The solutions of many important problems in population genetics require tools from probability theory because the evolution of a population over time is modeled as a random process. In this talk, we focus on the question of how to describe the effect that beneficial mutations have on the genealogy of a population. We present two approximations. The first, which involves coin tossing, is simple but not very accurate. The second, which is based on a stick-breaking construction, leads to much more accurate results.