I have the following problem I would like to get some input for. Let say I have a set of numbers from 1 to 500 : and i am preforming the following experiment (sampling with replacement). I randomly pick 100 numbers from A. Since i am assuming the picking is done randomly i can assume that each pick happens with probability that is i have an equal chance to pick 1 and 40. since i am picking 100 numbers from those 500 (with replacement) i can get for a result 100 1's each with the same probability . eventually if i repeat this experiment many times i'll get two 1's two 2's, two 3's and so on... (is this clear so far). Now a single experiment is picking 100 numbers from those 500 allowing replacement. since in one experiment i can get four 1's and in the next (assuming i repeated the experiment) i can get ten 1's my question is will the frequencies of 1's be also uniformly distributed after i repeat the experiment many times. And how to prove that? If not how can i prove the opposite and figure out the distribution.