X: If you throw a coin eight times and the barrel falls out all eight times, it does not increase the odds of the eagle falling out the ninth time. It still remains 1/2.

Y: If the coin has fallen 8 times, it is a proof that the coin is shattered and the probability of an eagle falling out is less than 1/2. Not a guarantee, of course. But it is worth thinking. If a coin has fallen 100 times in a row, you have to see if there is an eagle on it.