Assume $I=(I_1,I_2,I_3)$ is a multinomially distributed random variable, say $\mathrm{Mult}(n;p_1,p_2,p_3)$. What is the expectation of the minimum of the first two components $\mathbb E[\min(I_1,I_2)]$?
Experiments suggest that in the limit, we have $$ \lim_{n\to\infty} \frac{\mathbb E[\min(I_1,I_2)]} {\min\{\mathbb E[I_1], \mathbb E[I_2]\}} = \lim_{n\to\infty} \frac{\mathbb E[\min(I_1,I_2)]} {n\min\{p_1,p_2\}} = 1 \,, $$ which allows an approximation at least. (Here the $p_i$ are constant, only the number of trials is increased.) Is this correct? How to prove it?
The direct attempt at computing the expectation by splitting the sum in two, one for the case $I_1\le I_2$ and one for $I_1>I_2$, yields two ugly sums that I could not simplify.
