Hey!
you mean P(A|B) right? P(A/B) could be mistaken for P(A\B) ..the difference.
Anyways, the concept behind this definition (debatably an axiom), is very simple.
Imagine a Venn Diagram with A and B as two circles crossing each other giving rise to a shared mid-section.
P(A) = [outcomes that give event A] divided by [the sample space [tex]\Omega[/tex]] (all possible outcomes)
P(B) = [outcomes that give event B] divided by [the sample space [tex]\Omega[/tex]] (all possible outcomes)
P(AnB) = [outcomes that are SHARED between A and B] divided by [the sample space [tex]\Omega[/tex]] (all possible outcomes)
When they ask you for P(A|B), what they are asking you is:
"what is the probability that A will happen given that we KNOW that B has happened". If we know that the circle B has been chosen the only part of A left that COULD happen is the intersection between A and B that they both share!
Basically: P(A|B) = "what is the probability of A, considering that the ONLY available sample space is now B?"
do you get it now? try and draw it out!