Exercise 3.3 Suppose that in the previous exercise k is initially unknown, but we know that the urn contains exactly 50 balls. Drawing out 20 of them, we find three different colors; now what do we know about k? We know from deductive reasoning (i.e. with certainty) that 3 ≤ k ≤ 33; but can you set narrower limits k1 ≤ k ≤ k2 within which it is highly likely to be? ( Hint: This question goes beyond the sampling theory of this chapter because, like most real scientific problems, the answer depends to some degree on our common sense judgments; nevertheless, our rules of probability theory are quite capable of dealing with it, and persons with reasonable common sense cannot differ appreciably in their conclusions.)
I guess the most likely answer is $ k = 3 $. We can solve this problem by iterating $3 \le k \le 33$, within each situation, this may use the maximum entropy to get the most likely distribution and then we can compute the maximum probability to get 3 kinds of colors. Through this method, We will get an optimized k. Am i right, help, Thank you.