Your mailbox gets no mail today

Question

A clerk sorts 4 letters into 5 mailboxes, dropping each letter into a uniformly random box, independently of the rest. What is the probability that your mailbox receives none of the four letters?

Accepted Answer

Consider your mailbox alone. It is empty exactly when all 4 letters go to other boxes. One letter misses your box with probability \frac{4}{5} (4 of the 5 boxes are "not yours"). The letters are independent, so \left(\frac{4}{5}\right)^4 = \frac{256}{625} \approx 0.4096. With fewer letters than boxes, an empty box is quite likely, over 40% here. Takeaway. The empty-box probability \left(\frac{n-1}{n}\right)^m grows as the number of items m shrinks relative to the number of bins n.

Your mailbox gets no mail today

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