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 inclusion-exclusion 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. <https://www.math.hmc.edu/funfacts>.
Fun Fact suggested by:
Francis Su