I have the following conditional probability expression:
$$ \frac{\frac{P(E|D)P(D)}{P(E)-P(E|D)P(D)}}{\frac{P(\bar{E}|D)P(D)}{P(\bar{E})-P(\bar{E}|D)P(D)}} $$
I want to simplify it to:
$$ \frac{\frac{P(E|D)}{1-P(E|D)}}{\frac{P(E|\bar{D})}{1-P(E|\bar{D})}} $$
I already did this on my own but in an overly lengthy way and I was told this should be much more simple. We could remove $P(D)$ from both numerators but other than that I'm pretty lost. I also don't know how to "move" the negation sign from E to D in the denominator part.
Any help will be welcome. Thanks!
Facts that might help: \begin{align} P(\bar E\mid D)P(D) &= (1 - P(E\mid D))P(D) \\[1ex] P(E) - P(E\mid D)P(D) &= P(E) - P(E \cap D) \\ &= P(E \cap\bar D) \\ &= P(E\mid\bar D)P(\bar D) \end{align}
In case you need a derivation of the first fact: \begin{align} P(\bar E\mid D)P(D) &= P(\bar E \cap D) \\ &= P(D) - P(E \cap D) \\ &= P(D) - P(E\mid D)P(D)\\ &= (1 - P(E\mid D))P(D) \end{align}