Sharp probability bounds for the binomial moment problem with symmetry

Sharp bounds in terms of the first few binomial moments are found for the probability of a union of events, when the random variable denoting the number of events that occur follows symmetric distribution. Connection between the bounds of this paper and the bounds from a special case of the binomial moment problem of Prekopa (1995) is shown. As a special case, bounds for the probability when the underlying probability distribution is unimodal-symmetric are also found.