This thesis studies semilinear wave equations with quadratric derivative non- linearity |∇u|2 (qDNLW) from the probabilistic perspective. We first adapt a method of Bjoern Bringmann in [Bri21] to the d = 2 setting. This method goes beyond the linear-nonlinear decomposition due to Bourgain ([Bou94] and [Bou96]). This is contained in Chapter III. We improve local-in-time well-posedness results in the probabilistic setting in spatial dimensions 2 and 3 by constructing the Random Averaging Operators for (qDNLW). Local well-posedness is proven for data in the spaces H^{3/2+}(\mathbb{T}^3) and H^{11/8+}(\mathbb{T}^2). The space H^{11/8+} (\mathbb{T}^2) is supercritical with respect to the deterministic scaling. Both these results improve over both the best probabilistic and best deterministic results. These thresholds however lie 1+ and 3/8+ above the respective probabilistic scalings for the problem (qDNLW). The argument is constructive in that it is shown that the solution has an explicit expression as the linear combination of a Gaussian sum with adapted random matrix coefficients and a smooth remainder term. This is contained in Chapter IV.