Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$, $$ \text{Prob}(|X-E(X)|\geq k) \leq \text{Prob}(|Y - E(Y)|\geq k). $$ Note that this is no duplicate of this question.
I know Chebyshev's inequality relates the above probabilities to the variance, but it doesn't say anything about the above, at least to my understanding.
Finally, if the above is true, how does it change for $|E(X) - E(Y)|$ being small but strictly positive?