• Understand the basic idea of the law of large numbers
Deeper study
• Distinguish between the strong law and the weak law of large numbers
Real-world practice
• Apply the law of large numbers in real-world practice
At a glance
The law of large numbers is a fundamental theorem of probability stating that as the number of independent trials n increases, the sample mean converges to the theoretical mean (the population mean, or expected value).
The weak law guarantees convergence in probability (lim P(|X̄ₙ−μ|>ε)=0) and the strong law guarantees convergence with probability 1 (P(lim X̄ₙ=μ)=1); the weak form follows from the fact that the variance of the sample mean shrinks as σ²/n, combined with Chebyshev's inequality.
In an insurance example with a 0.1% annual mortality rate, the observed rate across 1,000 policies swings widely between 0.05% and 0.2%, but across 1 million policies it settles into the 0.099%–0.101% range, which is what makes risk pooling possible.
Because each trial is independent, the gambler's fallacy — believing that earlier outcomes influence the next one — is a misapplication of the law of large numbers, and the law does not apply to distributions with infinite variance.
You need the concepts of expected value and probability
Basic: What Is the Law of Large Numbers?
Difficulty 2/5
About 20 min
The law of large numbers says that as the number of trials grows, the sample mean converges to the theoretical mean (the expected value).
Keep flipping a coin and the proportion of heads gets closer and closer to 50%. Early on the gap can be large, but the more flips you make, the more the proportion settles toward 50%.
Key points
•
As the number of trials increases, the sample mean converges to the population mean
•
It gives probability its practical meaning
A simple example
Flip a coin 1000 times and the proportion of heads will be close to 0.5.
Check your understanding
Answer:
The law of large numbers is about the convergence of the mean; the central limit theorem is about the shape of the distribution (normal distribution).
The law of large numbers comes in two forms.
The weak law of large numbers (WLLN) is a statement about convergence in probability: as the sample size grows, the probability that the sample mean lies more than some fixed distance from the population mean goes to 0.
The strong law of large numbers (SLLN) is a statement about almost sure convergence: with probability 1, the sample mean converges to the population mean.
Derivation:
1) Weak law (via Chebyshev's inequality)
- Sample mean: Loading...
- Expected value: Loading...
- Variance: Loading...
- Apply Chebyshev's inequality: Loading...
- As Loading...: Loading...
2) Strong law (via the Borel–Cantelli lemma)
- Condition: Loading...
- By the Borel–Cantelli lemma: Loading...
- Therefore: Loading...
Mathematical formula
The weak law expresses "convergence in probability"; the strong law expresses "almost sure convergence". X̄ₙ is the mean of n observations, μ is the population mean, and ε is any positive number.
Learn through examples
Example 1: A Casino's Revenue Model
Computing a casino's expected return on roulette
A simple even-money bet on European roulette (37 pockets):
• Player's probability of winning: Loading...
• Casino's win rate: Loading..., or about 51.4%
• After 100 spins, the actual win rate fluctuates between roughly 49% and 54%
• After 10,000 spins, it converges to the 50.8%–52.0% range
Answer:
By the law of large numbers, the more spins are played, the closer the casino's actual win rate gets to its theoretical expected value of 51.4%, which guarantees its long-run profit.
Any single spin is uncertain, but over a large volume of play the casino has a stable and predictable revenue model.
Example 2: Quality Control in Manufacturing
Managing defect rates in a factory
A smartphone production line with a theoretical defect rate of 2%:
• First 100 units: observed defect rate fluctuates between 0% and 5%
• 1,000 units: observed defect rate in the 1.5%–2.5% range
By the law of large numbers, the observed defect rate converges to the designed defect rate as production volume increases, so quality standards and cost forecasts become accurate.
Quality fluctuates widely in small production runs, but in mass production quality becomes stable and predictable.
Example 3: Accuracy of Opinion Polls
Why sample size matters in election forecasting
Polling a candidate whose true support is 45%:
• Sample of 100: results fluctuate between 35% and 55%
• Sample of 1,000: results converge to the 42%–48% range
• Sample of 10,000: accuracy improves to the 44%–46% range
• A margin of error of ±3% requires a sample of at least 1,067
Answer:
By the law of large numbers, a larger sample brings the sample proportion closer to the true population proportion, so a reliable poll needs a sufficiently large sample.
The reliability of the poll results you see in the news depends directly on the sample size.
Common mistakes
Mistake:
Applying the law of large numbers too eagerly to small samples
Why is it wrong?
The law of large numbers is an asymptotic property that holds when the sample is sufficiently large; it cannot be applied to small samples.
Correct approach:
Apply it only when the sample is large enough, and accept that outcomes in small samples are genuinely variable.
Mistake:
The gambler's fallacy — believing that earlier outcomes influence the next one
Why is it wrong?
If a coin lands heads 10 times in a row, the probability of tails on the next flip does not go up.
Correct approach:
Remember that each trial is independent, and that convergence to the theoretical probability happens only in the long run.
Mistake:
Confusing the law of large numbers with the central limit theorem
Why is it wrong?
The law of large numbers is about the convergence of the mean; the central limit theorem is about the shape of the distribution (normal distribution).
Correct approach:
Keep them apart: the law of large numbers answers "what does the mean converge to?", while the central limit theorem answers "how does it converge?"
The law of large numbers comes in two forms. The weak law of large numbers (WLLN) deals with convergence in probability: as the sample size grows, the sample mean converges in probability to the population mean. The strong law of large numbers (SLLN) deals with almost sure convergence: under stronger conditions, the sample mean converges to the population mean almost surely.
Formally, the weak law states Loading..., and the strong law states Loading.... These results are the theoretical foundation of Monte Carlo simulation, actuarial science, financial risk management, quality control, and many other fields.
Real-world applications
Finance
Pricing derivatives with Monte Carlo simulation
Example:
When the price of a complex option is computed from millions of simulated paths, the law of large numbers makes the estimate converge to the theoretical price.
Why it matters:
It makes accurate pricing possible for complex financial products that have no closed-form solution.
Insurance
Pooling risk across a large number of policies and setting premiums
Example:
Individual deaths cannot be predicted, but the overall mortality rate across hundreds of thousands of policies can be predicted with statistical stability.
Why it matters:
It lets insurers avoid the pitfalls of small numbers and run a stable business.
Manufacturing
Quality control and defect-rate forecasting
Example:
Whether any single unit is defective is random, but in mass production the defect rate converges to a steady level.
Why it matters:
Quality standards and cost forecasts become accurate.
Online Services
Analyzing A/B test results
Example:
A test on a handful of users is unreliable, but with data from tens of thousands of users the true effect can be measured accurately.
Why it matters:
It enables reliable, data-driven decision-making.
Case study: Risk Management at an Insurance Company
Background
Company A, a large life insurer in South Korea, manages 1 million life insurance policies a year. The mortality of any individual policyholder is hard to predict, yet the company needs to understand overall mortality patterns to set premiums and manage reserves on a stable footing.
The problem
Whether an individual policyholder lives or dies is a random event that is difficult to predict. With only a small number of policies, the actual mortality rate can differ greatly from its expected value, exposing the insurer to large losses. How can the insurer run its business on a stable basis?
The data
The company holds life insurance policy data on 1 million men aged 30–40. According to the national statistical office's life tables, the annual mortality rate for this age group is 0.1% (1 in 1,000). The average benefit per policy is KRW 100 million, and premiums are set according to the level of risk.
Method
Apply the law of large numbers to verify that the actual mortality rate converges to its expected value as the number of policies grows. Run simulations at scales of 1,000, 10,000, 100,000, and 1,000,000 policies and analyze the variability of the mortality rate at each scale.
Solution and results
1,000 policies: actual mortality rate swings widely between 0.05% and 0.2%
10,000 policies: variability shrinks to the 0.08%–0.12% range
100,000 policies: stabilizes in the 0.095%–0.105% range
1,000,000 policies: nearly constant in the 0.099%–0.101% range
As the number of policies grew, the actual mortality rate converged to its expected value of 0.1% and its variability fell sharply. At the 1 million-policy scale, the standard deviation of the annual mortality rate was below 0.001%, an extremely stable pattern.
Conclusion:
Thanks to the law of large numbers, an insurer can spread risk across a large number of policies. This is the core principle of the insurance business, and it shows that events that are unpredictable individually can be managed reliably in aggregate.
References
Bernoulli, J. (1713), Ars Conjectandi, Basel: Thurneysen Brothers — the classic of probability theory containing the first proof of a law of large numbers for binomial trials
Kolmogorov, A. N. (1930), "Sur la loi forte des grands nombres", Comptes Rendus de l'Académie des Sciences (Paris) 191, 910–912 — the original paper establishing the conditions for the strong law of large numbers (SLLN)
Etemadi, N. (1981), "An Elementary Proof of the Strong Law of Large Numbers", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 55(1), 119–122 — an elementary proof of the strong law assuming only pairwise independence
Metropolis, N. & Ulam, S. (1949), "The Monte Carlo Method", Journal of the American Statistical Association 44(247), 335–341 — the original paper on the Monte Carlo method, which applies the law of large numbers to numerical computation
