If you decide on the set of 130 hands ahead of time, you get a different number than if you ask what the probability that you will get AA 5 times in some 130 hands, since there are many different opportunities for streaks.

The chance that there will be 5 AA hands within a particular 130 is (130 choose 5) (1/221)^5 (220/221)^125 = 0.000308 or about 1/3247. The chance to get 5 or more is about 1/2944.

You can estimate this type of probability using a Poisson distribution. If there are a large number of independent events which each have a low probability, and the average number of occurrences is L, then the probability there are k found is L^k/k! exp(-L). L=130/221 so this approximation is 1/3068 here, about 6% away from the actual value.

Again, since in a set of 10k hands there are many disjoint sequences of 130 hands, the chance that you will see 5 AA hands in some set of 130 is much higher, at least by a factor of [10k/130]. How much higher actually seems like an interesting mathematical problem.