Law of Large Numbers Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Law of Large Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

As the number of trials increases, the experimental probability (sample average) converges to the theoretical probability (population mean).

Flip a coin 10 times: maybe 7 heads (70%). Flip 100 times: closer to 50%. Flip 10,000 times: very close to 50%. More trials = more reliable averages. Short-run luck evens out.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: As the number of independent trials grows, the sample average converges to the theoretical expected value. Short-run results are unpredictable; long-run averages are stable.

Common stuck point: Students commit the gambler's fallacy — thinking that after several tails, heads is 'due.' Each flip is independent; past outcomes do not change future probabilities.

Worked Examples

Example 1

easy
A coin is flipped and the running proportion of heads is recorded: After 10 flips: 0.70, after 50: 0.56, after 200: 0.52, after 1000: 0.498, after 10,000: 0.5012. What does this pattern illustrate?

Solution

  1. 1
    Step 1: The proportion of heads starts far from 0.5 (at 0.70 with just 10 flips).
  2. 2
    Step 2: As the number of flips increases, the proportion gets closer and closer to 0.5: 0.70 → 0.56 → 0.52 → 0.498 → 0.5012.
  3. 3
    Step 3: This illustrates the law of large numbers: as the number of trials increases, the experimental probability (relative frequency) converges toward the theoretical probability.

Answer

This illustrates the law of large numbers — the proportion of heads converges toward the theoretical probability of 0.5 as the number of flips increases.
The law of large numbers states that the average of results from a large number of independent trials gets closer to the expected value as more trials are performed. This is why experimental probability becomes more reliable with more data.

Example 2

medium
A casino offers a game where you win \1 with probability 0.48 and lose \1 with probability 0.52. (a) What is the expected value per game? (b) Explain why the casino is guaranteed to profit in the long run using the law of large numbers.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A basketball player has a career free-throw percentage of 80%. She misses 5 free throws in a row in one game. A commentator says she is 'due' to make the next one. Is this reasoning correct? Explain using the law of large numbers.

Example 2

hard
An insurance company insures 100,000 homeowners. The probability of a claim in a given year is 0.02, with an average claim of \50,000. They charge \1,200 per policy. (a) What is the expected number of claims? (b) What is the expected total payout? (c) What is the expected profit? (d) Explain why this business model works using the law of large numbers.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probability basicmean