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Calculate Expected Day for First Selection in Sampling

Last updated: Mar 29, 2026

Quick Overview

This question evaluates a candidate's understanding of probability theory and expected value in finite sampling processes, focusing on contrasts between sampling with and without replacement.

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

Without replacement (no repeats until everyone is chosen), what is the expected day on which a given person is first selected?
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.

Solution

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