Zoox·Machine Learning Engineer·Technical Phone Screen·Senior

Senior

Jun 2026

Summary

Zoox ML engineer interview with a probability question that looks straightforward but has three parts that build on each other. The Bayesian twist in part (b) is where things get interesting.

Questions Asked (1)

You pick either a 6-sided or 8-sided die with equal probability and roll it. What is the probability of rolling a 3? Then, given you rolled a 3, what is the probability the die was the 6-sided one? Finally, using that posterior as your new prior, what is the expected value of the next roll on that same die?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

Part (a) is just total probability: (1/2)(1/6) + (1/2)(1/8) = 7/48, fine.

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AI HintsAI Generated

Suggested Approach

Break the problem into three parts: first compute the marginal probability of rolling a 3 using the law of total probability, then apply Bayes' theorem to find the posterior probability that the die was 6-sided given a 3, and finally use that posterior as the prior for the next roll to compute the expected value as a weighted average of the expected values for each die type.

Pro tip: Clearly state your assumptions (e.g., fair dice, independence of rolls) and show your work step-by-step; this demonstrates rigor and helps the interviewer follow your reasoning, especially important for ML roles where probabilistic reasoning is key.

1. Compute marginal probability of rolling a 3

Use the law of total probability: P(3) = P(3|6-sided)*P(6-sided) + P(3|8-sided)*P(8-sided). Since each die is chosen with equal probability, P(6-sided)=P(8-sided)=0.5. For a fair 6-sided die, P(3|6)=1/6; for a fair 8-sided die, P(3|8)=1/8. Thus P(3) = (1/6)(0.5) + (1/8)(0.5) = 7/48.

2. Apply Bayes' theorem for posterior

Compute P(6-sided|3) = P(3|6-sided)*P(6-sided) / P(3). Substitute the values: (1/6 * 0.5) / (7/48) = (1/12) / (7/48) = 4/7. So the posterior probability is 4/7.

3. Use posterior as new prior for next roll

After observing a 3, the updated probability that the die is 6-sided is 4/7, and that it is 8-sided is 3/7. This becomes the prior for the next roll on the same die.

4. Compute expected value of next roll

The expected value of a roll given a 6-sided die is (1+2+3+4+5+6)/6 = 3.5. Given an 8-sided die, it is (1+2+3+4+5+6+7+8)/8 = 4.5. The overall expected value is the weighted average: (4/7)*3.5 + (3/7)*4.5 = (14/7) + (13.5/7) = 27.5/7 ≈ 3.9286.

Key Points to Mention

Law of total probability for marginal probability
Bayes' theorem for posterior probability
Assumption of fair dice and independence of rolls
Expected value calculation for discrete uniform distributions
Use of posterior as updated prior for sequential decision making
Clear communication of steps and intermediate results

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