We present an approximate sampling framework and discuss how risk-limiting audits can compensate for these approximations, while maintaining their "risk-limiting" properties. Our framework is general and can compensate for counting mistakes made during audits. Moreover, we present and analyze a simple approximate sampling method,"$k$-cut", for picking a ballot randomly from a stack, without counting. Our method involves doing $k$ "cuts", each involving moving a random portion of ballots from the top to the bottom of the stack, and then picking the ballot on top. Unlike conventional methods of picking a ballot at random, $k$-cut does not require identification numbers on the ballots or counting many ballots per draw. We analyze how close the distribution of chosen ballots is to the uniform distribution, and design different mitigation procedures. We show that $k=6$ cuts is enough for an risk-limiting election audit, based on empirical data, which would provide a significant increase in efficiency.