Posterior Probability of a Double-Headed Coin

A bag contains 19 fair coins and 1 double-headed coin. You pick a coin uniformly at random and start flipping it. After observing $k$ heads in a row, you want to be at least 95% sure you picked the double-headed coin. What is the minimum number of consecutive heads $k$ needed for the posterior probability that your coin is double-headed to exceed 95%? Derive the posterior as a function of $k$ and solve.