First is, does this question even make sense? I know that infinity is a concept of endlessness but people seem to use it a lot, assuming infinity can be reached, to get a solution. Can it be applied here? I'm not a mathematician. I'm just curious.
By equal, I mean that the two sequences have all the same numbers from the beginning to infinity. Also does it matter to the question if each item in the sequence is finite or is infinite? For example, if two people flips a coin an infinite number of times you only have two possibilities for each item in the two sequences (heads or tails). Does that make a difference to the answer compared to having no limit for each item in the sequence?
The chance the first digits are equal is $P_1 = 1/10$. The chance that the first $n$ digits are equal is $P_n = (1/10)^n$. The chance that an infinite number of digits are equal is:
$$P_\infty = \lim\limits_{n \to \infty} \left( \frac{1}{10} \right)^n \to 0$$