Deriving Bayes Rule from conditional probability Bayes Rule and Conditional Probability look so similar to me.
I'm having a hard time figuring out how to derive Bayes from the conditional probability equation.
If I start with $$P(A,B) = P(A|B)P(B)$$, how do I get to
$$
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
$$

Decomposing the joint probability both ways, you get:
$$\mathbb{P}(A,B) = \mathbb{P}(B|A) \mathbb{P}(A) = \mathbb{P}(A|B) \mathbb{P}(B).$$
Consequently, you get:
$$\begin{align}
\mathbb{P}(A|B)
= \frac{\mathbb{P}(A,B)}{\mathbb{P}(B)} 
= \frac{\mathbb{P}(B|A) \mathbb{P}(A)}{\mathbb{P}(B)}.
\end{align}$$
For more information on the intuition of this result, and a way of expressing it via a symmetry principle, see this related answer.
