Differential Privacy and appropriate noise distribution In differential privacy solutions and more specifically for queries that they do entail counting the proposed solutions define the Laplace distribution that is best calibrated for low error. Other solutions for non-numerical queries propose exponential distribution. However, it is not clear why noise from a Laplace distribution is chosen for numerical? What is the special property that best suits the Laplace distributrion?
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.