This is just a problem I have thought of.
Let us say we have a square matrix of dimension $n$ with possible entries $1, 2, 3,..., n^2$ and we can freely distribute the entries in the matrix, each one has to be used exactly once.
The numbers then are drawn without replacement.
The probability that number $k$ is drawn is $0 \leq P(k) \leq 1$ with $\sum_{k=1}^{n^2}P(k)=1$
Now, we know the probabilities of the draws before we fill in our matrix. Let us say we sorted them such that $0 \leq p_1 \leq p_2 \leq ... \leq p_{n^2}$. We want to cheat and optimally distribute the numbers in order to minimize the expected number of draws until we cross out a whole row or column.
Is this a known problem? Is there an optimal strategy on how to distribute the probabilities $p_i$ in our matrix to minimize the expected number of draws until we cross out a whole row or column, and what is one such strategy?
I was thinking of determining the expected value of number of draws to get a certain number (then determine expected draws to get whole rows and columns and minimize those), but I do not think I can define it only given the $P(k)$, I would need to define how they change after each draw.
I was thinking that it is reasonable to remodel the probabilities after each draw for nonzero $P(k)$. Let us call $D$ the set of already drawn numbers. Then for $k \notin D$ with $P(k)\neq 0$ we do $$P'(k)=\frac{P(k)}{1-\sum_{ d \in D}P(d)}$$
But it sounds horribly complex!
Then again, it shouldn't matter, because if we have some remaining nonzero probabilities, their new probabilities are in the same order, that is if $P(k) \geq P(l)$ and both were not drawn, then $P'(k) \geq P'(l)$.