Problem 2. Suppose there are N fish in a pond, where you don't know what N is. Say you catch 100 fish and tag them, so the pond now has 100 tagged fish and N-100 untagged fish. (a) Suppose you go back to the pond and catch fish, one at a time. Suppose catching each of the N total fish is equally likely. You do this 20 times. If there were N = 200 total fish, what is the probability you see 5 tagged fish out of the 20? If there were N = 1000 total fish, what is the probability you see 5 tagged fish out of the 20? (b) Suppose you go back to the pond and catch fish one by one until you see one that is tagged. If there were N = 200 total fish, what is the probability you see a tagged fish within the first 10 catches? If there were N = 1000 total fish, what is the probability you see a tagged fish within the first 10 catches? (c) Suppose you go back to the pond and catch 50 fish in one go. From a distribution perspective, suppose that each fish is equally likely to be in the 50 that are caught, like balls in an urn. If there were N = 200 total fish, what is the probability that 20 out of the 50 fish caught are tagged? If there were N = 1000 total fish, what is the probability that 20 out of the 50 fish caught are tagged?