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  1. 068A box holds ten coins: one has heads on both sides and nine are fair. You pick a coin at random, flip it five times and see five heads. What is the probability you picked the double-headed coin?Conditional probability and BayesWarm upJump TradingChicago · 2018

    Try it first

    After five heads, roughly how likely is the double-headed coin?

    Show the worked solution

    32/41, about 78%. Before flipping, the odds are 1 to 9 against the double-headed coin. Five heads happen for certain with it and with chance 1/32 with a fair coin, a likelihood ratio of 32. Multiply: posterior odds are 32 to 9, which is 32/41. Each extra head doubles the odds, so a sixth head would take it to 64/73, about 88%.

    Why is the answer not close to certain?

    Imagine a rare illness and a decent test. A positive result raises the chance you have it, but if the illness is rare enough, most positives still come from healthy people. Evidence is weighed against how common each explanation was to begin with, so five heads, which a fair coin produces only once in 32 tries, still has to overcome nine fair coins for every double-headed one. The 97% instinct takes 1 minus 1/32 and forgets the nine-to-one start.

    Each head doubles the odds on the double-headed coinPick1/10Double-headed5 heads: 1joint 1/10 = 32/3209/10Fair5 heads: 1/32joint 9/320Odds 32 : 9 for the double-headed coinP = 32/41 = 78.0%Chance it is the double-headed coin10%018%131%247%364%478%5heads seen in a rowodds 1:9, 2:9, 4:9 ... 32:9
    Starting from odds of 1 to 9, five heads multiply the odds by 32 to give 32 to 9, so the chance of the double-headed coin rises from 10% to 78%, roughly doubling the odds with each head.

    How do you run Bayes in odds form?

    Odds form is the fastest way to say it in the room. Posterior odds equal prior odds times the likelihood ratio: (1 to 9) times 32 gives 32 to 9. Converting back, 32 out of 32 + 9 is 32/41, about 78%. The long form gives the same thing: the joint chance of picking the special coin and seeing five heads is 1/10, the joint chance of a fair coin and five heads is 9/10 x 1/32 = 9/320, and the posterior is (32/320) / (41/320).

    The relationship
    P(D∣5H)=110⋅1110⋅1+910⋅132=3241≈0.78odds=19×32=329P(D \mid 5H) = \frac{\tfrac1{10}\cdot 1}{\tfrac1{10}\cdot 1 + \tfrac9{10}\cdot\tfrac1{32}} = \frac{32}{41} \approx 0.78 \qquad \text{odds} = \frac19 \times 32 = \frac{32}{9}
    Dthe event that the double-headed coin was picked
    5Hthe observation of five heads in five flips
    32the likelihood ratio: 1 divided by 1/32
    What it says in wordsPrior odds of one to nine, multiplied by a likelihood ratio of thirty-two, give odds of thirty-two to nine.

    What does the odds picture tell you about more flips?

    Each head is twice as likely under the double-headed coin, so every head doubles the odds and every tail ends the question, since the special coin never shows tails. After 0 to 5 heads the chance runs 10%, 18%, 31%, 47%, 64% and 78%; it passes 50% only after the fourth head. The useful follow-up is the next flip: it lands heads with chance 32/41 + (9/41)(1/2) = 73/82, about 89%. On a desk, the same arithmetic tells you how many winning days it takes before a new strategy's record says anything about skill.

    Where candidates lose it

    The trap is answering 31/32, about 97%, by looking only at how unlikely five heads are from a fair coin. That ignores the prior; with nine fair coins in the box, the base rate matters as much as the evidence.

    The second slip is the reverse: staying near 10% because the coin was chosen at random. Say the odds form, prior times likelihood ratio, and both errors disappear.

    What the interviewer asks next

    • What is the probability that the next flip is heads?
    • How many heads in a row would you need to be 99% sure?
    • If one of the ten coins were double-tailed instead, how would five heads change the answer?

    Asked at Jump Trading, Quantitative Research, Chicago, 2018 (Wall Street Oasis): Then he asked one question of probability which can be solved by Bayesian formula.