[uuDevil]

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The possible starting hands are not all equal in frequency. Any pair occurs 1 in 221 times, not 169. By my calculations any offsuited combination occurs about 1 in 108 hands, while any suited combination occurs 1 in 352 hands. So after 15,000 hands, AA should come up about 68 times.

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I think a couple of these numbers are slightly off.

There 4 ways to be dealt suited cards:
P(suited hand)=4/C(52,2)=1/109.5 or 108.5:1

There are 12 ways to be dealt an unsuited hand:
P(unsuited hand)=12/C(52,2)=1/331.5 or 330.5:1

For the OP, the NUMBER of AA hands you are dealt will not necessarily "average out" after some number of hands. It is the PROPORTION of AA hands that will approach it's expected value. So after 15,000 hands if you have gotten AA 53 times (15 less AA hands than you "should" have), then after 150,000 hands you may end with 600 AA hands (80 fewer than expected). You have now been shorted more AA hands, but the proportion of AA you received has actually gotten closer to 1/221 (.004525). The proportion has gone from 53/15000 (.003533) to 600/150000 (.004000).