Given 8 people choosing a random number between 1 and 100, with repetition allowed, what's the chance of being the one to choose the highest number? Assuming it is truly random.

Obviously the chance of choosing one particular number is $ 1/100 $

If rank didn't come into play, I would think it would simply be $ 1/8 $

However, even with the choices being independent, the results rely on all the others, so would they technically be dependent?

That and the rank of the numbers coming into play is what's confusing. It's been several years since I touched a probability textbook and have forgotten most of all of it.

Now, assuming that implies they are dependent, how does one chain 8 conditional probabilities into one calculation?

My best guess for notation is $ P((A | B)|C...) $ but I'm not sure where I would start with calculations.