Semantics on percentile chances aside, if the original poster is just talking about copying files to another drive, then the chances of problems are functionally nil past what you usually have when working on a computer (lighting strike, massive spontaneous hardware failure, etc); it just doesn't happen. I routinely move hundreds of thousands of files around between drives (both locally and over a network) for institutional server backups without any issue.
If something DID go wrong, the OS should tell you, though the issue is always some sort of permission issue, not hardware problems. To be extra paranoid, any number of backup or synchronization programs will double-check the files after copy to make sure there were no problems; I use Sync personally (also because it can be set to only copy changed files, which with a volume containing 200,000 files is nice).
If we're talking about optical disc, the answer is "if there's an error burning the disc, the Finder or Disk Utility will tell you when it verifies it". The verification process reads back the entire burned disc and compares it to the original files to make sure there aren't any problems.
Optical discs can, of course, go bad later, either due to scratching or (if you wait long enough) media degradation due to sunlight, manufacturing error, whatever, but that is unavoidable with any media format (eventually SOMETHING is going to go bad--even paper rots). Generally if you have a good burn, and decent media, and you store it carefully, it'll be good for quite some time.
When I'm being extra-paranoid, I burn the same archival copy onto two different brands of media, and every once in a while (maybe annually) check an individual disc from each of the spindles I bought, figuring that if one goes, they're probably all going, so I'll notice it, and that two brands from two different spindles aren't likely to both spontaneously "rot" at the same time.