Here we must distinguish between what a mathematician means by choosing a point in the square and what the computer programmer means.
The mathematician wants all the bits of X and Y while the programmer wants only enough initial bits to fill a floating point number.
Here we relate the notion of choosing a real X in the unit interval [0, 1] with the notion of sequentially choosing the bits b_{i} of the binary representation of X.

X = Σ_{i>0} b_{i}2^{−i}

If the bits are unbiased and independent then the chosen real will be uniformly distributed and conversely.
The mathematician may be satisfied with the notion that this process can be carried out to any degree of accuracy.
Computer science offers the idea of an algorithm or oracle that will produce any finite number of bits and the mathematician will, for some purposes, consider the matter closed.
See Biased Radicies