If you are saying the probability of sampling something with half ones and zeroes is greater than the probability of getting 0000 0000 or 1111 1111, that is a correct statement because you’re sampling the output, not the generation. The probability of generating 1010 1010 is the same as the probability of generating 0101 0101 which is the same probability as generating 0000 0000 or 1111 1111 or 1111 0000 or 0000 1111. You’re looking at the difference between (8 choose 4) vs (8 choose 8).

However, in a truly random system, 0000 0000 will appear as often as 1010 1010. The distinction of “solid patterns” is meaningless at scale. You’re the one differentiating between the two. Now as we analyze the randomness, we would expect more samples that have four ones than have eight ones, but we would also expect an equalish number of each pattern to appear (eg 0011 1100 appears the same amount as 1100 0011 which appears the same amount as 0000 0000).

You’ve conflated the number of patterns with the number of ones or zeroes that appear in the patterns.