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.
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.