Hello,

The probability "B given A" is the probability for B to happen if A already happened before. So weknowA happened before and we're studying the probability for B to happen under this condition.

If you multiply it by the probability of A to happen, you'll have the probability :

- that A happens

- that B happens, since you suppose B happens if A happens

So that's why basically P(A and B) is P(B|A)P(A).

Maybe it helps to see it this way, dunno...

Imagine you have a soccer match. Teams A and B have the same probability to score (0.5). A player from team A scores (event A). This will make the players of team A very happy and they will be less careful. On the other hand, players from team B will do their best in order to score back, otherwise they'll be fired by their coach. So given the fact that team A scored first, the probability for team B to score (event B) will be superior, for example 70% !!

These 70% are the probability of "B given A".

And we have P(A)=0.5, P(B|A)=0.7

My explanations in English are just awful so if you have any questions...