In the textbook from which I am teaching a Discrete Math course, the authors propose randomly generating an infinite sequence of decimal digits $d\_1, d\_2, \dots$. We are to think of this as the decimal expansion of a real number in the unit interval. 


They propose that "it is overwhelmingly likely" that the resulting sequence is non-periodic, and thus that the number in question is irrational. This is intended to motivate the idea that there are more irrational numbers in [0,1] than rationals in [0,1]. 


This argument is appealing until you realize that it's nonsense, since we are somehow taking a uniform probability distribution on an uncountable set. 


My question is whether there is any way to interpret this so that it is not nonsense. 


Let me make a couple of quick points, in the hopes that this question will not be closed out of hand. I know the irrationals are uncountable, and I know the rationals are a set of measure zero. What I am asking is whether *the above argument* can be put on a legitimate footing. I don't insist that it be possible to put it on a legitimate footing which I could explain to my Discrete Math students --that seems clearly out of reach-- but whether or not there is some legitimate footing will alter how I want to present this argument. 


The textbook in question, if anyone is interested, is A Discrete Transition to Advanced Mathematics, by Richmond and Richmond. I like it, and will probably use it again. 



Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved?
You can make sense of the uniform probability distribution on lots of infinite sets, notably any compact topological group $G$, where "uniform probability distribution" should mean "normalized [Haar measure](http://en.wikipedia.org/wiki/Haar_measure)." Here the group is an infinite direct product of copies of $\mathbb{Z}/10\mathbb{Z}$, which is compact by Tychonoff's theorem. You've probably also considered the uniform probability distribution on $S^1$ at some point in your life. 


The failure of groups like $\mathbb{Z}$ and $\mathbb{R}$ to have a uniform probability distribution isn't just due to infinitude, it's also due to lack of compactness. 


