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

Edit : This answer applies to the original question, that was : "for ANY random variable Y, what is the lower bound of (formula)"
The lower bound is 0.
Let's take $(Y_{n})$ a series following a Bernoulli law such as : $P(Y_{n} = 1) =1/n$.
Then, ${E(Y_{n})= \lambda = 1/n}$.
$P(\lvert Y_{n} \rvert > \frac{\lvert \lambda \rvert}{2}) = P(\lvert Y_{n} \rvert > \frac{1}{2n}) = P(Y_{n} =1) = \frac{1}{n} <  \frac{2}{n}\underset{n\to +\infty}{\longrightarrow} 0 $
Then, $0 < P(\lvert Y_{n} \rvert > \frac{\lvert \lambda \rvert}{2})  < \frac{2}{n}$ , so asymptotically, the lower bound is 0.
