Let $X=(X_1,\ldots,X_n)$ be an iid sequence of random variables, and let $\nu$ be a *uniformly random* integer in the range $1,\ldots,n$. Then $\xi_\nu$ is a random entry of $X$. Is it always true that after deleting such a random entry from $X$, the remaining sequence is still iid?

Due to the uniform randomness of the index $\nu$, it is tempting to answer that the leftover sequence always remains iid. This is correct, if we also assume that $\nu$ is chosen *independently* of $X$.

What happens, however, if no such independence is assumed? For example, consider a 0-1 valued iid sequence. Remove a random 1, chosen uniformly at random among all 1s. If there is no 1, then remove a random 0. In this situation, the position of the removed entry is still uniformly random, but not independent of the original sequence. Can such a removal destroy the iid property in the remaining sequence?