Slightly pedantic correction: For a problem to be in the class P means that a solution can be guaranteed to be found deterministically within polynomial time. NP-hard problems might have a deterministic polynomial-time solution for a specific instance, but it's not guaranteed for all instances. So, "NP" actually stands for non-deterministic polynomial, meaning that an answer can be verified in polynomial time, so if you get a lucky guess you could get a solution in polynomial time.

Yeah, that's what I always thought, too. Kind of like how a Red Mage is in between a Black Mage and a White Mage.