Double-headed coin after ten straight heads

A bag holds $100$ coins: $99$ are ordinary fair coins and one has a head on each side. You pull out a coin uniformly at random and toss it $10$ times, and every toss comes up heads. Given this, what is the probability that the coin in your hand is the double-headed one?

$1024/1123$. By Bayes' rule, the double-headed coin produces ten heads with certainty while a fair coin does so with chance $2^{-10}$. The posterior is $\dfrac{\tfrac{1}{100}}{\tfrac{1}{100}+\tfrac{99}{100}\cdot 2^{-10}}=\dfrac{1024}{1024+99}$.