# Law of Large Numbers >[!Abstract] The **Law of Large Numbers** states that if you sample a random variable independently a large number of times, the measured average value should converge to the random variable's true expectation. Stated more formally: $\bar{X} = \frac{X_1 + X_2 + ... + X_n}{n} \rightarrow \mu $ As $b \rightarrow \infty$ > A casino might experience a loss on any individual game, but over the long run should see a predictable profit over time Keep in mind that the law of large numbers applies - For averages and therefore also for percentages, but not for sums as their SE increases by a rate of $\sqrt{n}$, this is due to the way [[The Standard Error]] works - For sampling with replacement from a population, or for simulating data from a probability histogram (copied from [[Module 5 - Sampling Distributions and the Central Limit Theorem#The Law of Large Numbers|Introduction to Statistics by Stanford]]) >[!Cite] Related Concepts >[[self.stats/Statistics]]