A note on optimal probability lower bounds for centered random variables

In this note we obtain lower bounds for $\mathrm{\P }(\xi \ge 0)$ and $\mathrm{\P }(\xi >0)$ under assumptions on the moments of a centered random variable $\xi$. 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.