Again, what people here are asking is "
is pi
normal?". That's the only way that makes sense.
hamandpineapple says:
Or to put it a little more precisely, the probability that any given sequence isn't in the first N digits tends to 0 as N tends to infinity
[my links],
which is not the question at all. After all, the digits of
π are a
fixed string. Suppose I ask "do 20 consecutive
7's appear in the first
1019 digits of the
decimal expansion of π?" I don't know the answer to this question. But since there is no
randomness involved, the
probability of 20 consecutive 7's appearing in the first 10
19 digits of the decimal expansion of π is either
1 (if the answer to the question is "
yes") or 0 (if it's "
no").
It's best to make the question precise. Either ask if pi is a normal number or ask if every finite sequence of digits appears in the expansion of pi. And the answer to both questions (an affirmative answer to the first would imply an affirmative answer to the second) is unknown.