Would an infinite sequence of random numbers contain an infinite sequence of zeroes?

An infinite sequence of digits picked from set 0,1,2,3,4,5,6,7,8,9 with equal probability contains an infinite sequence of zeros if you strike out the non-zero digits. In other words: an infinite sequence of random digits has an infinite sequence of zeros hiding in it, but the zeros are not all together. An infinite sequence of random digits (as defined above) cannot contain an infinite sequence of zeros in a row.

In fact, the only sequences that contain an infinite squence of zeros in a row, is a finite sequence of digits followed by an infinite sequence of zeros. More.

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