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 Originally Posted by deadgoat
Last night at the same table I lost to set vs set twice when set vs set is suppose to happen 1 out of 100 times.
This is one of those vague numbers that people hear and it starts to stick, but it has no basis in fact.
Obviously, the chance of an overset on the flop is very dependent on what set you hold. If you hold AA, then there is no chance. If you hold 22 then any other set will be an overset.
Consider:
1) You are at a 10 man table and you are dealt 22.
2) There are 50 unknown cards and 48 of them are higher than 2 (and the 2 other 2s make 50). So, a player must be dealt 1 of the 48 out of 50 cards to have a possibility of a higher PP.
3) Then the player must be dealt 1 of the 3 other cards of that rank to have that PP.
4) So, the player on your left has a (48*3)/(50*49) chance of being dealt a higher pocket pair than 2's. That is a 5.8776% chance. (Quick check for sanity, in a 52 card deck the chance of being dealt a pair is 1/17. Our number should be close to that because we are saying all pairs except 2's and 2 of the 2's are already gone. 1/17 = 5.8824%. So everything looks good so far.)
5) And, in fact, each of the other 8 players at that table has exactly the same chance. So, the total chance of at least one person at that table having a higher PP is 1- ((1-0.058824)^9) = 42.0521%. (Quick sanity check, odds of being dealt a PP are 1/17. So the odds of 9 people all not getting one would be roughly 1-(16/17)^9 = 42.0518%. So everything looks good so far.) [And I realize this number is not exact, as it is not an entirely independent equation (if someone is not dealt a PP, the odds of the next person not getting a PP have changed a little due to the previous non-PP skewing the card distribution), but it is easily close enough.]
So . . . when you are dealt pocket 2s, there is a 42.0521% chance that at least one person at a 10 man table was dealt a higher pocket pair.
6) It is a given that you have flopped EXACTLY a set.
7) There are two other cards on the board and both will be higher than 2s. What are the odds that they do not give the higher PP a set? Well, we know 5 cards, your PP, his PP, and the other 2 for your set. So there are 47 cards to be concerned about and there are exactly 2 we do not want to see (the two that match his PP). So 45 of the 47 cards are okay for the 2nd card on the flop and 44 of the 46 are okay for the 3rd card on the flop. (44*45)/(47*46) = 91.5819%. So there is a 91.5819% chance that they missed, which means there is a 8.4181% chance that they hit a set or quads.
8) So, first someone has to have a higher PP (42.0521%) and then they have to hit the higher set when you do (8.4181%). 0.420521 * 0.84181 = 3.54%.
Did I forget anything? Yes, I did. There is of course the chance that there are multiple higher PPs out there which mean that it is even more likely that the other 2 cards match someone. But I am not going to worry about that right now. Because I can say with confidence that when your 22's hit a EXACTLY a set on a 10 man table, someone else would have hit a higher set (or quads) AT LEAST 3.54% of the time if they stayed in with their PP.
This is obviously much more likely than 1 in 100.
How do 8's hold up?
Chance of the guy to your left have 99+ = (24*3)/(50*49) = 2.9388%
Chance of any of the other 9 players having 99+ = 23.5439%
Chance of them hitting their set (or quads) when you hit yours (on the flop) is the same 8.4181%.
Chance of it all happening at once: 0.235439 * 0.084181 = 1.982%
So, even 8s are going to be cracked around 2 in 100.
How about Queens? 0.7137%
Kings? 0.354%
And remember that all of my numbers are a bit on the low side because I do not account for multiple over PPs. Could we get more exact numbers? Sure, I would need to compute the exact chances for 1 to 9 over pairs (with each additional PP being more unlikely) and multiple those by the chance that someone hit one of the other two cards on the flop (with each additional PP making it more likely). But this math should give you a pretty good rough estimate.
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When should you fold a set with no flush or straight posible
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