Double-headed coin among a thousand after ten heads

Among $1{,}000$ coins, $999$ are fair and one has two heads. You pick a coin at random and toss it ten times, getting ten heads. What is the probability that you picked the two-headed coin?

$1024/2023\approx0.506$. By Bayes, $\dfrac{\tfrac1{1000}}{\tfrac1{1000}+\tfrac{999}{1000}\cdot2^{-10}}=\dfrac{1024}{1024+999}$. Ten heads is roughly a 1-in-1000 event for a fair coin, matching the prior odds, so it ends up close to even.