As Paul said, absolutely; the purchases are tied to your Google account.
You'll be able to find them under 'My Apps' in the Play Store app or on the desktop site, and won't have to pay for them again. However, unless the game in question stores its appdata in the cloud (using a function like Google's 'Cloud Save' feature intended for game developers),you will lose your progress when the app in uninstalled. To avoid this (for example, to preserve any plague types you've completed in Plague Inc.) you can use backup/restore options like
Titanium Backup or Koushik Dutta's
Helium to save and replace this data if you need to reset your phone. Some of these tools, including Titanium, require root access to function. Others, including Helium, do not.