Law of Large Numbers (Intuition) Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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

Solution

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

Answer

Proportions: 0.60, 0.53, 0.498 โ€” converging to 0.5 as nn grows.
The Law of Large Numbers states that as sample size nโ†’โˆžn \to \infty, the sample mean Xห‰\bar{X} converges to the true population mean ฮผ\mu. Small samples can show large deviations from the truth; large samples reliably estimate it.

About Law of Large Numbers (Intuition)

The law of large numbers states that as the number of independent trials increases, the sample mean converges to the true population mean โ€” randomness averages out over many repetitions.

Learn more about Law of Large Numbers (Intuition) โ†’

More Law of Large Numbers (Intuition) Examples

Example 2 medium

A casino game has expected value [formula]0.05$ per play (house edge). A player plays 10 games vs. 1

Example 3 easy

A die is rolled repeatedly and the running average is tracked. After 5 rolls, the average is 3.8. Af

Example 4 hard

A student argues: 'I've flipped heads 10 times in a row, so tails is overdue.' Using the LLN correct

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