I see flaws in the statement
1. If we assume that tagged or untagged rabbit is equally likely to be caught the number of rabbits on island HAS to be 14 as during 2 days we have already tagged 7 rabbits. So, there are no smallest or greatest numbers - only 14 rabbits.
2. What kind of probability are we looking for? What catch is meant?
When I toss a coin the outcomes are assumed to be equally likely. Because
p(H) = p(T) = 1/2 [ or assumed to be very close to 1/2]. If we do an empirical test of 100 tosses and found 48H and 52T, we still would consider the coin fair.
If the outcomes were 80H and 20T then the coin would be considered skewed and the outcomes of tossing the skewed coin are not equally likely.
Probability of getting a head is no longer 1/2, it is approximately 4/5.
In the rabbit example, we know we have 7 tagged rabbits in the island. If we assume there are only 9 rabbits in the island then
p(untagged) = 2/9
p(tagged) = 7/9
and the result of any empirical test, of catching the rabbits, would be skewed toward the tagged rabbits. [not equally likely]
If we have 14 rabbits then p[U] = p[T] = 1/2.
If we assume we have only 13 rabbits, we know for a fact the probability of any catch is a little bit skewed toward tagged rabbits. We may consider the outcomes equally likely. But we know they are not.
If we have 15 rabbits then p[U] and p[T] are approximately equal to 1/2. We still may consider the catch as equally likely and we know the outcomes are skewed a little toward untagged.
Hence, 14 is the smallest number of rabbits there can be in the island, to be sure that a catch is equally likely.