What if we write 0. and then throw a coin and depending on the result continue the number with 1 or 0 and continue this process indefinitely. It seems like a recipe for producing irrational numbers.
Are these numbers really irrational? Are they transcendental? Are they normal? Maybe I should also ask: is anything well-defined by this recipe? :-)
since it may happen that the coin keeps showing tail and in this case the number (base 2) is an integer, I think that a definite answer cannot be done.
UPDATE if you are interested in a probabilistic / philosophical answer (i.e., a supernatural being which performs a supertask tossing the coin at times $t$, $t+\frac{1}{2}$, $t+\frac{3}{4}$, $t+\frac{7}{8} \ldots$, at time $t+1$ she will with near certainty ($p=1$) obtain a normal number. But this is not a "recipe", in the sense she cannot be sure it is. (Ok, if she is a supernatural being maybe she has some other way to know it :-) )