Put $m$ balls into $n$ bins. What's the probability that more than half of bins are empty?

Question

We put $m$ balls into $n$ bins once a time and uniformly at random. What is the probability that more than half of bins are empty?

Thank you!

Answer 1

Too long for a comment:

For $m\ge1$, it seems to be

• $\dfrac{3}{3^m}$ for $n=3$,
• $\dfrac{4}{4^m}$ for $n=4$,
• $\dfrac{10\times 2^m-15}{5^m}$ for $n=5$,
• $\dfrac{15\times 2^m-24}{6^m}$ for $n=6$,
• $\dfrac{35\times 3^m-84\times 2^m + 24}{7^m}$ for $n=7$,
• $\dfrac{56\times 3^m-140\times 2^m + 120}{8^m}$ for $n=8$

and it seems the leading term is something like $\displaystyle {n \choose \lfloor(n-1)/2\rfloor}\left(\dfrac{\lfloor(n-1)/2\rfloor}{n} \right)^m$, though this may only be useful when $m$ is large. The other terms may also have a combinatorial expression