Are paid carats worth the same as regular ones?

Less than 4 hors left on Agnes Digital's scout and I still dont have her... However i only need 20 more pulls to exchange points for her. There's no way i can get 3'000 carats in that time.

I want to buy some paid ones but idk how exactly the paid currency works:

• Is 1 paid carat the same as 1 free one?
• What deals generally offer more value? I saw one that gives carats every day but considering my situation i dont think it's for me.