In this work, the authors give some rigidity results for Bach-flat manifolds with positive constant scalar curvature under pointwise or integral pinching conditions which have the additional property of being sharp. They give some sufficient conditions for stochastically complete Bach-flat manifolds with positive constant scalar curvature to be either Einstein or of constant sectional curvature. We remember that every parabolic Riemannian manifold is stochastically complete. A metric $g$ is called Bach-flat if the Bach tensor $$ B_{ij}=\frac{1}{n-3}\nabla^k\nabla^lW_{ikjl}+\frac{1}{n-2}R^{kl}W_{ikjl} $$ vanishes.