A critical aspect of applications with Wireless Sensor Networks (WSNs) is network lifetime. Power-constrained WSNs are usable as long as they can communicate sense data to a processing node. Poor communication links and hazardous environments make the WSNs unreliable. Existing schemes assume that the state of a sensor covering targets is binary: success (covers the targets) or failure (cannot cover the targets). However, in real WSNs, a sensor covers targets with a certain probability. To improve WSNs' reliability, we should consider that a sensor covers targets with users' satisfied probability. To solve this problem, this paper first introduces a failure probability into the target coverage problem to improve and control the system reliability. Furthermore, we model the solution as the α-Reliable Maximum Sensor Covers (α-RMSC) problem and design a heuristic greedy algorithm that efficiently computes the maximal number of α-Reliable sensor covers. To efficiently extend the WSNs lifetime with users' pre-defined failure probability requirements, only the sensors from the current active sensor cover are responsible for monitoring all targets, while all other sensors are in a low-energy sleep mode. Simulation results validate the performance of this algorithm, in which users can precisely control the system reliability without sacrificing much energy consumption.