Spoiler Your sis probably went on some bad applications on facebook...
Before you actually play any game/app on facebook it actually asks you to have access to all your friends, pictures, videos and chat.
Yeah that's shitty. It's just like the terms of service in an installer, you never read them. you sister probably just clicked accept and just played ( might also be something else like a quiz ). That's why sometimes you post stuff on your facebook which you never posted ( The app posts the stuff.. )
So the thing is that it uses the chat and sends stuff to her friends. Friends click link because they trust her ( e.g. watch this nice video i found! Download this epic app i found!).
So basically your sis clicked a link and downloaded something.
Just install an antivirus. Anyone, they'll all at least protect her from retarded things like that...