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

This question already has an answer here:

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?

## marked as duplicate by Did, Adriano, Lord_Farin, Jared, azimutAug 3 '13 at 8:15

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Search for "coupon collector's problem"; you'll find loads of related questions on this site, and a Wikipedia article. – joriki Mar 1 '13 at 13:37

## 3 Answers

If a series of independent events each have probability $p$, then the expected duration until the first occurrence is \begin{align} &1\overbrace{p}^{\text{1 success}}+2\overbrace{(1-p)p}^{\begin{array}{l}\text{1 failure}\\\text{1 success}\end{array}}+3\overbrace{(1-p)^2p}^{\begin{array}{l}\text{2 failures}\\\text{1 success}\end{array}}+4\overbrace{(1-p)^3p}^{\begin{array}{l}\text{3 failures}\\\text{1 success}\end{array}}+\dots\\ &=\sum_{k=1}^\infty k\color{#C00000}{p}\color{#00A000}{(1-p)}^{k-1}\\ &=\frac{\color{#C00000}{p}}{(1-\color{#00A000}{(1-p)})^2}\\ &=\frac1p \end{align} The probability of getting a new grade after you have already gotten $k$ grades is $\frac{6-k}{6}$.

Thus, the expected duration to get your first new grade is $\frac66$; the expected duration to get your second new grade is $\frac65$; the expected duration to get your third new grade is $\frac64$; etc.

• Necessary, five months after one comment and two answers got posted? – Did Aug 3 '13 at 6:59
• @Did: You know, I really should have checked the date on the question when it showed up on the front page. My bad. – robjohn Aug 3 '13 at 7:07

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).

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

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.