That's weird, but not surprising considering this is EA. If you're missing the TS3W.exe file, and so is the other person who first replied to you, then it's likely that that is the source of the problem. And if that's the case, your only other option (besides trying to miraculously get a refund out of EA) would be to purchase a physical copy of the game. Considering you simply stated "The Sims 3" with no specification of an expansion pack/stuff pack or DLC content, I'm going to assume you purchased the base game. You can probably find its disc for only a few dollars. Also, it may just be the base game doing this. I bought two of the expansion packs within the past year, and they work perfectly fine. (As do a couple of the others bought before them. The rest I have physical copies for, so I can't attest to those). A warning, though: If you fix your situation, I strongly suggest NOT buying the DLC worlds through the EA App. They are glitched and do not work, because the key provided to you for installation via email will come up as "expired" or "already registered" even though it's supposed to be a new copy.
I'm sorry I can't give you a concrete solution to your problem, but I hope you find a way to get it taken care of. Best of luck!