Whenever you want to switch conditional probabilities, use this.

$P(A)$ comes from the law of total probability
$P(B)$ is the "prior probability" - this is an unconditional probability. In our example, probability of being male
$P(A|B)$ is probability of A with condition B - this must be known from collecting data. In our example this is the probability of left-handedness in males, which is known.

$P(B|A)$ is the posterior probability - this is what we are always trying to calculate. This is based on the prior probability and some data on this population. In our example, this is the probability that a left-handed person is male.

Bayes Factor

Week 2 Homework#Question 16. Bayes factor.

When you cannot calculate the total probability (due to lack of information), you can use the Bayes Factor to compare two hypotheses