Irrational number with known probability distribution on digits Is there any irrational number that is known the probability distribution of digits? 
Something like 0 appears 10% of time, 1 appears 10% of time, etc.
Probably irrational numbers that are defined by a construction on digits like:
1234567891011121314....
You can prove digits distribution but I am interested in numbers that is not defined this way like pi, e, or square root of prime.
Is there any advance on this topic? 
 A: If all you care about is the distribution of digits, it is easy to construct a number with a distribution arbitrarily close to a given one: just construct a rational number with a repeating block with close to the right distribution, then add to it a Liouville number (where the digits are zero, except for the ones in the $n!$ places. What is more, the above construction will work on the nose when the probabilities of digits are rational, and any number where the probability of some digit is irrational has to be irrational itself (exercise to the reader).
A: Your question belongs to the context of normal numbers, which are real numbers whose digits and blocks of digits are uniformly distributed in the sense that each digits (or block of digits) occurs asymptotically with the correct "fair" frequency. The number you suggest is called "Champernowne's constant", it is known to be normal in base 10.
For a first introduction, see for example here: https://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf
A number which is normal in all possible (integer) bases is called absolutely normal.
If the distribution of digits is not "fair", then the number is sometimes called "abnormal". See for example here:  Martin, Greg. Absolutely abnormal numbers. Amer. Math. Monthly 108 (2001), no. 8, 746–754.
It is not known whether $\pi$, $e$, $\sqrt{2}$ etc. are normal in any base. However, it is conjectured that all algebraic irrationals are absolutely normal. See for example here for more information: Bailey, David H.; Crandall, Richard E.:
On the random character of fundamental constant expansions. 
Experiment. Math. 10 (2001), no. 2, 175–190. 
