Law of Large Numbers (Intuition) Math Example 4
Follow the full solution, then compare it with the other examples linked below.
Example 4
hardA 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.
Solution
- 1 The student confuses LLN with the Gambler's Fallacy — LLN does NOT say tails is 'due'
- 2 LLN says: over many flips, the proportion approaches 0.5 — achieved by dilution, not compensation
- 3 Dilution example: after 10 heads (10H, 0T), if we then flip 990 more approximately fairly (≈495H, 495T), total = 505H/495T out of 1000 → proportion = 0.505 (close to 0.5 without any 'catching up')
- 4 The past excess is swamped by future fair results — not corrected by future extra tails
Answer
LLN works by dilution not compensation. The 10-heads streak is swamped by future fair flips, not corrected by extra tails.
This is the most important subtlety of LLN. Convergence to 0.5 happens because past deviations become negligible as grows — not because future outcomes compensate for past ones. Each flip is still independent with .
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 — randomness averages out over many repetitions.
Learn more about Law of Large Numbers (Intuition) →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 mediumA casino game has expected value [formula]0.05$ per play (house edge). A player plays 10 games vs. 1
Example 3 easyA die is rolled repeatedly and the running average is tracked. After 5 rolls, the average is 3.8. Af