There is a bag with mostly blue and some red marbles. If I decide in advance to randomly pick, record the color of, and replace marbles one at a time until I pick one red marble, what is the best estimate I can make about the color ratio in the bag, given that it ends up taking 10 picks to find the first red marble?
If I had decided to pick 10 marbles, and the 10th marble had happened to be the only red marble picked, then the best estimate given the data would have been that the bag is 10% red and 90% blue. Is the outcome different (due to some sort of selection bias or something) in the above example, where instead of picking a set number of marbles, I decide instead to pick until I get my first red result, and it happens to be on the 10th pick? If so, how can the answer be generalized to solve for the best estimate of the color ratio when the red marble is found on the Nth pick?