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