The probability of a number appearing in an approximation of an irrational number?

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I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 appearing is greater than for others? This question doesn't have to be specific to Pi alone, it can be applied to any irrational number or infinity I think.

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For $\pi$ the answer is not known...as far as I am aware, nothing out there would suggest that the digits are distributed non-uniformly. Here is an article which, among other things, gives a histogram of the occurrence digits in the first trillion digits of $\pi$:

http://www.ams.org/samplings/math-history/hap-6-pi.pdf

Still, a trillion is finite and nobody knows what happens as you go further and further out.

For other irrationals, there certainly can be non-uniform distributions. .101001000100001... for example uses only 0's and 1's (a lot more 0's!).