Suppose 60% of people have a credit card, 37% of people have a debit card and 30% of people have neither a credit card or a debit card. Call these sets $A,B$ and $C$ respectively.
I am assisting a friend of mine and her solution online says that the probability that a random person have both a credit card and a debit card given they already have a credit card or a debit card is given by
$$P(A\cap B| A)P(A\cap B|B)$$
but I cannot fathom why this is the case, in particular, why it is the multiplication of these two probabilities. These two conditional probabilities do not seem like they are independent at all. My guess would have been that the probability is given by
$$P(A\cap B| A\cup B)$$
but I have tried to justify both of these solutions enough times that I've lost a lot of confidence. I understand that it is not a good practice to base things off of black box solutions, but I am normally able to convince myself otherwise if they are wrong and this time I can't. Any help would be appreciated.