Conditional probabilty; dependent and independent events

This is my first post to this forum. I feel it's time I gained a proper grasp on probabilty and statistics, so I've started working my way through a text book on the subject... I made it as far as page 7!

P(B|A) is the probabilty of B given that A is true. I understand that to mean that the event occuring in A is certain (P(A) = 1), given that, what is the probabilty of the event also occurring in B. ie. the sample space is now A, what is the probabilty of the event being in the region A$\displaystyle \cap$B?

So, independent events; I take this to mean both occur in the same sample space, but there is no overlap between them. Is that correct?

I don't understand why P(B|A) = P(B), by my thinking it should be zero!

If anyone can understand my ramblings and help me understand what I am missing I would be grateful.

Many thanks