OK, that sounds as if the installation of Island Paradise has failed to properly add the expansion, but somehow the icon has made it into the Applications folder. Go back into Applications and find the The Sims 3 folder again.This is the location from which your game actually launches (as opposed to a similarly named folder within Documents, which is where your saves and other data are stored). In the folder in Applications you will see the shortcut icons for the base game as well as any expansions or stuff packs installed. If you can see the icon for the Island Paradise, try right clicking on it and this should give you the option to show the original file. You can move the original files to the trash from there. This may allow you to commence a reinstallation. (Many thanks go to Tremayne from TS3 Mac forum for this suggestion.)
If this does not work, you may need to completely uninstall and reinstall.
Before you start uninstalling, make copy of your Store downloads and any important saved games so that you don't lose them. These can be found in Documents, then Electronic Arts, then The Sims 3, then Saves and Downloads.
When you uninstall TS3 games on your Mac using the uninstaller, the game leaves behind stray files and isn't completely removed from your computer. These files may conflict with future installations. It is best to completely remove them. BluebellFlora's website provides a guide on how to do this:
http://bluebellflora.com/the-sims-3-on-a-mac/how-to-do-a-clean-uninstall-and-reinstall-on-a-mac/
After cleanly uninstalling, install the base game and patch it up to date. If you cannot update via the Launcher, download and apply the latest cumulative manual patch (the superpatch) for Mac.
Check that your game works properly. Then install any EPs and SPs that you own, in order of release date.
Mac-users definitely want to use patch 1.63. It fixed a lot of problems and made the game more stable on our operating systems, so it's worth sticking with it and getting the game running at that patch level.
Caspin