Quote:
Originally Posted by masque de Z
You have XX he has YY. You need a flop that gives sets and only sets for both at the same time. Only if X different than Y of course this is possible.
So you need flops that look like XYZ with Z non X,Y
Example 33 vs 77 you need 37Z flops.
Obviously this probability is the ratio of the ways you can deal from 52-2-2=48 cards 3 to produce XYZ over the total number of ways you can deal 48 cards to obtain a flop.
Since you have 2 X left and 2 Y left and anyone will do you have 2*2 possible XYZ for a given Z. Z now can run from any card in the rest 48-2-2=44 that are not involving any X,Y.
Additionally the way to deal from 48 cards a flop is C(48,3)=48!/3!/45!
Finally then the P(XYZ flop)= 2*2*44/C(48,3)=4*44*3!*45!/48!=4*44*3!/48/47/46=1/98.28=0.010175...
as previously stated by others.
It was important however to show you exactly how logically you can arrive there by considering the possible ways you can get XYZ flops over the totality of flops out of 48 cards. It is the most intuitive way to understand this.
Now of course that is given the fact 2 have xx yy already.
The problem is that this in itself is a rare event. First of all not all cases of 2 players that have 2 pairs see a flop . Say one has 33 the other QQ do you think the 33 will call the big raise the QQ will make unless it is a deep heads up game or the big blind has 33 capped at say 4bb size to call? Probably not , especially with others left to act after 33's spot in general. Basically its very complicated but you get the idea that many of those never see flop.
Without being too careful in the math from this point further since this is no longer an easy to handle problem since it depends on loose tight attitude and number of opponents etc in a full ring of 10 players you can expect each guy to have a pair about 5.85% of the time. So in a rather crude manner you can deal people as those with pair and those with nonpair type hands. Since we are concerned with the easier case and since the case of 3 pairs is a lot tougher its not too terrible to claim that each guy has almost independently 13*0.45%=5.85% chance for a pair.
So it is then as if you ask what is the chance in 10 people 2 to have 5.85% events and the other 8 1-5.85% events (pairs vs non pairs) . It will be roughly
10!/2!/8!*0.0585^2*(1-0.0585)^8~0.095 or 9.5%.
Just to give an idea if you demand 3 such pairs in 10 or 2 you have 11.1%. Now lets say that we consider the overall chance of 2 or 3 starting pairs as 11.1% . 4 or 5 and more are really negligible chances plus they need to be calculated with some correction because the fact multiple players have already pairs substantially alters the chance an extra one will have pair further down from 5.85%. That error in the first 2-3 is not huge but its there.
Anyway if we can settle that ~11% of the time you have 2 or 3 starting pairs and agree that since they can be all over the place and many of them will be top vs small lets settle on the fact that 50% of them survive preflop action and see flop. Say mostly the small vs another small and the big vs another big type of cases.
You are looking then at ~6% of the hands dealt in full ring ending in flops of XX vs YY.
Now combine this with the previous ~1% probability (which is not exactly 1 in 98.28 now because there is some kind of subtle correction coming from the fact that only 2 or 3 had preflop pairs meaning the distribution of the cards left is a bit different than totally random but this is a very delicate not materially important detail here on what is already a crude estimation) and you obtain your overall answer.
In a full ring of 10 players you can expect 1/98*6%= 1 in 1633 hands dealt will lead to set over set at flop. You can basically play live for a full week and never see it.
So full ring 1 in 1600 to 6max 1 in 2000 etc. Some crude numbers.
Only the 1/98 is accurate here but then again you need to arrive to a XX YY flop anyway and that is not trivial . Hence the small overall chance. Real small.
Now tell me what is the chance you flop trips with XY you upgrade to full house by river on a board like YYZWX and the other had XX for a higher full house hitting his 1 outer at river ? Thats another cooler stack killer situation . You have seen it in the opening scene or rounders...and someone in this site may have seen it also today...;-). Of course both in rounders and in real life the good guy comes ahead in the end of the day/movie. lol
Excellent reply, just what i was after. Thanks
