Jane Street–styleTraderTraderQuestion #61
What is the expected value of the maximum of two fair six-sided dice?
Free worked solution · Quant Blueprint
161/36, approximately 4.4722.
- Let M = max(X,Y), with independent fair dice. For k = 1,…,6, P(M ≤ k) = (k/6)².
- Subtract adjacent cumulative probabilities: P(M = k) = [k² − (k−1)²]/36 = (2k−1)/36.
- Then E(M) = Σk(2k−1)/36 = (1+6+15+28+45+66)/36 = 161/36.
Common mistake
Taking the larger of the two means gives 3.5, but taking a maximum after observing outcomes has additional value.
Follow-up
For the minimum m, M + m = X + Y in every outcome. Therefore E(m) = 7 − 161/36 = 91/36.
Original educational explanation for this practice question. Reviewed . Firm-style practice; the label describes the collection’s preparation context.
More Jane Street–Style Trader Questions
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