How many papers do you expect to hand in before you receive each possible grade at least once?

Question

A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, A-, B+, B, B-, C+}, with equal probability, independently of other papers. How many papers do you expect to hand in before you receive each possible grade at least once?

Answer 1

Before the first point at which you have received all 6 grades, you must have passed the first point at which you have received 5 different grades, which occurs after the first point at which you have received 4 different grades, and so on. Due to the linearity of expectation, you can break up your random variable $X$ (number of papers turned in before having received all 6 grades) into the sum of 6 geometric random variables (number of papers turned in before receiving a grade that you haven't received before).

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Edit: As joriki and Louis mentioned, this is the Coupon Collector's Problem. I had forgotten what it was called.

Answer 2

This called the "coupon collectors problem". You can search for it.

The way to see the expectation is the following. Break up the process into a sequence of rounds that end each time you get a new paper. Let $N$ be the total number of grades. The length of the $i$th round has geometric distribution with parameter $(N - i + 1)/N$. You can probably take it from here.