[Peter_Rus]

I've heard that set on set is so rare that seems everyone ignore it's possibility when getting one. So i try to exam it more accurate. So you have set of 2's on flop 27Q and get a great action - i really don't care you tell for yourself and keep betting to find you beaten on river for many many chips. How does it happen and how often it will be happen?

Let's see: we see 5 cards - so 47 are in deck. supposing everyone playing any pair PF lets look to probabilities:

Probability of pair of 7 in pocket is 3/47*2/46 resulting 0.28%. So probability of pair of QQ is the same (until no reads on players). P77+PQQ=0.56% (1 time in 180). When you see turn of 5 and get raised again. You probability become higher a bit and equals 3/46*2/45.

Ok, lets get more than 1 opponent so probability for 1 of 5 opponents making higher set on flop is: 1- (1-P)^5 where P=0.56% for flop. Using such method i've got table:

1 2 3 4 5
flop 0,56% 1,11% 1,66% 2,20% 2,74%
turn 0,58% 1,16% 1,73% 2,30% 2,87%
river 0,61% 1,21% 1,81% 2,40% 2,99%

so, supposing everyone would play any pair you have 2.7% (1 time in 37) chance with lowest set to be broken already on flop. Actually, depending on texture game you can tell with more than 80% sure that someone have a pair.

Usually you can say it with about 40% of confidence (any raise PF mean something like that). So in a raised pot we will see changed probability. We have 12 possible pairs in pockets probability of any pair the same except pairs improved to set - it's probability lesser twice times Pofanypair=4/47*3/46 while Pofsetimprovedpair is 3/47*2/46.

So p of set improved pair when we 100% ensure that someone has at least a pair is counted as P=3/47*2/46*2/(3/47*2/46*2+4/47*3/46*11) equals 8,33%. So because we sure only for 40% this will less - 8,03%*40% equals 3,3%.

so probability of higher set
for flop is: 3/47*2/46*2/(3/47*2/46*2+4/47*3/46*11) *40%=3.3%
for turn: 3/46*2/45*3/(3/46*2/45*3+4/46*3/45*10)*40%=5.2%
for river: =3/45*2/44*4/(3/46*2/45*4+4/46*3/45*9) = 7.6%

it's when you heads up. Increasing number of player our table will look now so:

1 2 3 4 5
flop 0,28% 3,33% 6,56% 9,67% 12,68% 15,59%
turn 0,29% 5,20% 10,13% 14,80% 19,23% 23,43%
river 0,30% 7,60% 14,62% 21,11% 27,11% 32,65%

so it's 1 to 4 on river that someone of three opponent catch his higher set if you suppose everyone for 40% of having any pair. Isn't it low? I don't think so. But your chances still high. But i wouldn't raise for cap even if there are no flush/str8 on board.

This is one more reason - why low pair distinct from another low pair and middle pair. Higher the pair - higher the chance to get low cards decreasing greatly probability of higher set.

I find that many people think set on set is very rare... hmm - yes it's rare but not so rare that you suppose i think and if it hits you this way usually you lost very much, decreasing you overall ev. Be sure - if there were active PF and you receive much action on turn especially from more than one opponent's don't be so sure and don't increase your variance especially if someone tight has become mad.