Ecologists use occupancy models to study animal populations. Ecologists at the Department of Natural Resources use helicopter surveying methods to look for otter tracks in the snow along the Mississippi River to study which parts of the river are occupied by otter. The occupancy rate is the probability that an animal species is present at a particular site. The detection rate is the probability that the animal will be detected. (In this case, whether tracks will be seen from a helicopter.) If the animal is not detected, this might be due to the site not being occupied or because the site is occupied and the tracks were not detected. A common model used by ecologists is a zero-inflated binomial model.
If the region is occupied, then the number of detections is binomial with n the number of sites and p the detection rate. If the region is unoccupied, the number of detections is 0.
Let α be the occupancy rate, p the detection rate, and n the number of sites.

(a) Find the probability of zero detections. (Condition on occupancy.)
(b) DNR ecologists searched five sites along the Mississippi River for the presence of otter. Suppose α = 0.75 and p = 0.50. Let Z be the number of observed detections. Give the probability function for Z.