I sometimes run sha256sum on large files after transferring them from one place to another, and will just skim the hash output to verify it's correct. But, I usually just look at the first/last 5 or 6 hex digits and call it good enough.
I know that the chances of collision are something like 1 / 2^64, but what are the chances of a "near collision"? E.g. only one or two hex digits are different.
As a related topic, If you have a binary sequence and just change one bit, you get a completely different hash, correct? So is it possible for random errors to result in nearly identical hashes? I'm aware that MD5 is 'cracked' such that a malicious agent could append whatever necessary data to a file to make it have the same hash output- but is this within any reasonable realm of possibility to have happen stochastically?
EDIT: This topic has spawned some discussion below that was just as (if not more!) informative than the scope of my original question. With that said- I am referring only to using hash functions as a file integrity check, not as protection against attacks.