Law of Large Numbers (Intuition) Math Example 3

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

Example 3

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 โ†’ โˆž?

Solution

  1. 1
    True population mean for a fair die: ฮผ=(1+2+3+4+5+6)/6=3.5\mu = (1+2+3+4+5+6)/6 = 3.5
  2. 2
    LLN states: as nโ†’โˆžn \to \infty, Xห‰โ†’ฮผ=3.5\bar{X} \to \mu = 3.5
  3. 3
    After 5 rolls: 3.8 (far from 3.5 โ€” small sample noise); after 500 rolls: 3.52 (very close)

Answer

As nโ†’โˆžn \to \infty, the running average Xห‰โ†’ฮผ=3.5\bar{X} \to \mu = 3.5.
The LLN guarantees convergence of sample means to the population mean. Deviations from 3.5 become smaller and smaller as more rolls are added. This principle is why large samples are more reliable than small ones.

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 1 easy

A fair coin is flipped. Show how the proportion of heads approaches 0.5 as [formula] increases, usin

Example 2 medium

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

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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