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  1. 031You roll two fair dice and are paid the higher of the two faces, in rupees. What is the expected payout?Expected value and dice gamesWarm upWolverine TradingChicago · 2025

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    Your quick estimate:

    Show the worked solution

    161/36, about Rs 4.47. The higher face equals k in 2k minus 1 of the 36 equally likely rolls: 1, 3, 5, 7, 9 and 11 rolls for k from 1 to 6. Multiply each value by its count, add to 161, and divide by 36. The lower face averages 91/36, about 2.53, and the two add to 7, the average total of two dice, which is a quick check.

    How many of the 36 rolls give each maximum?

    Think of two runners and a prize for the faster time: the winning time is better than a typical single runner's because you always keep the better of two. The higher face is at most k in k x k of the 36 rolls, so it equals exactly k in k squared minus (k minus 1) squared, which is 2k minus 1 rolls. That gives 1 roll with a maximum of 1, 3 with a maximum of 2, and on up to 11 with a maximum of 6. On the grid those cells form L shapes that grow as you move towards the corner.

    The 36 rolls, each showing the higher face: big maxima own more cells123456223456333456444456555556666666112233445566rows: first die, columns: second dieCells with each maximummax 11max 23max 35max 47max 59max 611(1x1 + 2x3 + 3x5 + 4x7 + 5x9 + 6x11) / 36= 161 / 364.47
    Of the 36 equally likely rolls, the higher face is 1 in just one roll and 6 in eleven rolls, so the expected maximum is 161/36, about 4.47, well above the 3.5 of a single die.
    The relationship
    E[max⁡]=∑k=16k⋅2k−136=16136≈4.47E[\max] = \sum_{k=1}^{6} k \cdot \frac{2k-1}{36} = \frac{161}{36} \approx 4.47
    kthe value of the higher face
    2k - 1the number of rolls, out of 36, whose higher face is exactly k
    What it says in wordsWeight each possible maximum by how many of the 36 rolls produce it.

    How do you check 4.47 in ten seconds?

    Use the pair. The higher face plus the lower face always equals the total of the two dice, so their averages must add to 7. The lower face is at least k in (7 minus k) squared rolls, which gives an average of 91/36, about 2.53. 4.47 plus 2.53 is 7.00. A second method that lands exactly is what makes an interviewer stop checking your arithmetic and move on to the follow-up.

    The follow-up is usually a game. If you could pay to roll one die or to roll two and keep the higher, the second is worth about Rs 0.97 more. That gap, the value of a free second look, is the same idea as an option: the right to choose after seeing the outcome is worth paying for.

    Where candidates lose it

    The common loss is answering 3.5 plus something vague, or 5, from instinct. Both skip the count of how often each maximum occurs, which is the whole question.

    The second is listing all 36 rolls one by one under time pressure. Say the 2k minus 1 rule, give 161 over 36, and use the lower face check to show the number is right.

    What the interviewer asks next

    • What is the expected higher face with three dice?
    • What would you pay to roll two dice and keep the higher, if you could reroll both once?
    • What is the expected value of the lower face, and why do the two add to 7?

    Asked at Wolverine Trading, Equity Hedge, Chicago, 2025 (Wall Street Oasis): Typical dice questions that you can find in most probability textbooks