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.

There are long lists of digits for $\pi$ online. You could write a computer program that calculates different types of statistics to test your hypothesis.

One interesting statistic could be to measure $$P(X_{i+1}=1|X_{i},\cdots,X_{i-n})$$That is: the conditioned probability of one bit being $1$ given that we know the binary number of length $n$ preceding it. This conditioned probability should converge to the stationary $P(X_{i+1}=1)$ for all such $n$ bit numbers if there is no short term-memory in our source.
