Deriving Probability from logit or log-odds

Question

How can I motivate the derivation of the $p$ below?
From: https://en.wikipedia.org/wiki/Logistic_regression#Logistic_model

$l = log_b\frac{p}{1-p}=\beta_0 + \beta_1x_1+\beta_2x_2 $

$p = \frac{b^{\beta_0 + \beta_1x_1+\beta_2x_2}}{b^{(\beta_0 + \beta_1x_1+\beta_2x_2)} +1}= \frac{1}{1+b^{-( \beta_0 + \beta_1x_1+\beta_2x_2 )}} $

Given that:

$p $= probability of event $Y$=1

$l$ = logit or log-odd of $p$

$x$ = predictors or independent variables

I don't understand how the p above arrived at $ \frac{b^{\beta_0 + \beta_1x_1+\beta_2x_2}}{b^{(\beta_0 + \beta_1x_1+\beta_2x_2)} +1}= \frac{1}{1+b^{-( \beta_0 + \beta_1x_1+\beta_2x_2 )}} $

Answer 1

You start by assuming a linear form for the log odds, i.e., $$ \ln \left( \frac{p}{1-p} \right) = \beta^Tx $$ thus, exponentiating both sides you get $$ \frac{p}{1-p} = e^{\beta^Tx} $$ re-arranging the equation $$ e^{\beta^Tx} - pe^{\beta^Tx} = p\to p(1+e^{\beta^Tx})=e^{\beta^Tx} $$ finally, $$ p = \frac{e^{\beta^Tx}}{1+e^{\beta^Tx}} $$

Answer 2

You've copied the equations from the wikipedia page incorrectly. It should read \begin{align*} p = \dfrac{b^{\beta_0 + \beta_1x_1 + \beta_2x_2}}{b^{\beta_0 + \beta_1x_1 + \beta_2x_2} + 1} \end{align*} where $b$ is the base of the logarithm (e.g. $e$). $b$ is not a parameter nor is it related to $\beta$.