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  1. 075You roll two fair dice and are paid the product of the two faces in rupees. What is the expected payout?Expected value and optimal stoppingWarm upWolverine TradingChicago · 2024

    Try it first

    What is the expected product?

    Show the worked solution

    Rs 12.25. The two dice are independent, so the expected product equals the product of the expected faces: 3.5 x 3.5 = 12.25. A check on the 6 by 6 grid: row i averages 3.5i, and the six row averages, 3.5 to 21, average 12.25. If both numbers came from one die, the answer would be E[X squared] = 91/6, about 15.17, which is not the question.

    Why can you multiply the averages?

    Picture a shop whose daily takings are footfall times average spend, where the two have nothing to do with each other. Over a year, average takings are average footfall times average spend. When two quantities are independent, the average of their product is the product of their averages, because knowing one tells you nothing about how big the other will be. That fails the moment they move together: a crowded day with bigger spending lifts the product above the product of averages. Two dice thrown separately are the textbook independent pair.

    Average of the grid = row average x column average = 3.5 x 3.5123456second die1123456avg 3.5224681012avg 73369121518avg 10.544812162024avg 14551015202530avg 17.5661218243036avg 21first diedark cells: above 12.25 (13 of 36)Two independent diceE[XY] = E[X] E[Y]3.5 x 3.5 = 12.25Not the same die squaredE[X^2] = 91/6= 15.17, higher by the varianceThe gap, 35/12 = 2.92, is one die's variance
    Across the 36 equally likely cells of the multiplication grid each row averages its row number times 3.5, so the grand average is 3.5 x 3.5 = 12.25, even though 23 of the 36 products are below it; squaring a single die would give 15.17 instead.

    How do you check 12.25 by brute force quickly?

    Sum the grid by rows. Row i of the multiplication table sums to i x 21, so the whole grid sums to 21 x 21 = 441, and 441 / 36 = 12.25. That is the product rule made visible: the grid's total factors into the first die's total times the second die's total. Note too that the product is skewed: only 18 different values appear, most cells are small, and the big ones, 25, 30 and 36, pull the mean up, so {BELOW75} of the 36 outcomes pay less than the average.

    The relationship
    E[XY]=136∑i=16∑j=16ij=21×2136=44136=12.25=E[X] E[Y]E[XY] = \frac{1}{36}\sum_{i=1}^{6}\sum_{j=1}^{6} ij = \frac{21 \times 21}{36} = \frac{441}{36} = 12.25 = E[X]\,E[Y]
    X, Ythe two independent die faces
    21the sum of the faces 1 to 6
    441the sum of all 36 products
    What it says in wordsThe grid's total is the product of the two dice's totals, so the average product is the product of the averages.

    What is the interviewer likely to ask next?

    The usual next step changes the dependence. If the same die is used for both numbers, you are paid the square of one roll, and its expectation is 91/6, about 15.17, higher by exactly the variance of one die, 35/12. That gap is the covariance at work: E[XY] = E[X]E[Y] + Cov(X, Y). Then comes the price: if you would pay to play, you should quote around 12.25 and note that the payout's standard deviation is about 8.94, so a single play is very noisy relative to its mean.

    Where candidates lose it

    The trap is computing E[X squared] instead of E[XY], getting 15.17, usually because the candidate thinks of rolling one die and squaring. The other slip is answering 18 by taking the midpoint of 1 to 36.

    Say independent and give 3.5 x 3.5 inside five seconds; then offer the row-sum check, 21 x 21 / 36, so the interviewer sees you can verify it.

    What the interviewer asks next

    • What is the expected payout if you are paid the square of a single roll?
    • What is the variance of the product of two dice?
    • You may reroll one of the two dice once after seeing both. What is the game worth?

    Asked at Wolverine Trading, Sales and Trading, Chicago, 2024 (Wall Street Oasis): don't think I was what they were looking for. Questions on EV & Dice.