I found coincidentally that these 2 existing formulas are equivalent, by using real numbers as inputs:
- $x=1-\frac{a/c}{1-a/c}*\frac{1-b/d}{b/d}$ and
- $a=\frac{b/d*(1-x)}{1-b/d+b/d*(1-x)}*c$
Using output $x$ of formula 1 as input in formula 2, yields $a$ in formula 2, using the same $b$, $c$ and $d$. I was wondering if it is possible to mathematically prove these formulas are equivalent. If not, how could I state in a formal way that it is probable that these are equivalent? These simple formulas may have been used for a variety of applications already, and may be there is already some kind of proof (if yes, how could I find a source?). I am not a mathematician and my equation solving skills are very limited. I would be very pleased if someone could help. Many thanks.
Thank you Jean Marie, using your suggestion I solved it.
$\displaystyle \frac{yB}{(1-B)}=\frac{A}{(1-A)}$
$yB(1-A)=A(1-B)$
$yB-yBA=A-AB$
$yB=A-AB+yBA$
$yB=A(1-B+yB)$
$\displaystyle\frac {yB}{(1-B+yB)}\displaystyle=A$