An interesting coincidence happened today in class. My co-teacher was handing the students back their tests. She wanted each student to mark another’s test. However, she accidentally handed a student back her own test. She casually said to me “weird, what are the chances of that?” Her question implied that this was a rare event. The answer is quite the opposite.

My co-teacher had a 63%, or to be precise, chance of handing a student back his or her own test. If you are wondering, e stands for a special number:

e = 2.71828…

This percentage is not good. It means that, if you randomly distribute tests to students to mark, about two thirds of the time at least 1 student will end up with his or her own test!

How in the world did I come up with that answer? Let me back up a bit. Suppose we had 3 students and we wanted to give them back their tests to mark so that no student had his or her own test. What options do we have? One option is:

Student A: test B

Student B: test C

Student C: test A

Another option is:

Student A: test C

Student B: test A

Student C: test B

Hence, we have 2 options where we will hand back the tests with no matches. How many options do we have in total?

3! = 3*2*1 = 6

Thus, there is a 2/6 = 33.3% chance that everyone has a different test. In other words, there is a 66.6% chance that at least one student receives his or her own test to mark (which would not be good for academic standards).

What if we had 4 students? You can do the math, and you will find that there are 9 options where the students have different tests. For example:

Student A: test B

Student B: test C

Student C: test D

Student D: test A

Again, basic counting principles dictate that there will be 4! = 24 different possible combinations. Hence, there is a 9/24 = 38% chance each student will have a proper test to mark. We could continue on like this; painstakingly grinding out the probabilities for each situation until we reached a class size of 25. However, there is a pattern we can exploit. Consider the following table:

Number of Students |
Fraction of proper distributions |

3 |
2/6 = 33% |

4 |
9/24 = 38% |

5 |
44/120 = 37% |

6 |
265/720 = 37% |

7 |
1854/5040 = 37% |

8 |
14833/40320 = 37% |

Do you see the pattern? Once the class size gets to 5, the percentage becomes stable. Based on the above data, we could predict that a class of 25 students will have roughly the same percentage, 37%. Indeed, one can prove mathematically that this percentage will be true for any size class, large or small.^{1} Thus, my advice to teachers is as follows. Be careful distributing tests to students to mark. You are playing a game with a 37% chance of winning. And any gambler knows, those are not good odds.

^{1}See next blog post.