Hi Lea,
Note that your initial equation of:
P*=(1- (single survey p)^K)
isn't quite correct. It should be
P*=1- (1-p)^K
where p is the single-survey detection probability. Your P* would be the probability of not detecting the species in every survey (i.e., 1 minus the probability of detecting it in every survey), given presence.
Darryl