Math: No Wizards on Weekends?

I noticed that I haven't been starting as a wizard in Nethack for a while. Maybe the DevTeam have special code for weekend players just like they have for the new and full moon? But that's jumping the gun. There are 13 roles in Nethack, so assuming that any role is generated independently at the start of a game, what is the probability that I wouldn't get a wizard after n starts? In other words, what is P(k=0) after n trials?

Using the binomial distribution C(k, n) * p^k * q^(n-k), let p=1/13, q=12/13, k=0 …
if n=20, p(k=0) = 0.20 and if n=40, p(k=0) = 0.04. So, I should only start to wonder about special code if I don't see any wizards in 40 starts!