= Solution
The requested bounds are $\mathbb E[X]-\sigma=2/5$ and $\mathbb E[X]+\sigma=4/5$, both within the support. Since the <probability distribution> is continuous, endpoints have zero <probability>, and subtraction of the <cumulative distribution function> gives
$$
\boxed{\mathbb P\left(\mathbb E[X]-\sigma\leq X\leq\mathbb E[X]+\sigma\right)=4\left[\left(\frac45\right)^3-\left(\frac25\right)^3\right]-3\left[\left(\frac45\right)^4-\left(\frac25\right)^4\right].}
$$
This leaves the answer in rational fractions as requested.
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