Probability of Overpair Out There

[Huskiez]

For a 10-handed game:

I know the odds of someone having AA when you have KK is around 23:1 or so.

Just wondering what the odds of at least one person having a better pair are when you have QQ, JJ, and TT?

[schubes]

Since it looked like this post would die unanswered, and I'm a little burnt out on actually playing poker (a bit of a downswing right now), I tried to figure it out.

Explaining each step of how I did it is much more work than actually doing it, but it involves thinking about identities like
P(A or B or C or...)=P(A)+P(B)+P(C)...-P(A&B)-P(A&C)-...+P(A&B&C)+...
where for instance A might stand for "Player 3 has QQ." These identities simplify greatly if you notice that many terms vanish and there are many symmetries. So if you want to figure it out yourself or double check my answers that's a big hint.

The odds someone has AA when you have KK:
9*(4*3/(50*49)) - C(9,2)*(4*3*2*1)(46!/50!) = .0439

21.8:1 (in getting 23:1 did you just multiply the odds of being dealt AA by 9?)

...when you have QQ:
2*(.0439) - 9*8*(4*3*4*3)(46!/50!) + 9*8*7*(4*3*4*3*2*1)(44!/50!)
- C(9,2)*C(7,2)((4*3*2*1)^2)(42!/50!)=.0860

10.6:1

...when you have JJ:
3*(.0860) - 3*(.0439)+ P(you are facing each AA, KK, and QQ) ~ .126
we can ignore the third term because it is so small

6.9:1

...when you have TT:
6*(.860) - 8*(.0439)+ 4*P(facing AA, KK, and QQ) - P(facing AA, KK, QQ, and JJ) ~ .165

5.08:1

[Huskiez]

Thanks for the reply.

This information is definitely useful for me, as it gives me a good approximation as to how often I am beat with these hands going to the flop.