A linear sensor system is defined as a sensor system in which the sensor measurements have a linear relationship to source variables that cannot be directly measured. Evaluation of the reliability of a general linear sensor system is a #P problem whose computational time increases exponentially with the increment of the number of sensors. To overcome the computational complexity, Monte Carlo methods are developed in this paper to approximate the sensor system's reliability. The crude Monte Carlo method is not efficient when the sensor system is highly reliable. A Monte Carlo method that has been improved for network reliability, known as the Recursive Variance Reduction (RVR) method, is further adapted for the reliability problem of linear sensor systems. To apply the RVR method, new methods are proposed to obtain minimal cut sets of the linear sensor system, particularly under the conditions where the states of some sensors are fixed as failed or functional. A case study in a multistage automotive assembly process is conducted to demonstrate the efficiency of the proposed methods.