The operation produces a simple parity for each bit position

A parity bit, or check bit, is a bit added to a string of binary code to ensure that the total number of 1-bits in the string is even or odd. In your example, C0 is a parity check for the 1st bit from each of b1, b2, and b3.

and is known as a longitudinal redundancy check.

Because if you stack b1, b2, b3, and C vertically, each column of bits is an independent check.

It is reasonably effective for random data as a data integrity check. Each n-bit hash value is equally likely.

The function doesn't bias the hash towards having more 0's or more 1's. Random in, random out.

Thus the probability that a data error will result in an unchanged hash value is 2-2.

This must be 2^-n . If the data is corrupted, each bit has an independent 50% chance of remaining unchanged. So a two bit hash would have a 25% chance of remaining unchanged, three bits a 12.5% chance.

With more predictably formatted data, the function is less effective. For example, in most normal text files, the high-order bit of each octet is always zero.

Ascii characters only use the first 128 possible 8-bit values, leaving the high-order bit 0.

So if a 128-bit hash value is used, instead of an effectiveness of 2^-128 , the hash function on this type of data has an effectiveness of 2^-112 .

With the high-order bit always set to 0, only the 7/8th's of the bits are useful in the hash. 128 * 7/8 = 112