[vitus]

Don't forget the celebrated Bayes theorem that falls directly out of the definition of conditional probability P(A|B) = P(A & B) / P(B).

If you had to squint at it and turn that into P(B|A) = P(A & B) / P(A), you'd realize that you can simply multiply the top and bottom by P(A), then pull out the remaining P(A)/P(B).

       P(A & B) / P(B)
     = P(A & B) * P(A) / (P(A) * P(B))
     = P(A & B) / P(A) * P(A) / P(B)
     = P(B | A) * P(A) / P(B).