What is the probability that direct detection experiments have observed Dark Matter?

In Dark Matter direct detection we are facing the situation of some experiments reporting positive signals which are in conflict with limits from other experiments. Such conclusions are subject to large uncertainties introduced by the poorly known local Dark Matter distribution. We present a method to calculate an upper bound on the joint probability of obtaining the outcome of two potentially conflicting experiments under the assumption that the Dark Matter hypothesis is correct, but completely independent of assumptions about the Dark Matter distribution. In this way we can quantify the compatibility of two experiments in an astrophysics independent way. We illustrate our method by testing the compatibility of the hints reported by DAMA and CDMS-Si with the limits from the LUX and SuperCDMS experiments. The method does not require Monte Carlo simulations but is mostly based on using Poisson statistics. In order to deal with signals of few events we introduce the so-called "signal length" to take into account energy information. The signal length method provides a simple way to calculate the probability to obtain a given experimental outcome under a specified Dark Matter and background hypothesis.