The Two-Headed Penny

A jar holds $999$ fair pennies and one two-headed penny. You pick one coin at random, flip it $10$ times, and get all heads. What is the probability that the coin is the two-headed one?

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#Bayes setup

Before flipping, the coin is two-headed with probability $\tfrac{1}{1000}$ and fair with probability $\tfrac{999}{1000}$ . Ten heads is certain for the two-headed coin and has probability $\left(\tfrac12\right)^{10} = \tfrac{1}{1024}$ for a fair one.

#Posterior

\PP(\text{two-headed} \mid 10\text{H}) = \frac{\tfrac{1}{1000} \cdot 1}{\tfrac{1}{1000} \cdot 1 + \tfrac{999}{1000} \cdot \tfrac{1}{1024}}. \tag{1}

Multiplying through by $1000$ clears the prior,

= \frac{1}{1 + \tfrac{999}{1024}} = \frac{1024}{1024 + 999} = \frac{1024}{2023} \approx 0.506. \tag{2}

Ten heads multiplies the rare trick coin's tiny prior up to a joint weight of 1024, against 999 for all the fair coins combined. The two are nearly equal, so seeing ten heads leaves the coin only just more likely to be two-headed.

#Result

The chance is $\tfrac{1024}{2023} \approx 50.6\%$ , barely better than a coin flip. Ten heads is strong evidence, a likelihood ratio of $1024$ to $1$ , but the trick coin is so rare at $1$ in $1000$ that the two effects nearly cancel.