I'm reading Introduction to Probability (Blitzstein and Hwang) and cannot wrap my head around an example they are using to provide intuition behind the formula for conditional probability.
In essence, we have a standard deck of shuffled cards and draw two cards randomly (one at a time, without replacement). We define A to be the event that the first card is a heart, and B the event that the second card is red. Then we are asked to find P(A|B) and P(B|A).
Where my confusion lies is in two calculations: how they calculated P(A ∩ B) and how they calculated P(B). They say "By the Naive definition of probability and the multiplication rule, P(A ∩ B) = (13/52) x (25/51)." Why is this valid when A and B are clearly not independent? Logically, I understand the math. But P(A)P(B) is clearly not equal to P(A ∩ B). How can you explain why this is valid, and is there a more clear way to calculate P(A ∩ B) (maybe more conditioning)?
I also don't know how you would go about calculating P(B) since it is not the first draw - any thoughts on the intuition / methodology to go about calculating this?
Much appreciated in advance!