Thought these up, there's no doubt in my mind they've been thought of long before I ever thought of them since they're quite trivial but I don't know who came up with them or what they're called. Hence why I have to ask /sci/ if they're even correct in the first place.

Probability of getting something that happens 1 in every N tries to actually happen within N tries:
1 - ((N - 1)/N)^N

Average number of tries needed to get something that happens 1 in every N tries to happen once (common sense says it should be N):
= Sum of (1 - ((N - 1)/N)^M)(M/N) for every integer M in [1, N]
= (i forgot how to evaluate this any further)