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, 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!