Suppose there are $m$ balls to be randomly thrown into $n$ bins ($m>n$). Let $X\_i$ be the number of balls ending up in bin $i$.


Let $X\_{max}$ be the heaviest bin and $X\_{min}$ be the lightest bin. In [Raab and Steger's paper](https://doi.org/10.1007/3-540-49543-6_13), the authors state that
$$
Pr[X\_{max}≥k]≤o(1).
$$
However, is there anyway that I can figure out how small $o(1)$ can be? Say, I want the probability to be bounded to some particular value $\frac{1}{n^w}$ ($w≥1$)
$$
Pr[X\_{max}≥k]≤\frac{1}{n^w},
$$
then, what should $k$ be in order to achieve the bound? (I just cannot figure it out in their paper.)


Many thanks for your help!



Balls and bins: Exact probability
You can find the answer in [this paper by Reviriego, Holst and Maestro](http://www.nebrija.es/~jmaestro/esa/papers/JDA2011.pdf). 
Especially interesting should be formula 23 giving the exact probability distribution. 
Of course it can be easily seen that your term “heaviest bin” is equivalent to the term “Longest Length Probe Sequence” used in the paper.


