Being able to integrate and differentiate is less important than understanding that P(B|A) does not, in general, equal P(A|B), and you can learn that latter fact without learning how a probability distribution is defined. Wikipedia has a proof of Bayes' Theorem without calculus:

https://en.wikipedia.org/wiki/Bayes%27_theorem

Yes, it is ultimately founded on calculus. But we don't teach numbers to kindergarteners starting from Peano's axioms, and we don't need to gate the essentials of conditional probability behind calculus. In fact, doing so is positively harmful to society at large.