Distance bounding protocols are proposed based upon the round trip time measurements of the executed messages to protect RFID systems against the relay attack. The existing distance bounding protocols employed by RFID systems are divided into two categories generally called, with random challenges or with mixed challenges. Since RFID systems and distance bounding protocols are particularly susceptible to noise, in this paper, the security analysis of distance bounding protocols with random challenges is performed over a noisy channel. This analysis is achieved by computing an attacker's success probability due to mafia fraud and distance fraud attacks in a noisy environment. In this case, the analysis as well as simulation results show that increasing the number of iterations (rounds) makes the attacker's success probability to decrease and adversely it can causes the rejection probability of a valid tag due to channel errors to increase. Therefore, the proper values for the number of iterations and the total number of errors that are acceptable are computed to obtain the optimal false-accept and false-reject probabilities in a noisy environment.