How to calculate lower bound on probability? Let $Y$ be a random variable such that $E[Y] = \lambda$,  $\lambda \in \mathbb{R}$ and $E[Y^2]<\infty$. Then the  problem is to find a lower bound on the probability
$$
P \left[|Y| > \frac{|\lambda|}{2} \right].
$$
Any leads would be appreciated.

In general, without further conditions there is non. For example, let $\lambda=1$ and $Y_n$ be a random variable given by $\mathbb{P}(Y_n=0)=1-1/n$ and $\mathbb{P}(Y_n=n)=1/n$ for each $n\in\mathbb{N}$.
Then $\mathbb{E}Y_n = 1$, but
$$\mathbb{P}(|Y_n|>1/2) = 1/n \overset{n\to\infty}{\longrightarrow} 0.$$
