Derivatives Foundation puzzles, solved step by step

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  1. 026I will pay you if at least 60% of the flips of a fair coin come up heads. Do you want 10 flips or 100 flips?Distributions and statisticsWarm upHudson River TradingNew York · 2020

    Try it first

    Before you count anything: which do you take?

    Show the worked solution

    Take 10 flips. You are betting on luck, and luck averages out as the trials pile up. With 10 flips, 6 or more heads happens 386 times in 1,024, about 37.7%. With 100 flips, 60 or more heads happens about 2.8% of the time. The share of heads settles towards 50% at the rate of one over the square root of n, so the 60% line gets harder to reach with every extra flip.

    Why does the number of flips change the odds at all?

    Think of a school with two cricket teams, one of eleven and one of a hundred and ten. If someone offers a prize for a team whose average height is 10 cm above the national average, the small team is the one that can win: one or two tall players move its average, while the big team's average is pinned down by sheer numbers. The same coin, flipped more often, produces a share of heads that sits ever closer to a half, so a payout that needs an unusual share wants the fewest flips you can get. This is the law of large numbers working against you, and the question is testing whether you know which side of it you are on.

    The same 60% line, 10 flips against 100: the share of heads tightens around a half10 flips01234567891060% of 10 = 6 headsP(6 or more) = 37.7%the line sits 0.6 sd from the meanheads out of 10100 flips304050607060% of 100 = 60 headsP(60 or more) = 2.8%2.0 sd from the meanheads out of 100 (30 to 70 shown)A bet that needs luck wants the fewest trials: the spread of the share of heads shrinks like 1 over the square root of n
    With 10 flips the bars at 6 heads and above hold 37.7% of the probability, but with 100 flips the bars at 60 heads and above hold only 2.8%, because the share of heads tightens around a half as the flips increase.

    How do you put a number on it without a table?

    Count the small case exactly: 6 or more heads in 10 flips means adding the ways to get 6, 7, 8, 9 and 10 heads, which are 210, 120, 45, 10 and 1, a total of 386 out of 1,024, so 37.7%. For 100 flips use the normal approximation. The standard deviation of the share of heads is 0.5 over the square root of n: 15.8% for 10 flips, 5% for 100. The 60% line is 0.6 standard deviations out in the first case and 2.0 in the second, and two standard deviations in one tail is about 2.3%. The exact binomial answer is 2.8%; the approximation gets you to the right decision in one breath.

    The relationship
    σXˉ=0.5nz10=0.10.158=0.63,z100=0.10.05=2.0\sigma_{\bar{X}} = \frac{0.5}{\sqrt{n}} \qquad z_{10} = \frac{0.1}{0.158} = 0.63, \quad z_{100} = \frac{0.1}{0.05} = 2.0
    sigma of X barthe standard deviation of the share of heads
    nthe number of flips
    zhow many standard deviations the 60% line sits from the mean of 50%
    What it says in wordsThe 60% line gets further from the centre, measured in standard deviations, as the flips increase, so it becomes rarer to cross.

    What is the interviewer listening for after the answer?

    Say the general rule and then the exception. The rule: whenever a payout needs the sample to look unlike the population, choose the smallest sample. The exception: if the payout were for landing between 40% and 60%, you would want the most flips, for exactly the same reason. Read the sign of the bet before you choose the sample size: a bet on luck wants few trials and a bet on the average wants many. If the interviewer changes the wording to more than 60%, 7 or more heads in 10 is 17.2%, still far above the 100-flip figure. And if the choice is between 100 and 1,000 flips, 600 or more heads happens about 1.36e-10 of the time, which is as close to never as a desk needs.

    Where candidates lose it

    The fast wrong answer is 100 flips, because more flips feel like more chances. They are more chances for the average to assert itself, not for luck. Candidates who say it have the law of large numbers backwards, and the interviewer hears it immediately.

    The second loss is saying 10 without a number. Have the 386 out of 1,024 ready, then the standard deviation argument for 100, so the answer sounds reasoned rather than remembered.

    What the interviewer asks next

    • I pay you if the share of heads is between 45% and 55%. Now which do you want?
    • What if the coin has a 60% bias towards heads? Does the answer flip?
    • Roughly how many flips make the 60% line a three standard deviation event?

    Asked at Hudson River Trading, Prop Trading, New York, 2020 (Wall Street Oasis): Questions on EV for coin tosses, law of large numbers, Bayes theorem