How does this easy transformation in logistic regression work?

Question

I know it's easy, but unfortunately not for me:/

I have:

$$ p(X) = \frac{e^{\beta_0+\beta_1X}}{1+e^{\beta_0+\beta_1X}} $$

and this turns out to be:

$$\frac{p(X)}{1-p(X)} = e^{\beta_0 + \beta_1X} $$

How does this work?

Answer 1

Consider this equation : $$x=\frac{y}{1+y} \tag 1$$ I suppose that it is easy for you to solve it for $y$. You get : $$y=\frac{x}{1-x} \tag 2$$ Let $x=p(X)$ and $y=e^{\beta_0+\beta_1X}$ $$ p(X) = \frac{e^{\beta_0+\beta_1X}}{1+e^{\beta_0+\beta_1X}}=x=\frac{y}{1+y}$$ This is the above equation $(1)$ which solution is above $(2)$ : $$y=\frac{x}{1-x}=e^{\beta_0+\beta_1X}=\frac{p(X)}{1-p(X)}$$