In general, Simpson's Paradox occurs because situation such as following occurs for some arbitrary events $A,B,$ and $C$:
$P( A | B , C) < P(A| B^c,C) \tag{1}$
$P( A | B , C^c ) < P(A| B^c,C^c) \tag{2}$
Can someone show me a step-by-step way to arrive at $P( A|B) > P(A|B^c)$ from (1), (2)?
The Law of Total Probability
$P( A | B ) = P( A | B , C ) P( C | B) + P( A | B, C^c) P(C^c | B)$
appears somehow involved but I don't see how. Any help would be appreciated.
There is no derivation of the third equation from the first two. If there were, then it would be the case that Simpson's paradox occurs for $all$ $A,B,C$. This is clearly not true. The correct statement is that $there$ $exists$ $A,B,C$ such that those three conditions hold. The fact that this occurs highly depends on the events $A,B,C$