The authors study the quadrature errors \[ \text{Err} (f;Q):=I(f)-Q(f) \] in the approximation of the multidimensional integral \[ I(f)=\int_{\mathcal X} f(\mathbf{x}) dF(\mathbf{x}) \] in \(\mathcal{X}\subset R^s\) (with respect to a probability distribution \(F\)) by quadrature rules \[ Q(f)=\sum_{i=1}^na_if(\mathbf{x}_i). \] They consider three error measures: the worst-case error, the random-case error, and the average-case error that are respectively defined by \begin{align*} e^{\text{worst}} &:={\text{rms}}_{Q\in\mathcal{Q}} \sup_{f\in\mathcal{F}}|\text{Err}(f;Q)|,\\ e^{\text{rand}} &:=\sup_{f\in\mathcal{F}} {\text{rms}}_{Q\in\mathcal{Q}}|\text{Err}(f;Q)|,\\ e^{\text{avg}} &:={\text{rms}}_{Q\in\mathcal{Q}}\sup_{f\in\mathcal{H}}|\text{Err}(f;Q)|, \end{align*} where \(\mathcal{H}\) is a separable Hilbert space and \(\mathcal{F}\) its unit ball; quadrature rules \(Q\) are randomly taken from a sample space \(\mathcal{Q}\) with a probability measure \(\mu\). Three explicit formulae are derived for these errors; they show the relative pessimism of the three approaches. The first one is the trace of an Hermitian nonnegative definite matrix \(\Lambda^\mu_{\mathcal Q}\), the second one is the spectral radius of the same matrix, and the third one is \(\text{trace}(\Sigma\Lambda^\mu_{\mathcal Q})\) where \(\Sigma\) is an Hermitian nonnegative definite matrix with \(\text{trace} (\Sigma)=1\). Several examples are studied including Monte Carlo quadrature and shifted lattice rules.