Here is my interpretation: Venn diagram shows the set of outcome of an experiment. But in many real world situation the set of outcome of an experiment is too

Here is my interpretation: Venn diagram shows the set of outcome of an experiment. But in many real world situation the set of outcome of an experiment is too large or difficult to work with. I think Venn diagram approach is very helpful if you finite set of outcome of an experiment that you can write/generate and draw/color those sets. It's right that P(B|A)P(A) is not intuitive, but if it is replaced by P(B cap A) or P(A cap B) suddenly you are back to the definition of probability of an event. Calculating P(B) using the law of total probability was very illuminating for me.