  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.
