gen-state-the-law-of-large-numbers-precisely-and-explain-why-it-fd76

State the law of large numbers precisely and explain why it matters in 2-3 sentences.

# Law of Large Numbers **Precise Statement:** As the number of independent trials *n* approaches infinity, the sample mean $\bar{X}_n$ converges to the expected value $\mu$. Formally: $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i \xrightarrow{P} \mu$ as $n \to \infty$ (weak form) or $\bar{X}_n \to \mu$ almost surely (strong form). **Why It Matters:** This theorem justifies using observed frequencies to estimate probabilities and guarantees that long-run averages stabilize to predictable values. It's the mathematical foundation for statistical inference, insurance, polling, and any field where we use data to draw conclusions about populations.