seems like an active, even central/core area of current research under the heading of "exact power of derandomization". its basically an open question with various open complexity class separations whether randomization adds power, or doesnt. theres also a deep connection to the $\mathsf{P \stackrel{?}{=}NP}$ problem in the natural proofs result of razborov/rudich which shows that, loosely, a $\mathsf{P \neq NP}$ proof would likely allow one to "break" what are now conjectured as "secure" random number generators.
[1] is a broad survey on derandomization literature by Impagliazzo. (think there are other surveys out there but cant find them this moment.) [2] is a rough outline of natural proofs paper. there is also a lot of connection of randomization (via pseudorandom generators) across the semifamous "5 algorithmic worlds" introduced also by Impagliazzo.[3]
basically various complex algorithms have been successfully derandomized in significant advances, but others resist efforts. one famous case study would be the AKS primality algorithm which for decades was a probabilistic test that was cleverly derandomized by AKS & proven to run in P time.
[1] Can every randomized algorithm be derandomized? by Impagliazzo 2006
[2] Natural proofs explained by chris calabro
[3] 2009 workshop on impagliazzos 5 worlds