In the new “Conditional Probability” course (very nice course) we learn about the Bayes Theorem. We define this theorem as follows:

`P(A | B) = P(A ∩ B) / P(B)`

and

`P(A ∩ B) = P(B | A) * P(A)`

so after replacing of `P(A ∩ B)`

we get the Bayes Theorem:

`P(A | B) = P(B | A) * P(A) / P(B)`

But before we learned that:

`P(A ∩ B) = P(B ∩ A) -> P(A ∩ B) = P(B | A) * P(A) = P(A | B) * P(B)`

so I can rewrite the Bayes theorem as follows:

`P(A | B) = P(A | B) * P(B) / P(B)`

But this solution is wrong. Where is my mistake?