The law of large numbers states that as the sample size of a random variable increases, the average of the sample approaches the true mean of the population. In other words, the more data we collect, the closer our sample mean will be to the true population mean.

For example, let’s say we want to determine the average height of adult men in the United States. We can take a sample of 100 men and measure their heights, calculating the sample mean. However, this sample may not accurately represent the true population mean because it only includes a small number of individuals. If we were to increase our sample size to 1,000 men, our sample mean would likely be much closer to the true population mean. This is because the larger sample size allows us to account for more potential variations and increases the likelihood that our sample accurately represents the population.

Another example of the law of large numbers is in the realm of probability. If we flip a coin 100 times, we may not necessarily get exactly 50 heads and 50 tails. However, if we flip the coin 1,000 times, the ratio of heads to tails will likely be much closer to 1:1 because the larger sample size allows for more accurate representation of the true probability of the coin landing on heads or tails.

Overall, the law of large numbers is a fundamental concept in statistics and probability that allows us to make more accurate predictions about a population based on a larger sample size. It is important to keep in mind that the law of large numbers only holds true for random variables and does not apply to non-random or biased samples.