Stochastic decoding is a hardware and energy-efficient approach to implement iterative decoders. We use this decoding method in an Automatic Repeat reQuest (ARQ) scheme to improve the performance of Low-Density Parity-Check (LDPC) decoders. We show that, aside from the simplicity in hardware implementations, stochastic decoding results in improved error rate performance and requires fewer retransmissions than other iterative decoding algorithms when used in a Hybrid Automatic Repeat reQuest (HARQ) system. We present an analysis of the error rate improvement and decoding latency of the HARQ scheme using stochastic decoding. We also provide a criterion to efficiently allocate the available decoding cycles to the decoding attempts in the HARQ system in order to achieve lower error rates and less decoding latency.