# Confused about Pi & Normal Numbers?

So, my (naive) understanding is that in order for Pi to contain every single possible number combination, Pi has to be a normal number, where each digit in base 10 appears with equal probability. However, I don't really see why this is the case? If you look at the first 50 digits of Pi, you'll see every digit from 0-9. Doesn't that mean that there is a non-zero probability of seeing the digits 0-9 for each place? Since there is a non-zero probability, wouldn't every single sequence show up as Pi is infinite then? I'd appreciate someone showing me what I'm not understanding. Thanks guys!

I don't understand how is it that you jump from “If you look at the first 50 digits of Pi, you'll see every digit from 0-9” to “there is a non-zero probability of seeing the digits 0-9 for each place”. Consider the sequence of $$0$$'s and $$1$$'s such that
• the first term is $$1$$;
• all other terms are $$0$$.
So, you see a $$0$$ and a $$1$$ in the first two terms. However, the probability of having a $$1$$ in the $$n$$th term is $$0$$ for every place other than the first one. And $$1$$ appears only once, not infinitely often, and therefore I don't see why is it that you ask “wouldn't every single sequence show up as $$\pi$$ is infinite then?”