Upstart·Software Engineer·Technical Phone Screen·Senior

Senior

May 2026

Summary

Research Scientist interview at Upstart that went deep into probabilistic modeling and simulation. The core problem was meaty enough that it probably took up most of the session on its own.

Questions Asked (1)

Build a Monte Carlo simulation to estimate the expected number of players who get banned in a coin-flip game where each player makes up to 100 guesses and gets banned after 10 consecutive correct guesses. Then compare your simulation result to an analytical solution and talk through how many trials you need for stable estimates, plus any variance reduction techniques.

Algorithms & Data StructuresTechnical Trade-offsA/B Testing & Experimentation

Author's notes

This one had a lot of layers.

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AI HintsAI Generated

Suggested Approach

Start by clearly defining the problem and the analytical solution, then outline the simulation design including trial count and variance reduction. Walk through the code structure and statistical considerations, and finally discuss how to validate results and ensure stable estimates.

Pro tip: Mention that you would use the analytical solution to validate the simulation, and discuss how variance reduction techniques like antithetic variates or control variates can be applied to this specific problem to improve efficiency.

1. Clarify the problem and analytical solution

Restate the problem to ensure understanding: each player flips a fair coin up to 100 times, and gets banned if they achieve 10 consecutive heads (or correct guesses). Derive or state the analytical expected number of banned players.

2. Design the Monte Carlo simulation

Outline the simulation: for each trial, simulate N players, each with up to 100 flips, tracking consecutive correct guesses. Count bans per trial, then average over many trials. Choose N and number of trials based on desired precision.

3. Determine required trials for stable estimates

Discuss how to compute the standard error of the mean ban count and use it to determine the number of trials needed for a given confidence interval. Mention that the required trials depend on the variance of the ban count per trial.

4. Apply variance reduction techniques

Suggest techniques like antithetic variates (using complementary random streams) or control variates (using a correlated variable with known expectation) to reduce variance and speed up convergence.

5. Validate and interpret results

Compare simulation results with the analytical solution, discuss discrepancies, and ensure the simulation is correctly implemented. Talk about how to present results with confidence intervals.

Key Points to Mention

Analytical solution: expected number of bans can be computed using Markov chain or dynamic programming; for a single player, probability of ban within 100 flips.
Simulation structure: loop over trials, within each trial simulate players, track consecutive successes, and count bans.
Law of Large Numbers and Central Limit Theorem: justify why averaging over many trials gives stable estimates.
Standard error and confidence intervals: how to compute and use them to determine required number of trials.
Variance reduction: antithetic variates, control variates, or importance sampling can be applied to reduce the number of trials needed.
Computational considerations: vectorization, parallelization, and efficient random number generation for large-scale simulations.

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