Sometimes, we know very little about our data—we might only know its average ($E[X]$) or its variance ($Var(X)$). In these "uncertain" cases, we use inequalities to set a strict "worst-case scenario" on probability.

Chebyshev's Inequality: The Power of Variance

Chebyshev's inequality is more powerful because it uses both the mean ($\mu$) and the standard deviation ($\sigma$).

Intuition: This tells us that for any distribution, at least 75% of the data MUST fall within 2 standard deviations ($k=2$) of the mean, and at least 89% must fall within 3 standard deviations ($k=3$). Unlike the Normal distribution's 95/99% rules, this works even for weird, non-Normal data.

The LLN is the "mathematical engine" of the world. It states that as the sample size $n$ goes to infinity, the Sample Mean ($\bar{X}_n$) will converge exactly to the True Expected Value ($\mu$).

The CLT is perhaps the most profound theorem in all of mathematics. It states that if you take the sum (or average) of $n$ independent and identically distributed (i.i.d.) random variables, the result will always look like a Normal Distribution as $n$ grows, regardless of the original distribution's shape!

The Mathematical Statement

If $X_1, \ldots, X_n$ are i.i.d. with mean $\mu$ and variance $\sigma^2$, then: