I am interested which software do people use in PBS? Final Cut Pro or Adobe Premiere?
Having a big company like PBS, they most likely use a mix of many pieces of software. They need to have a 3D tool like Cinema 4D for some of the title sequences and may even use custom made software. Some of the software they use may not even have been heard of before. So the answer to your question is most likely both.
PBS is made up of 354 different broadcasters and carry content from many different studios, it is highly unlikely that they all use one product and chances are good they use a product other than Final Cut or Premiere. Avid makes several very popular products and the software that drives actual broadcast studios isn't the same kind of software used by production studios.
I'll try to clarify further. PBS is not a content creator, they are a broadcaster. Content creation shops, such as studios, would be the ones to use Premiere or FinalCut, but generally studios prefer higher end rigs like one of Avid's platforms. A broadcast studio isn't going to use FinalCut or Premiere (at least not for anything beyond very limited and basic work if anything). They are going to use software that is concerned with sequencing up content and playing it back in order without interruption so that they can have programs run together smoothly. In some cases, this can even be a hardware system rather than any software component, particularly in someplace like a PBS station where finances are limited.