Let $\xi=(\xi_1,\ldots,\xi_n)$ be a sequence of independent random variables. Let us pick an index $\nu\in \{1,\ldots,n\}$, and replace the entry $\xi_\nu$ by a constant $c$. The rest of the $\xi_i$ remain unchanged.

**Question:** Is it true that the altered sequence remains independent, even if $\nu$ is random, possibly dependent on $\xi$?

*Note:* If $\nu$ is a fixed constant index, then the independence is clearly preserved. It is less clear, however, what happens if $\nu$ is random and dependent on $\xi$. For example, we may pick the first hit to some set, or apply any other data-dependent rule to pick the index $\nu$.