How to prove or disprove $\mathbb P[X\in[\mathbb E[X]-a\sigma(X),\mathbb E[X]+a\sigma(X)]]\leq\frac{1}{a^2} $?

Question

$X$ is a random variable in $\mathcal{L^2}$, $a>0$ and $\sigma(X)$ is the standard deviation of $X$.

Answer 1

False. As $a \to \infty $ LHS tends to 1 and RHS to 0.

Answer 2

Let $X$ be constant so that LHS equals $1$. Further let $a$ be large enough to satisfy $\frac1{a^2}<1$.