The reason it seems that way is because you are imagining holding the number of Ys constant. However, if the number of Ys is unknown, you have to figure out what proportion of the cards say Y as you go along, so you get a different result.
Maybe an analogy will help. Because you draw the slips of paper in random order, they will not be correlated with each other except through the total percentages that say Y and N. Analogously, if you flip a weighted coin, the flips will not be correlated with each other except through the bias of the coin. Drawing a slip of paper follows the exact same mathematical rules as flipping a weighted coin. And so since Laplace’s rule of succession works for the weighted coin, it also works for the slips of paper.
Since you’re already thinking about keeping the number of Ys fixed, you may object, “but the number of Ys is fixed in the case of the papers and not fixed in the case of the coin, so they must be different.” So we can go a step further and imagine someone else flipping the coin, and then writing down what they get. Now when we read the papers, there is a fixed number of Ys, but since it’s the same coinflips all along, the probability of seeing Y or N is exactly the same. This demonstrates that having a finite amount of stuff doesn’t really matter, what matters is the mathematical rules that stuff follows.
The reason it seems that way is because you are imagining holding the number of Ys constant. However, if the number of Ys is unknown, you have to figure out what proportion of the cards say Y as you go along, so you get a different result.
Maybe an analogy will help. Because you draw the slips of paper in random order, they will not be correlated with each other except through the total percentages that say Y and N. Analogously, if you flip a weighted coin, the flips will not be correlated with each other except through the bias of the coin. Drawing a slip of paper follows the exact same mathematical rules as flipping a weighted coin. And so since Laplace’s rule of succession works for the weighted coin, it also works for the slips of paper.
Since you’re already thinking about keeping the number of Ys fixed, you may object, “but the number of Ys is fixed in the case of the papers and not fixed in the case of the coin, so they must be different.” So we can go a step further and imagine someone else flipping the coin, and then writing down what they get. Now when we read the papers, there is a fixed number of Ys, but since it’s the same coinflips all along, the probability of seeing Y or N is exactly the same. This demonstrates that having a finite amount of stuff doesn’t really matter, what matters is the mathematical rules that stuff follows.
Thanks :)