This is adapted from a little game I've played recently:
Suppose that every day you get to select one sealed box out of nine and collect whatever item is in the box. How the $n>9$ items are placed in the various boxes changes every day. (Note that there is only one item in each box, and the $n$ items are unique.) Suppose that of the $n$ items you're really only interested in one of them, so that item is what you hope to pick every day, considered here as "winning the game". Here are two things I'm curious to know:
1) What is the probability of winning on a given day?
2) Over a period of $9<k<n$ consecutive days, what is the probability of winning? More precise perhaps, what is the probability of winning at least once?
I don't really have much to contribute on my own. There are $n!/(9!(n-9!)$ ways that the boxes could be filled, and the event of losing or winning has no bearing on the probability of losing or winning the next day, i.e. independence.
EDIT: The chosen item is replaced.