Differential Privacy and appropriate noise distribution
In order to bridge the gap between two worst case scenarios to produce similar distribution of privatized answers certain noise is added. For example the salaries of a CEO (max) and a line worker or who ever gets minimum wage (min) may not be produce similar distribution without the noise. 

Random values taken from Laplacian distribution with standard deviation is large enough to cover the gap. 

This link has a good beginners tutorial with detailed explanation. Around 15.33 minutes in the video the speaker discusses about the issues like Global Sensitivity and Laplacian Noise.  Subsequently she discusses the proof too.