Compute expected counters after color elimination

Quick Overview

Compute expected counters after color elimination evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Compute expected counters after color elimination

Company: Optiver

Role: Data Scientist

Category: Statistics & Math

Difficulty: medium

Interview Round: Technical Screen

A bag contains 10 red, 10 yellow, and 10 blue counters. Repeatedly draw one counter uniformly at random without replacement (remove it from the bag) until the bag contains counters of only two colors. At that stopping time, what is the expected number of counters remaining in the bag?

Quick Answer: Compute expected counters after color elimination evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

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Jul 17, 2025, 12:00 AM
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Compute expected counters after color elimination

Problem

You have a bag with 30 counters: 10 red, 10 yellow, and 10 blue. You draw counters uniformly at random without replacement. Stop the moment the bag contains counters of only two colors (i.e., right after the 10th counter of some color has been drawn and that color is exhausted).

What is the expected number of counters remaining in the bag at that stopping time?

Assumptions/Clarifications

  • Draws are without replacement and uniformly random.
  • The stopping time is the first time any color is exhausted (its 10th counter is drawn). Ties cannot occur because one counter is drawn at a time.

Clarifying Questions to Ask Guidance

  • Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
  • Show enough derivation for the interviewer to follow the reasoning.
  • Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers Guidance

  • A correct setup with definitions, formulas, and boundary conditions.
  • A step-by-step derivation or estimation plan.
  • Interpretation of the result, including uncertainty and practical limitations.
  • Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions Guidance

  • How would the result change if the assumptions were relaxed?
  • Can you verify the answer with a simulation?
  • What is the most likely source of estimation error?
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