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help needed understanding conditional probability

in http://www.thestudentroom.co.uk/showthread.php?p=4006575&highlight=conditional+probability#post4006575

jw writes

Define two events:

A = {first card is an ace}
B = {second card is not an ace}

P(A|B)
= P(A and B)/P(B)
= (4/52)(48/51) / (48/52)
= 4/51

recently the following question was asked:

One bag contains four white balls and another contains two white balls and two black. One of the bags is selected at random, and a ball is drawn from it which turns out to be white. What is the probability that the remaining balls in the bag are all white?

following above we have:
Define two events:

A = {bag1}
B = {white ball}

P(A|B)
= P(A and B)/P(B)

now i cant "get" the P(A and B) term. clearly it is not P(A)P(B) as in above example (or is it?)

But how do i think of what this P(A and B) means?
my limited skill with prob can "waffle" an answer if it was P(B and A)
ie P (bag 1) and P(white)=1/2x1 which gives the correct answer.

but how do i think of P(white) and P(bag1)? or does it not make sense this way round as we choose a bag first so it only makes sense to say

p(pick bag 1)=1/2 and so P(white)=1 so p(B and B)=1/2x1?

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