Heads needed to trust the two-headed coin

A bag holds 20 coins: 19 are fair and one has heads on both sides. You draw one coin at random and toss it repeatedly. What is the smallest number $n$ such that, after seeing $n$ heads in a row, your posterior probability that you hold the two-headed coin is at least $95\%$?

9. By Bayes, after $n$ straight heads the posterior is $\dfrac{1}{1+19/2^n}$. Requiring this to be at least $19/20$ means $2^n\ge 19^2=361$, and the smallest such $n$ is 9.