[ot]

It's not "probably polynomial time", it's polynomial time with bounded error probability. Derandomization means bringing the probability to 0.

The definition of a PRNG in complexity theory (for the purposes of this topic) is that no polynomial-time algorithm can look at its output and distinguish it from real randomness (with high confidence etc etc).

Now imagine you have a PRNG whose seed size is logarithmic in the output size. You can compose it with our BPP algorithm and iterate over all the possible seeds in polynomial time (because the seed is logarithmic), and take the majority result. You built a polynomial time algorithm that, if it gave incorrect results, could distinguish the PRNG from real randomness. So it must be correct.

[bodhiandphysics]

it can be probably polynomial time... that's ZPP! (ZPP is the complexity class of languages that are accepted by a randomized turing machine in expected polynomial time. However, in the worst case, the turing machine is allowed to take arbitrary time). In ZPP note the error probability is 0... it always returns YES or NO correctly.

It's unknown whether ZPP = P (but it almost certainly does). It is known that ZPP is the base of the random hierarchy... ZPP = RP u co-RP, where RP is the class of languages that are accepted with bounded error.

Any good complexity textbook beyond Sipser will discuss the randomized complexity classes at extreme detail, since they are crucial for constructing all the other interesting complexity classes (notably IP and MIP).

[ot]

Sure, there are plenty of complexity classes :) but TFA is about derandomization of BPP, even if it doesn't mention the class by name, it talks about Nisan-Widgerson.