Using the digits of $\pi$ to generate random numbers. Let's say I've been captured by Russian operatives and am locked in a room with only one object: a book listing the digits of $\pi$.
I'm told to generate a sequence of binary digits. If this sequence is random, they will cut off one of my arms and let me free; if this sequence is not random, however, I will be killed.
My first solution was to take the digits of $\pi \ \text{mod} \ 2$, so that:
$$3.1415926535897...$$
$$\downarrow$$
$$1.1011100111011...$$
And I would read the digits from left-to-right of the second number.
My Question
Is there any way to prove that the bits I generate are random (no discernible pattern)?
Are the digits of $p \ \text{mod} \ 2$ random for any transcendental $p$? How about any irrational $p$?
I feel like this should be a really easy question (with an affirmative answer), but I don't know how to show it.

What does "random" mean? In one natural sense (or class of senses, rather) coming from computability theory, your sequence is not random: even though it's hard to notice it, there is a pattern to the bits. 
While it's clear that in some intuitive sense "$\pi$ mod 2" is somewhat random, and "$\pi$" is somewhat less random, I suspect it will be very difficult to pin down exactly what you mean by this (and so very difficult to give a satisfying answer to your question!).
