Law of Large Numbers
In short: Law of large numbers says that as a chance process is repeated more and more times, the observed average or proportion gets closer and closer to the true theoretical value.
AP StatisticsProbabilityAP Statistics Unit 2
What is Law of Large Numbers?
The law of large numbers explains why casinos and insurers can predict totals even when single outcomes are unpredictable. Over a handful of trials, results wander widely. Over thousands, the sample proportion or mean hovers near the true value.
For a fair die, the proportion of sixes after 6 rolls might be 0 or 0.5, but after 60,000 rolls it will be very close to \(\frac{1}{6}\). Variation in the sample proportion shrinks like \(\sqrt{\frac{p(1-p)}{n}}\). The law describes the long run; it never says that a streak must be balanced out by the next trial.
How to spot it
- A problem asks what happens to a proportion or an average as the number of trials grows.
- A simulation of coin flips, dice or spins is run and results are compared with the theory.
- An insurance or casino context says the average per customer is predictable for large numbers of customers.
- Someone argues that a streak means the opposite result is now “due”.
The method
- Identify the theoretical value, such as \(p = \frac{1}{6}\) or the mean \(\mu\) of a random variable.
- Compute the typical size of chance variation with \(\sqrt{\frac{p(1-p)}{n}}\) for several values of \(n\).
- Observe that the variation shrinks as \(n\) grows, so results cluster more tightly near the true value.
- State the conclusion for the long run only, and reject any claim about the next single trial.
Law of Large Numbers: solved example
Problem
A fair die has \(p = \frac{1}{6}\) of showing a six. How large is the typical chance variation in the proportion of sixes for \(n = 60\) rolls and for \(n = 6000\) rolls?
Solution
Use \(\sqrt{\frac{p(1-p)}{n}}\) with \(p(1-p) = \frac{1}{6}\cdot\frac{5}{6} = \frac{5}{36} \approx 0.1389\).
For \(n = 60\): \(\sqrt{\frac{0.1389}{60}} = \sqrt{0.002315} \approx 0.048\). For \(n = 6000\): \(\sqrt{\frac{0.1389}{6000}} \approx 0.0048\).
Multiplying the number of rolls by 100 shrinks the variation by a factor of 10, which is why the proportion tightens around 0.167.
Try it
Question 1
A coin lands heads five times in a row. Is tails now more likely on the next flip?
Show the answer
No. Flips are independent, so the chance of tails is still 0.5. The law of large numbers concerns long-run proportions, not making up for past results.
Question 2
A free-throw shooter makes 40% of attempts. Find the typical variation in her proportion made after \(n = 2500\) attempts.
Show the answer
\(\sqrt{\frac{0.4 \times 0.6}{2500}} = \sqrt{0.000096} \approx 0.0098\), under 1 percentage point.
Law of Large Numbers vs. Central Limit Theorem
The law of large numbers says the sample mean or proportion approaches the true value as the sample grows. The central limit theorem goes further and describes the shape of the distribution of sample means: approximately normal. One speaks about where the average ends up, the other about how it is distributed on the way.
Keep going
Lessons
Worksheets and quizzes
- Probability Rules and Simulation – Worksheet
- Probability Rules and Simulation – Quiz
- Random Variables, Expected Value and Combining Variables – Worksheet
- Random Variables, Expected Value and Combining Variables – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Probability