Good attempt in simplifying the concept of Bayes Theorem. But I am afraid in the effort to over simplify you might have erred a little.

“It tells us how often A happens given that B happens, written P(A|B)”

I think the above statement is flawed, in statistics we interpret it as probability of Event ‘A’ happening given that Event ‘B’ has already happened. Also I would suggest, you use the word ‘often’ judiciously as in the context of probability things don’t occur ‘often’ rather it is a chance .