Infinite Sequence

Although there is a doubt whether the sequence generated in the way you have in mind could be called "random", usually it is thought possible to generate a sequence with the property you have specified (through Axiom of Choice), however there is no way to prove the Axiom of Choice from other axioms of mathematics.

Of couse, it is just a matter of luck, isn't.
Theoretically, the probability for that to happen is greater than 0.

The question is not if it is possible (probability greater than 0), but if it is guaranteed. (probability of 1)

Sure!---
If you pick a string long enough (infinite) somewhere you will find a sequence within that string that matches the 50 (or so) digits.

Look if you have a ten sided die (0 to 9) the probability of a sequence following your sequence is P(E)=(number of permutations=1)*(0.1)^n. Now this is a pretty small number, but it is not zero and you are rolling this die an infinite number of times. So at some time you are going to roll your sequence. Consequently, you're going to run into it.

So consider that the number of trials is infinity and the P(E) is not zero for n trials, and somewhere in time it will happen.

Rap