[AaronBrown]

When the probability of success is low, the Poisson distribution is a better approximation than the Normal for a binomial.

You expect to get X/221 pairs of Aces over X hands. Call this M. For a Poisson, the variance equals the mean. The probability of getting no pocket Aces is exp(-M). The probability of getting K pocket Aces is M^K*exp(-M)/K!.

Over 221 hands, for example, the exact probability of getting zero pocket Aces is (220/221)^221 = 0.3670. The Poisson approximation is exp(-1) = 0.3678. The probability of getting three pocket Aces is 0.6117. The Poisson approximation is 0.6131. The exact standard deviation is 0.9977. The Poisson approximation is 1.0000.