a lot, you'll have a good idea where you stand with your hands / time left alone.

I play poker full time, just kind of doing this mostly out of curiosity (plus the added benefit of knowing what break even selling points are to make some extra dough if I get a high hand (though that is pretty rare)). I thought it would be pretty obvious in the first place, but after doing the math for 0-2 hole cards and finding that hitting a high hand like 2222 has such a small expected value... only $60 bucks with 30 minutes left in an 11 table room with $500 high hand it seems like selling high hands could be much more profitable than expected. I would have never sold 2222 on a $500 high hand in a 0-2 card room for $100 with 30 min left. But now I know that $100 is +40 over EV so I would gladly sell it.

Quote: Sunchips

Hello! New to this forum. I've done some searching on the internet for the answer to this but can't find anything. I started trying to solve it myself but it looks like it's going to be very time consuming the way i'm doing it so I'm looking for a better method or if anyone knows a site where the probabilities are already listed.

1) What are the odds of making AAAxx (aces full of anything) or better using both hole cards in texas holdem? (so, AAAxx, quads, and straight flushes).

Question #1: do you want the odds based on what they are before you see your hole cards, or do you want 169 separate sets of odds, as there are 169 distinct pairs of hole cards (A2 offsuit, A3 offsuit, ..., QK offsuit, A2 suited, A3 suited, ..., QK suited, AA, 22, ..., KK)?

Question #2: what if you can make Aces full or better with both hole cards, but you can make a better hand with one or none - does that count? For example, if your hole cards are AQ and the table has AAAQ6, you can make Aces full of Queens with both hole cards, but four Aces with just one.

Pardon for being Captain Obvious, but for the purposes of the OP (if I understand it correctly) of course it really isn't just a math problem. Yes, what is asked in the OP, knowing what the probability would be in some alternative universe where everybody always held everything to the river could help as a useful starting point for getting an upper limit in estimating what I think you care about, but. An unknown and unknowable but definitely non-trivial number of hands that would have qualified, had they been held to the river regardless of the board and the betting action a player faced on earlier rounds, will instead quite sensibly and rightly be folded. That part is obviously dependent on the sort of players and their behavior, what games are running, often related to the stakes and structure, even the time of day (and player sobriety or lack thereof) and how tables are playing with the individuals at the time. But for high hand bonuses like this even the loosest little 2/4 limit table will of course be significantly affected by player decisions to fold.

Leroy in seat 5 is dealt Ac6h, seat 2 raises, seat 3 raises, seat 4 three-bets, seat 2 is already reaching for his stack perhaps to shove....

...and/or...

Flop is single suited Ks,Qs,6s, and action in front of Leroy is similarly frisky as in the pre-flop example...

Of course if you've played a lot you know that for poor Leroy the turn brought the Ad, and then the river came As, so the final board ran out: Ks,Qs,6s-Ad-As. Similar examples for quads and higher. Happens all the time, often enough to me. What are the chances Leroy is still in the hand? If he is, poker must be a VERY expensive hobby for him on most days, and he probably won't be able to afford playing it long, even with his luckbox high-hand bonus.

Quote: Sunchips

Hello! New to this forum. I've done some searching on the internet for the answer to this but can't find anything. I started trying to solve it myself but it looks like it's going to be very time consuming the way i'm doing it so I'm looking for a better method or if anyone knows a site where the probabilities are already listed.

1) What are the odds of making AAAxx (aces full of anything) or better using both hole cards in texas holdem? (so, AAAxx, quads, and straight flushes).

2) What are the odds for each hand individually (using both hole cards)?
I.e.
the odds of making AAA22 are a (you can have A2 in your hand on AA2xx or AA in your hand on A22xx or 22 in your hand on AAAxx)
the odds of making AAA33 are b
....
the odds of making 7777x are c (not necessarily using pocket pairs. So, the board can be 77xxx and you can hold 77 in your hand, or the board can be 777xx and you hold A7 in your hand)
the odds of making 8888x are d
...
etc.

In south florida most of the poker rooms have a 'high hand' promo where they give away $300-$1000 for making aces full or better in a 30 minute period. But if someone beats your high hand in that 30 minute period, they win. Lots of people will get a high hand with 5-30 minutes left in the high hand period and sell it to someone else. i.e. I have 7777 with 15 minutes left, high hand is $500 and I sell it to someone for 250 meaning that if I win it, they get the 500 and I keep the 250. If I lose it, I still keep the 250.
I'm trying to calculate the EV of selling a high hand for the different ranks of hands with different times left. I've already done it for using 0 hole cards, but i'm having trouble doing it using both hole cards.