The relationship between probability in classification and real values in regression problems? Say I have a problem of predicting the house price.
Originally it is a regression problem. However I consider two classes (expensive houses and not) by set the threshold of 100.000$.
Say I develop a logistic regression (model 1) to predict a house is expensive or not. The model outputs a probability.
I also develop a linear regression (model 2) to predict the price of a house. The model outputs real numbers.
Could I expect the output probability reflects the price of the house as well? If not what should be the relationship between the output of model 1 and model 2?
 A: There is a correspondence between regular linear regression and a generalized linear regression model with a probit link function. So this answer doesn't apply to the logistic regression model with a logit link.
Assume $\tilde{Y}$ is your home price, $x$ is your nonrandom predictor, $\beta_0$ and $\beta_1$ are your coefficients, and $\epsilon \sim \text{Normal}(0,\sigma^2)$ is one of your spherical error terms. If 
$$
\tilde{Y} = \beta_0 + \beta_1 x + \epsilon
$$
then $\tilde{Y} \sim \text{Normal}(\beta_0 + \beta_1 x, \sigma^2)$. We have normality, so we can come up with probabilities. 
Let $Y \in \{0,1\}$ be your binary predictor that tells you when a home price is greater than \$100,000. Or in other words, $Y=1$ whenever $\tilde{Y} > 100,000$. Then
\begin{align*}
P(Y=1 |x) &= P(\tilde{Y} > 1e5) \\
&= P( \epsilon > 1e5 - \beta_0 - \beta_1 x ) \\
&= P( \epsilon < -1e5 + \beta_0 + \beta_1 x ) \\
&= \Phi\left(\frac{-1e5 + \beta_0 + \beta_1 x}{\sigma}\right).
\end{align*}
Coming from the other end of things, a probit model estimates
$$
P(Y=1|x) = \Phi(\alpha_0 + \alpha_1 x).
$$
Both models, if used to predict the binary outcome, predict with a function formed by composing the normal cdf with a linear transformation of $x$. The coefficients are not quite the same, but you can see the relationship: $\alpha_0 = (-1e5 + \beta_0)/\sigma$ and $\alpha_1 = \beta_1 / \sigma$. This means the models should give you the same predictions roughly for whether a house is expensive, but not the same parameter estimates.
