TITLE: On an infinitely sided die, is it possible to roll and get the same number twice?
QUESTION [5 upvotes]: In music class, our teacher was saying that every sound we make is uniquely different and has never been heard before. So I kind of thought of a different problem, with the dice. Is there some type of way to calculate the chance? It can’t be zero or can it?

REPLY [2 votes]: I like this question a lot.
So lets say the die is a sphere, with a continuously changing and unique hue on every point of its surface.
The probability of getting a hue in a given range can be calculated by dividing the solid angle that contains that hue by $4\pi$ (the solid angle of the whole sphere).
In this case we are interested in an infinitesimal solid angle to capture one colour only. 
Let's call the 'area' $\delta$ and we will take $\delta \rightarrow 0$ in the end. Let us call the total number of ways the sphere could land (or exact colours) we could obtain $C$ and we will take $C \rightarrow \infty$ in the end. 
The number of ways two dice could fall is the product of each individual possibility; $C^2$ (infinite number of possibilities, like in the music example). The number of ways they could fall and have hues in the same area is;
$$ C * \left(C\frac{\delta}{4\pi}\right) $$
That is, the number of ways the first die can land times the number of ways the second die can land times the fraction of positions of the second die that would be the same hue. 
So now the probability of getting the same hue on the two dice is the number of same hue outcomes divided by the total possible outcomes;
$$ \left(C * \left(C\frac{\delta}{4\pi}\right)\right)/(C^2)= \frac{\delta}{4\pi} $$
Directly proportional to the area of the die that we consider to be the same hue. As we take $\delta \rightarrow 0$ to require an exact colour match this probability also goes to $0$.