From the Fun Fact files, here is a Random Fun Fact, at the Easy level: Matching Problem
Suppose I return N homeworks randomly to my N students.
What is the chance that no student gets back her
own homework? In a class of 30? Would it be more or
less if I had more students?
Surprising answer: the chance that no one gets back
her own homework is approximately (1/e), which is
approximately 36.8 percent of the time!
And the answer is about the same, no matter how many
students you have! (It gets closer and closer to 1/e
the larger N gets, and is already a very good
approximation for 5 students.)
Gee, where did the number e come from?
Presentation Suggestions:
Poll the class about the probabilities
before you tell them the answer, just to
see how good their intuition is.
The Math Behind the Fact:
This interesting fact is usually proved in classes in
discrete mathematics or probability,
using something called the inclusionexclusion principle. The answer
turns out to be the partial sum of an infinite series
for 1/e.
How to Cite this Page:
Su, Francis E., et al. "Matching Problem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
The Link for this Fun Fact:
is directly accessible here.
