Please provide complete and well-written solutions to the following exercises. D

Question

Please provide complete and well-written solutions to the following exercises. Due February 18, in the discussion section. Suppose there are ten separate bins. You first randomly place a sphere randomly in one of the bins, where each bin has an equal probability of getting the sphere. Once again, you randomly place another sphere uniformly at random in one of the bins. This process occurs twenty times, so that twenty spheres have been placed in bins. What is the expected number of empty bins at the end? Exercise 2. You want to complete a set of 100 baseball cards. Cards are sold in packs of ten. Assume that each individual card in the pack has a uniformly random chance of being any element in the full set of 100 baseball cards. (In particular, there is a chance of getting identical cards in the same pack.) How many packs of cards should you buy in order to get a complete set of cards? That is, what is the expected number of packs of cards you should buy in order to get a complete set of cards? Note that N_1 = 1. Define N_0 = 0. Then

Explanation / Answer

There are ten separate bins and each has an equal probability of getting the sphere. Now since each bin can accomodate all the spheres, hence there could be an expectation of 9 empty bins at the end. Since one has to accomodate all the spheres.

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