Daily Sampling: First-Selection Day

Context

• There are 1,000 people. Each day, 10 distinct names are drawn uniformly at random.
• We consider two regimes across days:
  1. Without replacement across the entire campaign: once a person is selected, they are not eligible again until all 1,000 have been selected (i.e., a random permutation of 1,000 split into 100 days of 10 each).
  2. With replacement across days: each day is an independent fresh draw of 10 distinct names from the full 1,000 (so someone can be drawn on multiple days).
• “Day” counting starts at 1.

Question

1. Without replacement (no repeats until everyone is chosen), what is the expected day on which a given person is first selected?
2. With replacement (names can repeat across days), what is the expected day on which that person is first selected?

Hints

• No-replacement waiting time is uniform on 1…100.
• With replacement it follows a geometric distribution with p = 0.01.