Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 4 / b / Solution

The intercept is the midpoint of the two log odds:Equivalently $e^{\alpha }$ is the geometric mean of the two odds. The logistic function evaluated at $\alpha$ represents a central event probability, and approximates the average of the two group probabilities when $\beta$ is small. It is not generally their arithmetic average, nor is $\alpha$ specifically the control-group log odds.