Estimate first selection round with/without replacement

There are 1000 people. Each round, 10 selections are made. Case 1 (without replacement across the entire process): in total 100 rounds are run to select everyone exactly once (10 per round); a given person's position in the random permutation determines their round. For a fixed person, compute the expected round index of their first selection. Case 2 (with replacement within and across rounds): each round consists of 10 independent draws with replacement from the 1000 people; a person is counted as 'selected' in a round if they are drawn at least once that round. For a fixed person, derive exactly the per-round selection probability p and the expected number of rounds until their first selection (a geometric random variable). Provide a first-order approximation to p.