Ten heads and the double-headed coin

Question

A jar contains 1{,}000 coins: 999 are fair and one is double-headed. You pick a coin uniformly at random, flip it 10 times, and see 10 heads. What is the probability that you picked the double-headed coin? And what is the probability the next flip is a head?

Accepted Answer

Apply Bayes' theorem. The prior is P(	ext{fake}) = \frac{1}{1000}. The likelihoods of ten heads: P(10H \mid 	ext{fake}) = 1, \qquad P(10H \mid 	ext{fair}) = \frac{1}{2^{10}} = \frac{1}{1024}. So P(	ext{fake} \mid 10H) = \frac{\frac{1}{1000} \cdot 1}{\frac{1}{1000}\cdot 1 + \frac{999}{1000}\cdot\frac{1}{1024}} = \frac{1024}{1024 + 999} = \frac{1024}{2023} \approx 0.5062. In odds form this is transparent: prior odds 1{:}999 times likelihood ratio 1024{:}1 gives posterior odds 1024{:}999, slightly better than even that the coin is fake. Next flip. Average over the posterior: P(H{11} \mid 10H…

Ten heads and the double-headed coin

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