$\pi$ normal to the base $10$

Question

If $\pi$ is normal to base $10$, why would we expect to find a string of ten $0$'s in its decimal expansion?

Answer 1

If $\pi$ were normal to base $10$, that would mean we would expect to see every string of digits equally often. So, we would expect to see "0000000000" and "7182818284" just as much as each other when we look through the digits of $\pi$. There are $10^{10}$ possible strings of ten digits, so, the hypothesis "$\pi$ is normal" says that if we chose a string of ten digits, and then random location in $\pi$ and then read off the next then digits, the probability of the string read from $\pi$ and the string chosen earlier being the same would be $\frac{1}{10^{10}}$ - that is, every single string would have equal probability of appearing. Thus, if we sampled enough positions, we would almost surely find a string of ten zeros - if we didn't, then the probability distribution of the $10$ digit strings would not be uniform, contradicting the very definition of "normal"