Clarification on my understanding of Bayes theorem?

Question

Do I have the proper understanding of Bayes theorem?

The difference between conditional probability and Bayes Theorem of P(A|B), is

For Conditional Probability, it is the probability of A given B $$ P(A|B) =\frac{ P(A \cap B)}{P(B)} $$

While for Bayes Theorem, we utilize "A" as a parameter with B unknown $$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$

Answer 1

I don't know what you mean by "utilize 'A' as a parameter with B unknown." In the example you give, it looks like we need to know quite a bit about both A and B in order to apply the theorem.

And the left-hand side of the equation in your Bayes' Theorem example, $P(A\mid B),$ is the exact same "A given B" as in your Conditional Probability example; Bayes' Theorem is a theorem about conditional probability.