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In this note we obtain lower bounds for P(ξ ≥ 0) and P(ξ > 0) under assumptions on the moments of a centered random variable ξ. The obtained estimates are shown to be optimal and improve results from the literature. The results are applied to obtain probability lower bounds for second order Rademacher chaos.doi:10.4064/cm113-2-5 fatcat:3znscpvu6bbtrjdsww6puqwjuy