A medical internship is mandatory for all medical students graduating from a Danish university. The internships are distributed fairly even amongst most Danish hospitals, m.a. in order to secure a steady inflow of new doctors to all regions of Denmark. When allocating the graduates to an internship by lottery, the National Board of Health takes into account only the candidates preferences over the various internships, for reasons discussed in the paper. However, a problem, relating to the lottery which the NBH uses, is that the lottery itself is not efficient in an economic sense. By way of simulations this article shows how an alternative lottery built on the so-called EPS-algorithm developed by Katta and Sethuraman [Journal of Economic Theory 131, 231-250] performs better than the lottery used today. This, along with other problematic aspects such as black-market trading of internships, indicates that a replacement of the currently used lottery may be beneficial for the NBH as well as the graduating medical students