Two coins, one fair and one double-headed: you draw one blind and flip three heads, so what is the chance you hold the biased one?

## Setting up the hypotheses Two hypotheses, mutually exclusive and exhaustive given the setup: - `D`: you are holding the double-headed coin. - `F`: you are holding the fair coin. The **prior** is `P(D) = P(F) = 1/2`, because the draw was blind and there is one of each. The **evidence** is `E = HHH`, three heads in a row. The **likelihoods** are the probability of that evidence under each hypothesis: ``` P(HHH | D) = 1 x 1 x 1 = 1 P(HHH | F) = (1/2)^3 = 1/8 ``` ## The calculation, two ways Bayes' theorem directly: ``` P(D | HHH) = P(HHH|D) P(D) / [ P(HHH|D) P(D) + P(HHH|F) P(F) ] = (1 x 1/2) / (1 x 1/2 + 1/8 x 1/2) = (1/2) / (9/16) = 8/9 ~ 88.9% ``` Or in odds, which is quicker and less error-prone: prior odds `1:1`, likelihood ratio `1 / (1/8) = 8`, posterior odds `8:1`, and odds of 8:1 are a probability of `8/(8+1) = 8/9`. A general form is worth carrying: after `n` heads with equal priors, ``` P(D | n heads) = 2^n / (2^n + 1) = 1 / (1 + 2^-n) ``` One head gives 2/3, two give 4/5, three give 8/9, seven give 128/129 which is about 99.2%. The evidence accumulates fast but asymptotically -- you can never reach 1 by flipping heads, because heads are never impossible for a fair coin. ## The asymmetry of a single tail A single tail ends the inference instantly: `P(tail | D) = 0`, so the posterior for `D` collapses to exactly 0 and stays there forever. No amount of subsequent heads can revive it, because the update is multiplicative and it has been multiplied by zero. This is the practical statement of a general rule: a hypothesis assigned zero likelihood by any observation is dead, and by the same token a hypothesis given **prior** probability zero can never be resurrected by evidence. It is the reason you avoid putting hard zeros in a prior for anything you are not certain is impossible. ## Changing the prior The likelihood ratio measures the strength of the evidence and is a property of the flips alone: three heads is eight times more likely from a two-headed coin than from a fair one, whatever box you drew from. The prior is a property of the box. Change the box and the conclusion changes even though the evidence did not: - One double-headed coin among 1,000 coins: prior odds `1:999`, posterior odds `8:999`, about **0.8%**. Three heads is nowhere near enough to overcome a rare-coin base rate. You would need about ten heads (`LR = 1024`) just to reach even odds. - Both coins double-headed except one fair among 10: prior odds `9:1`, posterior odds `72:1`, about 98.6%. That is the base-rate lesson in miniature, and it is why the puzzle is asked: candidates who reason only from 'three heads is very unlikely for a fair coin' produce a confident number without ever consulting the prior, which is the same mistake as reading a rare-disease screening result off the test's accuracy. ## Common errors - **Answering 7/8.** That is `1 - P(HHH | F)`, the probability that a fair coin would *not* have produced three heads. It is a statement about the fair coin's behaviour, not about which coin you hold, and it ignores the prior entirely. - **Answering 1/2**, on the grounds that the draw was 50-50. That treats the flips as uninformative when they carry a likelihood ratio of 8. - **Answering 1**, treating three heads as proof. Three heads is perfectly possible for a fair coin -- it happens one time in eight. - **Forgetting to weight the likelihoods by the prior** when the priors are unequal. With a 1:1 prior the weights cancel and the shortcut works; with any other box it does not. ## How to answer in the room State the two hypotheses and their priors, write down both likelihoods, produce 8/9 by the odds route in one line, then volunteer the two extensions that show understanding rather than memorisation: what a single tail does, and what happens when the box is not 50-50. Those two remarks are what separate a recited answer from a reasoned one.