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.