[BruceZ]

Let's try this one more time. Forget the scraps of paper. Throw them in the trash, they're confusing you. You're sitting at the table, and everyone gets two cards. Nobody has looked at their cards yet. You peek at yours and see KK. Now each of those other players cards have a probability of exactly 6/1225 to be AA. If you were to just add these 9 6/1225's together to get 9*6/1225, that would be extremely close to what we are looking for, the probability of at least one more AA. The only thing that happened is that we double counted all the times that two people got AA. But those times are very rare, which is why this is so close. If we really wanted an exact answer, we would compute the probability of two people getting AA, and we would find this is 0.01%, and we would subtract this little bit off from 9/1225 to get the exact answer. At the bottom of my original post, there are combinatorics for both of these terms, and the first will evaluate to exactly 6*9/1225.