What is the probability of a pair of characters present in 72 choose 9

I was hoping you could walk me through the solution to the problem described below.

Background: While using an online pseudo-random number generator to create a 9-digit password, I hit the [Generator password] multiple times and observed a surprisingly high frequency of pairs of numbers or pairs of letters contained in a string of 9-digit passwords I created; e.g., 3 out of 4 passwords had a pair present. I defined a pair as two of the same number or two of the same letter without differentiating between upper and lower case; i.e., A-A and A-a were considered pairs.

72: A-Z, a-z, 0-9, 10 symbols

Mmmm... I never did see a pair of symbols, but I did see pairs of numbers. I would have expected the frequency of number pairs to be "equal" to the frequency of symbol pairs.

To repeat my request with a little more detail now that I have put my question above in context.

What is the probability of observing a "pair" as defined above in a 9-digit password?

Thanks!