My friends and I came up with the question while we are discussing some board game rules, but we cannot figure out the solution, so I am posting here for help.
The problem can be abstracted as:
We have $x$ boxes, each box contains $y$ items. In each turn, a player would randomly pick a box and attempt to take out 1 item from the box and does NOT put it back. Assume that before picking, each player does not know if the box is empty or not, and he could not remember it. Hence the players might pick from an empty box and waste their turns.
The game has $x \cdot y$ turns in total, so if all players make the right decision, they could just pick out all items from all boxes. In the worst case, they could still pick out $y$ items in total (assuming they are repeatedly picking from the same box).
For simplicity, assume $x > 0$ and $y > 0$.
The question is, what's the expectation $\mathop{\mathbb{E}}$ of number of items picked out in a game?
I am not very familiar with random process and does not know which field this problem belongs to. I don't expect someone to give me the solution directly (but certainly would welcome that if you have time), but hope someone to point me to a direction where I could look into it.
Really appreciate your help!