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Can someone explain for the non-statisticians?
P(A|B) means the conditional probability of event A happening if B has happened. It is defined as P(A∩B)/P(B).
The equation show is the Bayes' Rule. It shows the relationship between P(A|B) and P(B|A). This is useful in Bayesian statistics (of course why would it not be) as you change your probability distribution based on the data observed, and almost every time it's easier to find the value by swapping the events.