Chaotic mesoscopic cavities with ideal leads have been studied extensively using the powerful formalism of random matrix theory. Consequently, the statistical behavior of quantum transport observables, such as Landauer conductance and shot noise power, is fairly well understood. However, these have been explored far less in cavities with nonideal leads, i.e. having tunnel barriers. Here, we consider chaotic mesoscopic cavities with one ideal and one nonideal lead, and investigate the statistics of conductance and shot noise power under the condition of broken time-reversal symmetry. We derive exact expressions for the average and variance for both conductance and shot noise power. Moreover, we provide exact characteristic functions of these quantities, which are subsequently Fourier-inverted using a simple yet effective numerical procedure to obtain the corresponding distributions. We examine these quantities by varying the tunneling probabilities. We also compare our analytical results with Monte Carlo simulations and find very good agreements. © 2022 Elsevier B.V.