I recently tried to engage one of my professors with the idea that every application of mathematics to the real world can only involve irrational numbers if perfect measurement is allowed (with the notable exception of counting loosely defined or symbolically perfect ideas, like, how many apples are in this basket, or, a football field is exactly 100 yards long, etc.). For example, if you had a cartesian coordinate system set up on a dart board, and threw a dart at it, the coordinates of the impact must necessarily be irrational and nothing else. Or that any two arbitrarily chosen intervals of a continuum, i.e. space or time, must necessarily be incommesurate; say, the ratio of the precise length of an episode of The Golden Girls to the length of the very unit used to measure it--a minute. To refute this, he cited probabilty. I contended that the chance of the result of a coin flip coming up heads is not 1/2, but slightly less because of the unlikely event that the coin lands on its side, and the rediculously unlikely event that the coin lands on the corner joining the side and face. Even certain events that intuitively yield a 100% probability, like the chance that the coin will hit the ground after being flipped are rendered irrational due to quantum weirdness. Anyway, to avoid a purely philosophical post, this got me thinking, even in purely conceptual terms, where a nickel is a perfect cylinder, is there a way to calculate the true odds of a coin toss coming up heads given all the possible outcomes (faces, side, edge), neverminding quantum weirdness? Also I would like to get some more in depth insight into the aforementioned idea of irrationals. Thank you.