Earlier quoted context omitted.
http://en.wikipedia.org/wiki/Normal_number#Properties_and_ex... Sure... the infinite string of 1s, or the infinite string of 0s, or the infinite string of 0s and 1s that is made up of only decimal 3s and 7s. There are an uncountably many number of non-normal real numbers like this, even though the infinite amount of normal real numbers is bigger.
> There are an uncountably many number of non-normal real numbers like this, even though the infinite amount of normal real numbers is bigger. How sure are you with that one? Mind to supply a proof?
From what I've been able to gather, I think the cardinality of the normal and non-normal numbers are the same, even though the non-normal numbers are measurably greater because of probability distributions. This is a paradox that I don't really understand. http://forums.xkcd.com/viewtopic.php?f=3&t=4270