Flop is paired, what are the chances someone has a trip?

[Tharpab]

AK flop is K33, assuming there are 5 opponents
what are the chances they hold the three assuming random hands, also how much it varies by the number of opponents?
Is there any rule of thumb(it doest need to be that accurante) like +10% -10% for every person seeing the flop?

[aloiz]

Given that you have AK and the flop is K33 and there are 5 other opponents in with random hands, the odds that at least one will have a 3 = 1 - P(none have a 3) = 1 - C(45,10)/C(47,10) =~ 38.4%

The function of the odds that some has a 3 for a variable number of players is not linear, so there's no fixed percentage you can add/subtract. However a rough approximation would be to add/sub 7% for each player.

aloiz

[Cerril]

One thing to remember is that at least some of the players won't have random hands. Depending on the card a person could be weighted as more or less likely to be holding that card. For example a flop containing paired tens or higher - especially A,J, or T - is a lot more likely to have made someone trips than paired 6s, 7s, or 8s (say). This is true even in quite loose games. That also applies when you've got trips and wondering if someone has the case Jack.

But but it doesn't hurt to know either.

[Precision1C]

Assuming you have AK and the flop is K33 the odds of no one having a 3 is listed below with the left hand column indicating the number of opponents, the second column is the odds against if you have X3.

9 .376 .617
8 .430 .660
7 .480 .702
6 .550 .745
5 .616 .787
4 .685 .830
3 .759 .872
2 .835 .915
1 .916 .957