Scheduling fast uplink grant transmissions for machine type communications (MTCs) is one of the main challenges of future wireless systems. In this paper, a novel fast uplink grant scheduling method based on the theory of multi-armed bandits (MABs) is proposed. First, a single quality-of-service metric is defined as a combination of the value of data packets, maximum tolerable access delay, and data rate. Since full knowledge of these metrics for all machine type devices (MTDs) cannot be known in advance at the base station (BS) and the set of active MTDs changes over time, the problem is modeled as a sleeping MAB with stochastic availability and a stochastic reward function. In particular, given that, at each time step, the knowledge on the set of active MTDs is probabilistic, a novel probabilistic sleeping MAB algorithm is proposed to maximize the defined metric. Analysis of the regret is presented and the effect of the prediction error of the source traffic prediction algorithm on the performance of the proposed sleeping MAB algorithm is investigated. Moreover, to enable fast uplink allocation for multiple MTDs at each time, a novel method is proposed based on the concept of best arms ordering in the MAB setting. Simulation results show that the proposed framework yields a three-fold reduction in latency compared to a random scheduling policy since it prioritises the scheduling of MTDs that have stricter latency requirements. Moreover, by properly balancing the exploration versus exploitation tradeoff, the proposed algorithm can provide system fairness by allowing the most important MTDs to be scheduled more often while also allowing the less important MTDs to be selected enough times to ensure the accuracy of estimation of their importance.