on a 224 flop AA is beaten only by 44, and a 2. Let's say you are playing 3 handed in omaha. Your opponent's have 8 cards in total. How often will one of them have a two? Let's ignore for the moment the selection bias issue that people are less likely to play two's and just look at the raw probability.

We know 7 cards. So there are 45 more. So would you

2/45 to the 8th power?
Or would you express 2/45 in % form (or the inverse probability actually, which is roughly 95.5556%) and then multiply that by 2/44 (well inverse) 95.4546% and then 2/43, 2/42, 2/41 ... 2/38?

You're on the right track. You're correct to use the complements (what you called inverses). Write the product with the complements instead of 2/_, and then ask yourself, what probability does that product give you?

Yes, your second approach is correct.

We just had a very similar thread on virtually the exact question last week. There are several ways to arrive at the answer, some easier than others, some more direct, and some more detailed.

Oh, I see heehaww posted some guidance while I was typing my reply so I will put my reply inside a spoiler.

thanks you guys are awesome

And there's a slicker way find the chance no one has it: what's the chance the remaining deuces are in the deck rather than somewhere in the villians' cards? Ie, how many ways can the deuces be in the deck vs how many possible locations are there for the deuces?

Then when you get through that, don't you also wanna know about the chance of someone having 44?

There is also 5s5c6c6s and some 3s5s XX (and the like) for open ended and flush or other outs too if the suits fit. I bet also if one has flush and gut straight they are also ahead of AAXY technically in terms of probability to win the hand. Imagine further one of the flush draw cards they have is some Ace removing your own outs.

Unless this is a prop bet freezing any action at flop, ending the hand there forever, the true equity the hand has is more important than who is currently ahead in hands formed.

See these examples, there are many hands better than some of these decent starting AAXY hands you may think are ok at this flop but prove really behind without the others having technically a better hand yet;

http://propokertools.com/simulations...d+kh&s=classic

http://propokertools.com/simulations...d+kh&s=classic

http://propokertools.com/simulations...d+kh&s=classic

http://propokertools.com/simulations...c+5s&s=classic

http://propokertools.com/simulations...c+5s&s=classic (3355 has you at 66% already hard to ignore as not better hand by far)

http://propokertools.com/simulations...c+6s&s=classic (3366 also at 58%)

http://propokertools.com/simulations...c+ts&s=classic (KKTT as tie lol)

http://propokertools.com/simulations...c+6s&s=classic (3s4s5c6c at 64% lol)

http://propokertools.com/simulations...c+6s&s=classic (even 4567 55%)
and so on if one keeps looking for over 50% equity opponents heads up if you go on to see turn/river vs them.

(you probably can use the programming tool of that same site to see exactly how often you are ahead in a flop vs some broad starting ranges but i have forgotten how to do it)

wtf are you doing??

Imo masque didn't spoil anything for OP. They're just stats as opposed to solutions, and they're stats for a different question than OP asked (equities as opposed to chance he's ahead).

Masque is right that equities are what matter in poker, not the chance you're ahead. Like if you're 55% likely to be ahead but are either slightly ahead or way behind (and nothing in between), that's a fold. But the question asked ITT serves fine as a math problem nonetheless.

I always like to deal with the way things are in the real world in the broadest sense i recognize the problem asked.

The OP never asked us what is the chance if we randomly deal hands and all 3 see a flop that at least one of them have a better hand if the game ended there. He asked for a chance to be beaten if we ignored the bias preflop, ie the AA guy will reraise or something and then action will change to filter some stupid badly suited X2YZ out etc. Unrealistic as this is to ask in a real game he made the specification about it so it becomes an exercise and must be treated as such.

It does make it a probability question in a combinatorial exercise sense if we define beaten as having a better hand if all stopped there. He didnt say that game ended there though, he spoke about being beaten without defining it.

I dont know about how others see this but a hand is beaten every time its sub 50% vs someone if this is not a prop bet to freeze action and end it all there.

If an opponent has some of the hands i mentioned that go over 64% it does feel like badly beaten. It is not as beaten as if they have 2 or 44 (up there in 80%+ levels) but its far enough from 50% to be a problem in some of these hands.

So i updated the list of hands that are beating some AAs at this kind of flop sometimes to remind OP that you can be behind in Omaha in far worse way than in holdem even if you have the best up to that point formed hand.

We should realize that we can answer both questions or suggest that both deserve an answer or define beaten in a restricted sense that doesnt help the broader understanding of things. Doing both is the best approach. Next we can even turn to real life poker and see exactly how likely to be beaten one can be at this spot anyway before acting and your equity is the way to define that. If your parents were watching you play live and seeing cards and % in their monitor/TV set they would be scared to see you are against this 3355 hand for example http://propokertools.com/simulations...c+5s&s=classic that is ahead if all went all in there.

And let me shock you possibly also with this hand that leaves them even with a 2 in their hand behind vs a special kind of non 2,4 AA in some flops;

http://propokertools.com/simulations...d+8d&s=classic


board: 2c 2s 4s
Hand Equity Wins Ties
As Ac 3s 5c 58.05% 476 0
2h 8h Qd 8d 41.95% 344 0

My exasperation with the sixth post in this thread has nothing to do with spoilers. It is probably obvious to everyone. I will leave it that.

Hah someone has to fill me in.

whosnext how do you like to argue your As Ac 3s 5c on 2c 2s 4s flop vs 2h 8h Qd 8d that someone claimed is beating you, has in fact 58% and is not at all beaten unless someone defined what beaten means in a way that many poker players that play the game will find questionable. Certainly good players wouldnt feel bad about that flop at all a second after the instant shock of a 2 seen in the other guy's hand.

I have no problem that some define beaten in terms of made hands if all ended there if they say so and i dont have any big deal issue if they think so and never define what they mean by it, imagining others would see it the same way. Why should you have a problem with anyone defining it either that way or as probability to win the hand at this point or both (as i do) pending clarification. Apparently you have a problem with it though. I would imagine a wise man has problems with things that can damage a discussion or a process etc. What is the damage here that warrants having a problem that leads to an expletive?

If that shocking result that trips are not beating you always with that non 2/4 AAXY is not seen by you as a contribution to the thread (a thread that never officially defined what beaten means in a game called Omaha that never anywhere ends evaluating hands at flop in non prop bet formats) then i suppose you have issues with someone called masque de Z that go back at least 10^50 Planck time units! Know that this guy has no issues with you though.

Intelligent people know that you can define beating in many ways (as a made hand already up to that point that is better or as probability to win the hand at this moment being larger than the opponent's). However until my posts in this thread intelligent people may not have known that some time even trips are not ahead of your non 2/4 AAXY here. I prop bet you if you asked casual poker people in a party they would bet against it being possible or admit at least that they would have to go for it since being asked, but find it kind of amusing as well to be the case. I find it a positive in any discussion which counts hands that beat a made hand up to the flop, to mention for the benefit of poker knowledge/appreciation of the structure of the game that one of them is not actually ahead in probability at all and that even others exist that are non 2,4 or AA hands that are ahead.

So objecting to contributions to the overall situation discussed is rather unwise, unkind and irrational because it wins absolutely nothing for anyone, especially when the guy at #6 already explained the various interpretations and didnt criticize anyone before for what they did!

Glad to see heehaww clearly shows the proper way to see things in all his generalized, open minded, kind attitude.

And i bet also someone called OmahaFanatical enjoys learning funny things about their favorite game.

I worked it out and get 16.57% as the percentage of times a 4 card combo will contain one of the unseen two's. There is a 16.57% for Villian A and a 16.57% chance for Villain B.

Below I show how I worked this out. There are some real experts on 2+2 and maybe one of them might take a look at it. I'd like to know if there is a quicker way. Actually, it did not take long to do the math, but it seemed like it took forever to write it out.


There are 45 unseen cards. Two 2's are among those 47 cards. Two villians hold 4 cards each. What are the odds that a two might be among those 47 cards?

45 cards can be dealt in 148,995 4 card combinations. How many of those combinations include 2s?

Let's say the two 2s are 2c and 2d.

I am going to follow this method:

Method:
1. Figure out how many 3 card combinations there are for 43 unseen cards (45 cards minus 2c and 2d = 45).
2. Add 2c to those three card combos. Add 2d to each three card combo.
3. Divide the number of combos with 2s by 148,995 to get the percentage of times the villians in this game will hold a 4 card combo that includes a 2.
4. Item 4 is the reconcilation, or check of the results.
a. Determine the total number 4 card combinations that can be made from 43 cards.
b. Determine the number of 4 card combinations that include both 2c and 2d.
c. Add the number of combos that include only 2c and the number of combos that include only 2d to 4a and 4b. The result should agree with the total number of 4 card combos that can be made from 45 cards, 148,995.

Results:
1. There are 12,341 three card combinations of 45 unseen cards.
2. If we add the 2 of clubs to all 3 card combinations and the 2 of diamonds to all three card combinations, which is to double 12,341: 24,682. This is the number of 4 card combos that include 2c and 2d.
3. 4 card combos that contain either 2c or 2d make up 16.57% of total combos.

Reconciliation:

a. 43 cards, deal 4: 123,410
b. 4 card combos with one 2: 24,682
c. 4 card combos with 2c and 2d: 903

a, b, and c add up to 148,995

When the results reconcile, I assume the solution is correct.

I see that I typed "47" rather than "45" at a few places at the beginning of my response. "47" shouldn't be there.

The important numbers are the number of unseen cards: 45
The number of unseen dueces: 2
The number of unseen cards less the two unseen dueces: 43

Here are two ways to look at it that are more simple:
This may be a quick way to look at this question, while at the table. There are 45 unseen cards. Two of those cards are dueces. If the dueces were evenly distributed in a deck of 45 cards, there would be 22 and 23 cards between each duece. If you dealt out an 8 card sequence, beginning with each card, 16 of the 45 8 card sequences would contain a duece. The ratio--16 to 45--is very close to the solution.

This is even more simple: you can deal out 5.625 8 card hands from a 45 card deck. Assuming the two dueces are distributed (or at least 8 cards apart), 2 of the 8 card hands will include a duece. That ratio, 2 to 5.625, is very close to the solution. And it's identical to 16 to 45.

I did OK at 5 card draw years ago calculating and making comparisons at the table. I based decisions on those ballpark results alone and did pretty good. It did help that the the guys I played played by gut instinct and that one or two or so of them was tilting every night.

I sat down at a table last month, a home game around here, playing Omaha hi lo for the first time ever, thinking I could calculate at the table. So how did that work out? This is how it worked out: It didn't.

Hey, I am sorry about the typos in my first response. I looked to see if I could edit the post and I didn't see any way to do it.

I think you can only edit your post within 30 minutes (maybe it is 20 minutes) of posting.

When you post and there is no heavy rapid exchange with others your post may stay as last post for a while. If you kept a tab that has the last post you made with the indication edit and used other tabs to continue to work that browser and ran other programs etc and you found out something needed editing and its over 30 min, you can use that old tab that still has edit showing. After selecting edit the old post ( it will allow you even if past 30' because the edit option is live even many hours later if you never hit reload) you can copy the entire post and then delete it, which it will allow you to do (it wont allow to edit only delete) (you may reload that page in another tab with your old post and no edit option to make sure you wont miss the old post by the deletion in case the copy/paste failed). In the original tab the delete will take place and now you can paste/repost as if brandnew the copied material and buy another 30 min time that way. If nobody else has posted since then responding to you it will be as if nothing happened (some people may have read the post already but its ok they will understand why you did it).

I think i have gone that way even 4-5 hours without a problem. In principle if you let the computer on and the browser doesnt crash and the old tab is never reloaded you may do it even a day later, although i havent tested it.

So if you have material that is heavily edited you can load the fresh post in 2 computers or 2 browsers in multiple tabs and buy that way a few hours provided others dont quote you or start a discussion that refers to your original post and you need to then respect that development and not delete the old post or if you do apologize and explain what you did, so that it makes sense to a new viewer lol and the guy that responded.

I think I understand what you are saying--and it would have worked if I had done it like you say here. I looked at the FAQ and I thought I read that you could edit--except I didn't have that button available.

Or I could just wait a day, but that would really slow things down, if everybody waited a day. Thing is, if I am writing about numbers, I am guaranteed to have something wrong. I rechecked all the numbers in that post except those two--I didn't scroll up far enough. I do not like it one bit when I am trying to follow somebody's math and they have wrong numbers all over the place.

I just thought--this is not the correct way to solve that problem, but the results are plenty close enough: the odds of picking a duece are 2/45. Multiply by because villains hold 4 cards each, resulting in 8/45, for each villain.

The thing I like about combinations is that you know you've got the right answer.

Thanks for telling me all that.

If there were only one deuce then this method would give the exact answer. You're right that it's close, and that's because its only error is that it double-counts the chance the villain has quads, which is a low probability and therefore small error. What this means is that inclusion-exclusion is another way to get the exact answer. For one villain you'd start with 8/45 and then subtract the chance he has quads.

This is a simplistic example: http://propokertools.com/simulations...0%25&s=generic

I assume hand one 1 was: A, A, something between 3 and K, something between 3 and K.

I assume the other two hands were totally random.

51% of the time we have 50% equity or more.
If the pot is going to be 3-way then 95% of the time we have 33% equity or more.