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.

I suspect you are wondering if you could begin arbitrarily at a point somewhere within the infinite pool of decimal numbers and then start creating One Time Pads (OTP).
Although it would be hard to detect by most people, and it does emulate randomness, it's not truly "pure random" because it does technically have a discernable pattern. It would fool most, but not all. It likely could be used for non-cryptographic applications. You would not want to use Pi(π), Phi(Φ), Euler's Constant(e), the Bible, War and Peace, Moby Dick, or any other known, discernable sources if the intent is for "pure random" crytography.
