P(A | B) = P(A and B) / P(B), defined when P(B) > 0.
Conditioning is the most reliable attack on a probability question you cannot see through: split on the first event, solve each branch, recombine with the law of total probability.
The trap. P(A | B) and P(B | A) are different quantities, and conflating them is the single most common error in applied probability. A test that is 99% accurate tells you P(positive | disease), not P(disease | positive) - and when the disease is rare those numbers differ by an order of magnitude. See Bayes' theorem.