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  1. 077The bill at a restaurant comes to exactly pi rupees, and you can only pay in whole paise. How do you pay a perfectly fair amount?Expected value and decisionsCoreMillennium ManagementSheung Wan · 2025

    Try it first

    Which approach makes your payment exactly fair?

    Show the worked solution

    Randomise: pay Rs 3.15 with probability about 0.159 and Rs 3.14 otherwise. Pi is 3.14159, which is 15.9% of the way from 3.14 to 3.15. Weighting the two payable amounts by those odds gives an expected payment of exactly pi. Neither side is favoured on average, which is the only sense in which an amount you cannot pay can be paid fairly.

    What does fair mean when the exact amount cannot be paid?

    Five children and four mangoes: nobody can get four fifths of a mango each without a knife, so a fair parent draws lots and one child misses out, with every child facing the same chance. When the exact amount is impossible, fair means fair on average: the expected payment equals what you owe. Rounding to 3.14 short-changes the restaurant every time; rounding to 3.15 overpays every time. Only a random choice between the two neighbours can hit pi exactly.

    Pick the two amounts either side of pi, then weight them so the balance sits on pi84.1%15.9%Rs 3.14Rs 3.15pi = 3.14159, 15.9% of the way from 3.14 to 3.15pay Rs 3.14pay Rs 3.15Coin method: the target odds 0.1593 are 0.0010100011... in binary. Flip a coin for digits 0 and 1.Stop at the first flip that differs from that digit: below the target, pay 3.15; above, pay 3.14.
    Pi sits 15.9% of the way from Rs 3.14 to Rs 3.15, so putting 84.1% of the probability on 3.14 and 15.9% on 3.15 balances the payment exactly at pi, and a fair coin can deliver those odds by comparing random binary digits with 0.0010100011 and so on.

    How do you find the right odds?

    Call the chance of paying 3.15 by the letter p. The expected payment is 3.14 plus p times one paisa, and you want that to equal 3.14159. So p is the fraction of the gap you still owe: 0.00159 over 0.01, which is 0.1593. Say it as a balance point: the further pi sits towards 3.15, the more often you pay 3.15.

    The relationship
    3.14+p×0.01=π⇒p=π−3.140.01≈0.15933.14 + p \times 0.01 = \pi \quad\Rightarrow\quad p = \frac{\pi - 3.14}{0.01} \approx 0.1593
    pthe probability of paying Rs 3.15
    0.01one paisa, the gap between the two payable amounts
    pi - 3.14the part of a paisa you still owe after paying 3.14
    What it says in wordsPay the higher amount with a probability equal to the share of the paisa you still owe.

    How do you actually produce odds of 0.159 with a coin?

    This is the follow-up that separates people. Write p in binary: 0.0010100011 and so on. Flip a fair coin to generate your own random binary digits, one at a time, and compare each with p's digit in the same place. At the first flip that differs, you know whether your random number is below p or above it, and you pay 3.15 or 3.14 accordingly. Each flip has a one in two chance of settling it, so on average two flips decide, even though p itself is irrational. The same idea, called randomised roundingRounding a fractional quantity up or down at random, with odds set so that the expected result equals the exact fraction., is used to split odd lots of shares fairly across client accounts.

    Where candidates lose it

    Most candidates say pay 3.14 or 3.15 and argue about which side should bear the fraction of a paisa. That treats the question as etiquette. The interviewer wants the expected value framing: fair on average is the only fairness available.

    The second loss comes on the follow-up. Candidates who reach p = 0.159 then say roll a thousand-sided die, which only approximates it. The binary coin comparison hits the odds exactly and needs about two flips.

    What the interviewer asks next

    • You eat at the same restaurant every day. Is there a non-random way to be fair over time?
    • How would you generate a probability of exactly one third with a fair coin?
    • The waiter is risk averse. Does paying pi in expectation still feel fair to him?

    Asked at Millennium Management, Quantitative Research, Sheung Wan, 2025 (Wall Street Oasis): How to pay the restaurant fairly if I owe pi dollars. Need to pay with usual dollars and cents.