Suppose we have $n$ distinct balls labeled $1,2,\dots,n$ in a black box. Now we want to fetch $k$ balls from the box, one by one. Let event $E_i$ be that the label of the $i$-th ball we fetch is no more than $m$, where $m \leq n$.

By intuition I think events $E_1, E_2, \dots, E_k$ are negatively associated. However after some simple tries I feel that I cannot easily make a proof. Since this process is quite natural I believe it has been studied before or it is quite easy to people who are familiar with probability theory.

Question. Can we prove that $E_1, E_2, \dots, E_k$ are negatively associated?

This question arises in my research, in which I need to solve it to prove that my algorithm behaves as desired. However I am worried that it may not be a research-level question because of its simple form. Let me know if it is not appropriate to put this question here.


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