If I recall right this depends on the original message length and 1 bit is a bit of an edge case. If you transfer just 1 bit, you're very space constrained and it's hard to do much. 1 bit to 1 bit has nothing, 1 bit to 2 bits has single-bit error detection, and 1 bit to three bits has single bit error correction (--and 2 bit error detection-- (this isn't right, thinking about it for a second)). After this, the minimum required checksum length growth logarithmically, plus 1 or 2 for detection / correction - and that constant factor makes 1 bit so weird.

All error correction and detection are designed with an acceptable probability of error in mind, which depends on the medium. For example, if you know cosmic rays might flip 1 in a million bits, and you want your system to have 1 error per trillion bits, then you need to send every bit twice to be able to detect a one-in-a-million error.

It's not only about memory size, but the speed of processing too.

Higher level protocols would signal an error, and require a retransmission until no error is detected.

However, "majority rule" does indeed sound like an error checking mechanism designed by a non-engineer (or non-mathematician) commitee.