This paper derives an explicit approximation for the tail probability of a sum of sample
values taken without replacement from an unrestricted finite population. The approximation
is shown to hold under no conditions in a wide range with relative error given in terms of
the standardized absolute third moment of the population, β3N. This approximation is used to obtain
a result comparable to the well-known Cramér large deviation result in the independent
case, but with no restrictions on the sampled population and an error term depending only
on β3N. Application to permutation tests is
investigated giving a new limit result for the tail conditional probability of the
statistic given order statistics under mild conditions. Some numerical results are given
to illustrate the accuracy of the approximation by comparing our results to saddlepoint
approximations requiring strong conditions.