Single error detection rate
If a digit whose weight is 1 is changed to , the weighted sum will change by . The error will go undetected only if . But this happens only when , in which case there has not been an error after all, so all errors of this kind are caught.
What if the weight were 3? Then the error would be undetected if . But again, this cannot happen unless . Thus, this method has a 100% single error detection rate.