Manipulating inequalities and probabilities We have the Chebychev's inequality $$\mathrm{P}\left(|X - \mu| \ge k\sigma\right) \le \frac{1}{k^{2}}$$ which tells us $\frac{1}{k^{2}}$ is the upper bound. How do we find the lower bound of this probability using this information?

The lower bound is zero. Most of the time, you can construct examples that hit these bounds exactly. E.g., a uniform distribution has support on $\mu \pm \sigma \sqrt{3}$, so it places zero probability for $k\ge \sqrt{3}$. But Chebyshev's inequality does not know about it, and that's not its job, anyway. On the other boundary, as far as I recall, Chebyshev's inequality is exact for a two-point mass distribution when $k$ is aligned to hit these masses, so it cannot be improved, either.
