Probability - General Multiplication Rule

Question

Let's consider the General Multiplication Rule in Probability, defined as:

$P(A\cap B)=P(A)P(B|A)$.

I don't understand why this formula commutes, that is:

$P(A\cap B)=P(B\cap A)$

to me, according with the general formula these 2 probabilities can be different, what do you think ?

Answer 1

The sets $A \cap B$ and $B \cap A$ are the same.

Answer 2

$A \cap B$ and $B \cap A$ both mean both $A$ and $B$ happen, leading to equality of the probability.

This leads to Bayes' formula for reversing the order of conditioning. $$ P(B|A)=\frac{P(A|B)P(B)}{P(A)} $$