Your assumptions about digit distribution of transcendental
numbers are incorrect. There are 2 separate concepts:
Transcendental numbers are number like e and π and
a=10^-(1!)+10^-(2!)+10^-(3!)+10^-(4!)+... which are
not roots of equations with rational coefficients. Normal
numbers are numbers (like 0.12345678910111213141516...,
although that is hard to prove!) that have every finite
sequence of digits (either in their base-10 representation
(or some other base) or, if you take the stronger more
common definition, in all bases) appearing at the
"appropriate" frequency.
It is not known whether π is normal. (See
my
writeup on Everything2 for details and more links).
It is known that many transcendental numbers are
not normal. For instance, the number a above cannot be
normal, as the only digits appearing in its decimal
representation are 0 and 1.
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