1. I probably won't stay at home. (Is it correct?) == Yes it is! 2. I'll probably spend time with friends. (Is it correct?) == Yes! 3. I will probably stay at home. OR I probably will stay at home. Which one is correct? OR Both are correct? == Both are correct! You can say : I might stay at home instead of I will probably stay at home and you can also say I will more likely be at home as well. You know that probably means that it is more likely to actually happen?

It is likely I won't stay at home (then). possible, likely, perhaps, could be, ....similar but subtle distinctions to tell apart.

The only thing that isn't correct: "Both are correct?" needs to be "Are both correct?" Another alternative to probably is likely, but I think probably sounds better in your examples. You can also say unlikely, for example "It's unlikely I'll stay at home." or "I'll be unlikely to stay at home."