Let $K$ be a finite (but very large) positive integer. Let $P=\{p_{k};k=1,\cdots,K\}$ be a probability distribution on an alphabet $\mathscr{X}=\{\ell_{k}; k=1,\cdots,K\}$. Let $p_{\wedge}=\min \{p_{k}\}$. Can anyone please tell me how $p_{\wedge}$ is related to a practical problem, or how knowing $p_{\wedge}$ (or estimating it) is of practical interest in some scientific context (physics, information theory, security, etc.)?
The reason I ask this question is that I think I have a good way of estimating $p_{\wedge}$ (non-parametrically) but could not think of a motivating context.
In fact, I could not even find literature references on estimating minimum probability. If you have any idea on references to this problem, please share - I would much appreciate it.