I must be the world’s worst person to be offering an answer to the question posed in the title, and I will be happy to be slapped down by someone who actually knows probability. But it seems to me that if you want to talk probabilities, you need to specify a probability space, and you need to specify the probability measure on it. Until you do both these things, you are merely talking philosophy, not mathematics.
Consider the example of a unit square $S$ as probability space, and ordinary Lebesgue measure on it, so that the probability of a point being in a subset $A\subset S$ is the area of $A$. Now draw the line from one corner to the opposite corner, and consider this subset $D\subset S$. What is the probability of a point lying on the diagonal $D$? Zero, of course, since a line has zero area. But there are points on the diagonal.
Now, to amplify @Tunococ’s good answer, let me say that one must make a careful distinction between real numbers and computer numbers. There are only a finite number of (floating point) computer numbers in your favorite computer, but uncountably infinitely many real numbers. I once sat in a room where the speaker (correctly) stated that it’s impossible to determine on a computer whether two real numbers are equal, and a Respected Member of the computer science department of my university said “Of course it’s possible: take their difference and see if it’s zero.” But he was wrong. For instance, there’s no way to tell by comparing the numbers on your computer that $\arctan(1/3)+\arctan(1/2)=\pi/4$, even though Pure Thought shows that it’s true.