AA v KK

[KenProspero]

In my last 4500 hands, I've had KK and been up against AA 5 times. Back of the envelope, I figure I should have KK and see AA should happen about once in 5000 hands. (In a full table, I'll get KK once in 221 hands -- One of my 9 opps will have AA about 1 time in 23).

First: How close is this guestimate.

Second: What are the chances of this occurring 5 times in 4500 hands.

I suspect the odds aren't astronomical, but can't do the math. Thanks.