This is related to the question I have asked yesterday: Expected value of max/min of random variables.
Assume you have $n$ urns and $k$ balls. Each ball is placed uniformly at random in one of the urns. Let $X_i$ denote the number of balls in urn $i$ and let $X = \min\{X_1,\ldots,X_n\}$.
I am looking for a $k$ such that $ Pr[X < 2\log(n)] < \frac{2}{n}. $ Clearly
$ Pr[X < 2\log(n)] = Pr[\bigcup_{i=0}^n X_i < 2\log(n)] \leq n*Pr[X_1 < 2\log(n)]$
Here is where it stops for me. We have to find an upper bound for $Pr[X_1 < 2\log(n)]$ but as far as I am apt in applying Chernoff/Markov bounds, one can only get a lower bound for this kind of expression.
Am I missing something? Or is there perhaps another way to solve the problem?