Mavensecurities·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

May 2026

Summary

Quant engineer interview at Maven Securities with a probability/expected value question that's basically a textbook problem if you've seen it before, but probably rough if you haven't.

Questions Asked (1)

Antony collects toys found inside cereal boxes. There are 5 distinct toys, each box contains one toy chosen uniformly at random, and each box costs £4. What is the expected total amount Antony spends to collect all 5 toys?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

This is the coupon collector's problem dressed up in a cereal box costume.

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AI HintsAI Generated

Suggested Approach

Recognize this as a coupon collector problem where each toy is equally likely. The expected number of boxes to collect all 5 toys is 5 * H_5, where H_5 is the 5th harmonic number. Multiply by the cost per box (£4) to get the expected total spend.

Pro tip: Mention that the coupon collector expectation is a classic result and that you can derive it by summing the expected waiting times for each new toy. This shows you understand the underlying probability, not just memorized formulas.

1. Identify the problem type

Recognize that this is the coupon collector problem: collecting all distinct items when each trial yields a uniformly random item.

2. Compute expected boxes for each new toy

When you have k distinct toys, the probability of getting a new one is (5-k)/5, so the expected boxes to get the next new toy is 5/(5-k). Sum these for k=0 to 4.

3. Sum to get total expected boxes

The sum is 5/5 + 5/4 + 5/3 + 5/2 + 5/1 = 5 * (1 + 1/2 + 1/3 + 1/4 + 1/5) = 5 * H_5, where H_5 = 137/60 ≈ 2.2833.

4. Multiply by cost per box

Total expected cost = (expected boxes) * £4 = 5 * H_5 * 4 = 20 * H_5 = 20 * 137/60 = 137/3 ≈ £45.67.

5. Present the final answer clearly

State the exact fraction and decimal: £45.67 (or 137/3 pounds). Optionally mention that this is the expected value, not a guaranteed amount.

Key Points to Mention

Coupon collector problem and its expected value formula: n * H_n
Harmonic number H_5 = 1 + 1/2 + 1/3 + 1/4 + 1/5 = 137/60
Expected boxes = 5 * H_5 = 137/12 ≈ 11.4167
Total expected cost = 4 * (137/12) = 137/3 ≈ £45.67
Explanation that each toy is equally likely and independent
Clarify that this is the expected value, not a deterministic outcome

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