Suppose there are $N$ people and $N$ prizes, and only $M$ out of $N$ are valuable.
Every time one person is picked randomly, then he pick one prize randomly as well (this prize/person is then removed from the list). I would like to know what is the expected probability each person can get the valuable prize if
(i) the prizes all look the same (one doesn't know which one is valuable), and
(ii) those $M$ prizes look different (every one know what left are valuables)?
My intuition shows that people will get more chance to win the valuable prize if they know what among them are actually valuable in advance. However, it seems wrong since the sum of the probability is always 1. So I think this depends on when you were picked up to select the prize. Is there a mathematical expression on this probability on every person?
From my understanding of the question you are asking does picking at random or not effect the expected outcome for players of the game despite that there turns are randomized in the first place. The answer is no. Here is an analogy, imagine we pick 5 random numbers in a list. Then we make a random permutation of the list. We still have a list with 5 random numbers and the expectation of those numbers doesn't change.