The company RandomCereal places figurines in cereal boxes via the following process: in each box, with probability 1/2 it places no figurine in the box, and with probability 1/2 it places some figurine in the box. When it places a figurine, it chooses one of 4 possible types uniformly and random.
(a) Suppose you have collected k of the 4 types of figurines, where k E (0,1,2,3) (collecting a type of figurine means that you have at least one figurine of that type). What is the expected number of cereal boxes you need to open until you obtain a new type of figurine (that you have not already collected)? Your answer should be in terms of k.
(b) Suppose you do not have any of the figurines. What is the expected number of boxes you need to
open to collect all 4 types of figurines?
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