Odds question

Ok, so I've read over and over that the odds of having a pp is 221:1.
So here's my question, what are the odds of you holding a pp, while one of your opponents at a FR table (let's say all the seats are full, so 8 opponents) has AA?
I simply ask because while I know I win AA over KK as much as I lose KK to AA (so it all comes out in the wash), it seems like these hands happen far too often, don't y'all think?
I mean I typically play 400-1k hands a day lately, and I would guess more than half the days I play I see at least one hand where KK is against AA (sometimes several).
I know that tonight, in 1k hands, I ran up against AA 5 times (which can easily happen statistically since my opponents would have seen around 8K hands while I saw 1k), but 4 out of the 5 where I played against AA and it showed down were with KK, QQ, TT and 99. Certainly AA was in my opponent's hands far more often and didn't get showed down because no one else made enough of a hand to show down with. I understand that I'm more likely to show down with a strong starting hand, which includes more pp's. I just can't wrap my head around how often this happens I guess.
So, anyone know how to figure it out statistically? This is not a complaint whatsoever. I'd just like to have more realistic expectations and understanding.

I have been told that on a 9 person table the odds of KK>AA are about 680 to 1. That's just the odds of it happening on your table though, the odds of someone else holding specifically KK while you hold specifically AA are like 6,100 to 1?

ugh, maths. Will check those figures later, you've given me a headache.

At your table, the chance player 1 has AA is 1/221, player 2 has AA is 1/221 etc. If you have KK at a ten handed table, there are nine other players, so the chance of an AA hand at your table is 9 * 1/221 which is 0.04 or 1/25.

So when you have kings, you will be up against aces once in 25 hands.

There are 1326 different possible hands and 6 ways to make a pair. Thats how you get the 1/221 figure: it is 6/1326. It also how you get the chances of getting any pair, which is 13 *6/1326 = 1/17.

So if you want to know the chances of a higher pocket pair, you count up the number of possible higher pairs, divide by 1326 and multiply the number of other people at the table. eg.

If you have JJ, there are 18 possible overpairs (6 Aces, 6 Kings and 6 Queens). If you are dealt JJ at a nine handed table, the chances of a higher pair are:

18/1326 * 8 = 0.108 or approximately 1/9.

Originally Posted by linaker

At your table, the chance player 1 has AA is 1/221, player 2 has AA is 1/221 etc. If you have KK at a ten handed table, there are nine other players, so the chance of an AA hand at your table is 9 * 1/221 which is 0.04 or 1/25.

So when you have kings, you will be up against aces once in 25 hands.

There are 1326 different possible hands and 6 ways to make a pair. Thats how you get the 1/221 figure: it is 6/1326. It also how you get the chances of getting any pair, which is 13 *6/1326 = 1/17.

So if you want to know the chances of a higher pocket pair, you count up the number of possible higher pairs, divide by 1326 and multiply the number of other people at the table. eg.

If you have JJ, there are 18 possible overpairs (6 Aces, 6 Kings and 6 Queens). If you are dealt JJ at a nine handed table, the chances of a higher pair are:

18/1326 * 8 = 0.108 or approximately 1/9.

Well this way of counting is wrong, but its not that far away from the truth, and even if its not correct it still gives quite a good idea about the probabilities.

Your KK will run into AA roughly once every 27 times you get dealt it (in a 9 handed game).

Phil Gordons website has a quick (not extact) formula for coming up with the chances of running into a higher pocket pair...
http://www.expertinsight.com/Articles/92.html

Look at it like this...A random number generator spews out cards to random players. It doesn't know if it gave you AA or not. Its more likely that you see things like this than you don't.

Say at your table you get KK and loose to AA. If things like the OP above didnt happen. Then you could sit back and say 'wow, glad I got that out of the way. Now I wont have a cooler for a while"...I would start to think things were rigged if I didn't see AA vs KK,QQ,TT a few times every now and then.

I know you originally asked for math which I cant give. But I do know that this is probably more likely to happen than it is to not.

-Your KK will run into AA roughly once every 25.52 times you get dealt it (in a 9 handed game). (50x49/2 = 1225/6 = 204.16... = 204.16/8 = 25.52).
You get dealt KK every 221 Hands, so this should happen about every 5639.92 Hands, to you.

-Someone will be dealt KK while another player at the table will be dealt AA every 627 Hands. (221/9 x 204.16/8 = (24.5...) x (25.52) = 626.657...)

Thanks guys.
Ok, the way JR layed it out connects the most dots for me.

You get dealt KK every 221 Hands, so this should happen about every 5639.92 Hands, to you.

When I first switched to ring, I kept getting KK running into AA - after several times I went back after the 3rd one to look in PT3. In my first 7 times getting KK, I had lost to AA 3 of them! I forget now, but I'd say all that was in the first 3 or 4k hands. Sick huh? It made me want to puke and run back to SNGs.
So based on your answers, it certainly isn't as rare as you would think without crunching numbers (at least what I would expect).

I didn't include it in the original post, but part of what also made me want to ask the question was that on the same night, I chopped with KK vs. KK and watched an AA vs. AA - and one of the guys in the hand chatted that he'd already had the same that night once before! Just makes you think, that's all.
Thanks everyone for your maths/thoughts.

I don't know shit about math but I think it's 50/50 either you do or you don't