The interplay between turbulent bursts and transport barriers is analyzed with a simplified model of interchange turbulence in the Scrape-Off Layer of magnetically confined plasmas. The turbulent bursts spread into the transport barriers, and, depending on the competing magnitude of the burst and stopping capability of the barrier can burn through. Two models of transport barriers are presented, a hard barrier where all turbulent modes are stable in a prescribed region and a soft barrier with external plasma biasing. This process can be modeled on the basis of competing stochastic processes. For classes of probability density function of these processes one can predict the heavy tail properties of the bursts downstream from the barrier, either exponential for a leaky barrier, or with power laws, for a tight barrier. The intrinsic probing of the transport barriers by the turbulent bursts thus gives access to properties of the transport barriers. The main stochastic variables of the two models addressed here are the barrier width and the spreading distance of the turbulent bursts within the barrier together with their level of correlation. One finds that in the case of a barrier located in the Scrape-Off-Layer, the stochastic model predicts a leaky barrier with an exponential probability density function of escaping turbulent bursts in agreement with the simulation data.