Law of Large Numbers (Intuition) Examples in Math

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

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

Concept Recap

As sample size increases, the sample average approaches the true population average.

As the number of trials grows, the sample mean converges to the true expected value — randomness averages out over many trials, making the average predictable.

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: Randomness averages out over large samples—individual weirdness cancels.

Common stuck point: Doesn't mean outcomes 'balance out'—past results don't affect future trials.

Sense of Study hint: Try simulating: flip a coin 10 times, record the percent heads. Now flip 100 times. Notice how the percentage gets closer to 50%.

Worked Examples

Example 1

easy
A fair coin is flipped. Show how the proportion of heads approaches 0.5 as n increases, using simulation results: n=10: 6 heads; n=100: 53 heads; n=1000: 498 heads.

Solution

  1. 1
    n=10: proportion = 6/10 = 0.60 (60% — far from 0.5)
  2. 2
    n=100: proportion = 53/100 = 0.53 (53% — closer to 0.5)
  3. 3
    n=1000: proportion = 498/1000 = 0.498 (49.8% — very close to 0.5)
  4. 4
    Pattern: as n increases, \bar{X} \to 0.5 = \mu, illustrating the Law of Large Numbers

Answer

Proportions: 0.60, 0.53, 0.498 — converging to 0.5 as n grows.
The Law of Large Numbers states that as sample size n \to \infty, the sample mean \bar{X} converges to the true population mean \mu. Small samples can show large deviations from the truth; large samples reliably estimate it.

Example 2

medium
A casino game has expected value -\0.05$ per play (house edge). A player plays 10 games vs. 10,000 games. Explain how the LLN affects the likely outcome in each case.

Practice Problems

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

Example 1

easy
A die is rolled repeatedly and the running average is tracked. After 5 rolls, the average is 3.8. After 500 rolls, the average is 3.52. What does the LLN predict will happen to the average as rolls → ∞?

Example 2

hard
A student argues: 'I've flipped heads 10 times in a row, so tails is overdue.' Using the LLN correctly, explain why this is wrong, and what LLN actually predicts.
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Background Knowledge

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

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