I am trying to solve this problem. My idea so far is to compute what is the probability of each item on a particular shelf being tasted. But since the number of pots on a shelf could reach 10^5, this approach doesn't seem feasible. Since I couldn't find the editorial for this problem, I would really appreciate if someone could point me in the right direction.
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You should use the following idea: the number of untasted pots on any shelf can not be increased, so at any moment of time this number for i-th shelf is no greater than Ai. Any probability value or expected value can be calculated using only the number of untasted pots on any shelf and the total number of pots on them.