gen-define-the-law-of-large-numbers-and-give-one-short-worked-ex-6869

Define the law of large numbers and give one short worked example.

# Law of Large Numbers ## Definition The **Law of Large Numbers** states that as the number of trials or observations increases, the sample average converges to the expected value (theoretical mean) of the distribution. More formally: As n → ∞, the sample mean x̄ₙ → μ (the population mean). ## Short Worked Example **Coin Flipping:** A fair coin has a theoretical probability of heads = 0.5 Let's track the proportion of heads as we flip: | Flips | Heads | Proportion | |-------|-------|------------| | 10 | 7 | 0.70 | | 100 | 47 | 0.47 | | 1,000 | 512 | 0.512 | | 10,000| 4,998 | 0.4998 | **Observation:** With just 10 flips, we got 70% heads (far from 50%). But as we increased to 10,000 flips, our proportion converged to 0.4998, very close to the expected value of 0.5. This demonstrates the Law of Large Numbers: the sample proportion becomes increasingly accurate as a estimate of the true probability with more trials.