An unused shelf among four

Question

A librarian places 5 books one at a time, each on a uniformly random shelf out of 4 shelves, independently. What is the probability that the top shelf receives none of the books?

Accepted Answer

Look only at the top shelf. It stays empty precisely when every book lands elsewhere. Each book avoids the top shelf with probability \frac{3}{4} (3 of the 4 shelves are "not the top shelf"). With 5 independent books, \left(\frac{3}{4}\right)^5 = \frac{243}{1024} \approx 0.2373. Takeaway. A specific bin is empty with probability \left(\frac{n-1}{n}\right)^m; here \left(\frac34\right)^5 \approx 0.24.

An unused shelf among four

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