Difference between conditional probability and Bayes rule

Question

I know the Bayes rule is derived from the conditional probability. But intuitively, what the difference? The equation looks the same to me. The nominator is the joint probability and the denominator is the probability of the given outcome.

This is the conditional probability: P(A∣B)=P(A∩B)/P(B)

This is the Bayes' rule: P(A∣B)=P(B|A)*P(A)/P(B).

Isn't "P(B|A)*P(A)" and "P(A∩B)" the same? When A and B are independent, there is no need to use the Bayes rule, right? What's the difference intuitively between conditional probability and bayes rule?

Answer 1

Baye's thereom uses inverse or posterior probability and also it uses the total probability of an event. You are considering two events here, maybe considering a more general case would shed light on the difference

Answer 2

They are basically the same, yes. Bayes's rule is essentially the fact that $$\mathbb{P}(B \mid A) \mathbb{P}(A) = \mathbb{P}(A \cap B) = \mathbb{P}(A \mid B) \mathbb{P}(B)$$ which is just two applications of the law of conditional probability you state. That doesn't make it any less important, though.