This paper discusses two relatively separate problems in discrepancy theory, one being well-known and one being introduced. The reason for doing so is that our solution for the former was actually inspired by our investigating the average case for the latter. The paper is structured as follows: We introduce the Lp −discrepancies, known results and a motivation behind a probabilistic approach towards discrepancy in this section, give our results in the second section and provide proofs in the last part of the paper...