Distribution of the digits of Pi

Can anything be stated about the distribution of the digits of Pi, i.e., if I were to sample n digits of Pi, can anything be said about the probability to observe certain digits, or is there any reason to assume that they would not be evenly distributed?
This is purely a curiosity question.

Solution 1:

It is suspected that $\pi$ is a normal number, i.e. that its digits in any base $b$ are uniformly distributed in a certain precise sense (the link explains in more detail). However, this has not been proven yet. In fact, there is relatively little we know about the distribution of the digits of $\pi$; take a look at these posts (here and here) on MathOverflow.