I am confused..
It doesn’t seem to be true that P(A and B) for two events happening consecutively is the same as P(A and B) for two events happening simultaneously.
So we should never use the notation P(A ∩ B), the probability of the intersection of A and B for the first case? (I have been using that notation whenever seeing A and B)
For example, let A be the event you roll an even # on a die, let B be the event you roll a 2.
P(A and B), the probability of A happening and then B happening is (1/2)(1/6) = 1/12 by the rule for independent events..
But P(A∩B), the probability of the events happening simultaneously is 1/6. There is only one case out of 6 where you would roll a 2 and even.
So P(A and B) ≠ P(A ∩ B) here.. what’s wrong??