Calculate Expected Day for First Selection in Sampling
Quick Overview
Meta probability prompt comparing expected first-selection day under sampling without replacement versus with replacement, using random permutations, uniform day distribution, and geometric waiting time.
Calculate Expected Day for First Selection in Sampling
Company: Meta
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Onsite
##### Scenario
Daily sampling of 10 names from 1 000 people
##### 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 each day), what is the expected day on which that person is first selected?
##### Hints
The no-replacement waiting time is uniformly 1…100; with replacement it follows a geometric distribution with p = 0.01.
Quick Answer: Meta probability prompt comparing expected first-selection day under sampling without replacement versus with replacement, using random permutations, uniform day distribution, and geometric waiting time.
Calculate Expected Day for First Selection in Sampling
Meta
Jul 12, 2025, 6:59 PM
easyData ScientistOnsiteStatistics & Math
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Expected Day of First Selection in Daily Sampling
There are 1,000 people. Each day, 10 distinct names are selected uniformly at random. Day counting starts at 1.
Compare two regimes:
Without replacement across the whole campaign:
once a person is selected, they are not eligible again until all 1,000 have been selected. This is equivalent to a random permutation split into 100 days of 10 people each.
With replacement across days:
each day is a fresh draw of 10 distinct names from the full 1,000, so the same person can be selected on multiple days.
Constraints & Assumptions
The person of interest is fixed before the sampling starts.
In the with-replacement case, daily selection events for that person are independent across days.
Report expected day, not probability of selection by a given day.
Clarifying Questions to Ask Guidance
Does "without replacement" apply only within a day, or across the whole campaign?
Are we choosing exactly 10 distinct people each day?
Does day 1 count as the first possible selection day?
What a Strong Answer Covers Guidance
Without replacement, the person's position in the random permutation is uniform from 1 to 1,000.
Grouping positions into 100 days of 10 makes the first-selection day uniform on days 1 through 100.
The expected day without replacement is
(1 + 100) / 2 = 50.5
.
With replacement across days, daily selection probability is
10 / 1000 = 0.01
.
The first-selection day follows a geometric distribution with mean
1 / 0.01 = 100
.
Clear explanation of why the two regimes differ.
Follow-up Questions Guidance
What is the probability of being selected by day 30 in each regime?
How would the expected day change if 20 names were drawn each day?
What if a person could be selected more than once within the same day?