Law of Large Numbers

Probability theory is a field that enables the mathematical analysis of random events and forms the foundation of statistics and data science such as. One of the most fundamental theorems of Probability theory is the Law of Large Numbers. This law states that the average outcome of an event over a large number of trials will approach its expected value.

The Law of Large Numbers (LLN) states that when a random experiment is repeated a sufficiently large number of times, the average outcome of the experiments will converge toward the theoretical expected value. That is, while individual trials may show significant variation, as the number of repetitions increases, the average gets closer to the true probability.

Mathematically, let X be a random variable with expected value denoted as E(X). Let X₁, X₂, X₃, ..., Xₙ be independent and identically distributed variables. In this case, the arithmetic mean of these variables is

as n approaches infinity, the mean converges to the expected value (µ):

This is a statistical principle that guarantees the observed mean within a sample will approach the true population mean.

To illustrate with a simple everyday example: when a fair coin is tossed, the probability of heads or tails is 50%. However, if the coin is tossed only 10 times, a result such as 7 tails and 3 heads may occur. This deviation is due to natural variability and differs from the true probability. But if the coin is tossed 1,000 or 10,000 times, the ratio of heads to tails will approach 0.5. This is one of the most straightforward examples of the Law of Large Numbers: as the experiment is repeated, the average converges toward the true expected value.

1. Insurance: Insurance companies use this law to estimate the probability of certain events occurring within a large population group.
2. Gambling and Betting: Casinos rely on it to ensure long-term profitability. While individual players may win in single games, casinos consistently profit when large sample sizes are considered.
3. Surveys and Election Predictions: By collecting opinions from a large number of people, results close to the true public sentiment can be obtained.
4. Quality Control: In factories, random sampling allows for reliable estimation of product quality on a large scale.