The trick, and the reason for using Turing machines, is that '0000' and '1111' ought to be considered more likely than '0101' and '1010', because they're simpler; and the way this model represent simplicity, a sequence is simple if it's the output of short programs (where 'program' in this case means 'prefix to a Turing machine tape'). If you used the tape directly, then '0000000000' and '0100011011' would be equally likely, which, generally speaking, they're not.