Hint: Consider the inverse of a probability where probability of event = p and inverse = 1/p.
Remember that one interpretation of a probability is #Times_Event_Happens/#Total_Number_Of_Trials = N_p/N_total where N_total is total number of events and N_p is number times events event p happened.
If you invert this you get #Total_Events/#Events_for_p which means that you get the number of times an event happens within some period.
If you expect all events to be random from the distribution, then 1/p will tell you the expected number of trials you need to wait before that event actually happens. The lower the probability, the longer you have to wait on average and that should make intuitive sense. If p = 1, 1/p = 1 and if p=1/30, then 1/p = 30.