← Optiver Interview Insights

Optiver·Data Scientist·Technical Phone Screen·Intermediate

Intermediate
Jun 2026

Summary

Optiver data science interview had a timed probability question that was genuinely stressful. The 90-second clock made it hard to think clearly, and the extensions they asked for afterward were where things got interesting.

Questions Asked (1)

Q1

An urn has r red, b blue, and g green balls. You draw 3 without replacement. What's the probability of getting at least 2 reds? You have 90 seconds, and they also want a mental-math shortcut that generalizes.

Algorithms & Data StructuresProduct Analytics & Metrics
Author's notes

The 90-second cap is real and they mean it.

Create a free account to read the full note

AI HintsAI Generated

Suggested Approach

First, clarify that 'at least 2 reds' means exactly 2 or exactly 3 reds, and compute the probability using combinations: P = [C(r,2)C(b+g,1) + C(r,3)] / C(r+b+g,3). Then, for the mental-math shortcut, use the binomial approximation with p = r/(r+b+g) and note that when the total number of balls is large relative to 3, the without-replacement draws are nearly independent, so P ≈ 3p²(1-p) + p³ = 3p² - 2p³.

Pro tip: Optiver values speed and intuition: state the exact formula but emphasize the approximation and its error bound (O(1/N)), showing you can balance rigor with practicality.

1. Clarify the event

Restate that 'at least 2 reds' includes exactly 2 reds and exactly 3 reds, and that draws are without replacement.

2. Set up exact combinatorial formula

Write the probability as [C(r,2)C(b+g,1) + C(r,3)] / C(r+b+g,3), explaining each term.

3. Derive mental-math shortcut

Let p = r/(r+b+g). For large N, approximate with replacement: P ≈ 3p²(1-p) + p³ = 3p² - 2p³.

4. Quantify approximation error

Mention that the error is small when N is large (e.g., O(1/N)), and give a quick sanity check with numbers.

Key Points to Mention

  • Combinations formula for without-replacement probability
  • Binomial approximation for large populations
  • The shortcut formula 3p² - 2p³
  • Error scaling with sample size (finite population correction)
  • Sanity check with extreme values (e.g., r=0 or r=N)
  • Time management: state exact formula quickly, then focus on approximation

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.