Mobile crowdsourcing to smartphones has emerged as a compelling paradigm for collecting sensing data over a vast area for various monitoring applications. It is of paramount importance for mobile crowdsourcing to provide incentive mechanisms. State-of-the-art auction mechanisms for mobile crowdsourcing are deterministic in the sense that given the real costs of the smartphone users, a fixed set of smartphone users are recruited for performing sensing tasks. This leads to serious issues including reduced diversity of sensing devices and starvation of some users. In this paper, we propose an approximate-truthful randomized combinatorial auction mechanism for the social cost minimization problem, which is NP-hard. We design an approximate task allocation algorithm that is near optimal with polynomial-time complexity and use it as a block to construct the whole randomized auction mechanism. We carry out numerical studies and show that our randomized auction mechanism achieves approximate truthfulness, individual rationality, and high computation efficiency. Moreover, the proposed mechanism increases diversity of devices and prevents starvation of some smartphones.