holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. the counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. a natural question to ask is: how random are the holonomy vectors of a translation surface? we motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. no prior background on translation surfaces or gap distributions will be assumed.