Instructor Guide: Law of Large Numbers

Companion notes for the StatLens conceptual demo — why the long-run proportion settles, and why the running total does not

Launch in Presentation Mode

Overview ~10 min

Two experiments behind one toggle: tossing a coin and watching the proportion of heads, or rolling a die and watching the average roll. Trials are added in batches of 1, 10, 100 or 1000, and the running value is traced against the number of trials with a dashed line at what the model expects (0.5, or 3.5).

The page carries one option most LLN demos leave out: Also show the running total, which draws heads minus tails from the same tosses. That second trace is the one that refutes the “law of averages”, and it costs nothing to show because it is the same data.

The statement, said carefully

The law of large numbers says the sample proportion (or sample mean) converges to the probability (or expected value) as n grows. It says nothing whatever about the counts. Both facts are true at once and students routinely merge them into a third, false one.

The misconception: “a coin is due”. After a run of tails, students expect extra heads to arrive and restore the balance. Nothing arrives. The proportion recovers because the denominator grows — the early tails are outvoted, never cancelled. Tick Also show the running total: the proportion settles toward 0.5 while heads − tails typically drifts further from zero. Same coin, same tosses, both lines on screen at once.

Running it ~6 min

  1. Ask before you press. “How many tosses before the proportion stops moving much?” Collect a few guesses. They are usually far too small.
  2. Press +1 about ten times. The line is wild — 1.0, then 0.5, then 0.67. This is the part students never see in a static textbook figure.
  3. Press +10, then +100. Note out loud that the line is not being pulled toward 0.5; it is being weighed down. One toss out of 20 can move the proportion by 0.05. One toss out of 2000 can move it by 0.0005.
  4. Press +1000. Read the distance-from-0.5 figure. Ask what would happen to it at 100,000 tosses — and what would happen to heads − tails.
  5. Switch to Die rolls. Same shape of picture for an average rather than a proportion. Worth saying explicitly: the law is about both, and 3.5 is not a face the die can show.
Discussion question. The proportion of heads is 0.40 after 10 tosses. Over the next 1000 tosses, does the coin produce extra heads to make up for it? (The page asks this too, with the answer hidden behind a button in discovery mode.) Answer: no — the next 1000 give about 500 heads, which leaves roughly 504 of 1010, a proportion near 0.499. The four missing heads are still missing.

The two options

Spread out the early trials

Puts the trial count on a log scale, so trials 1–10 get as much width as 100–1000. On an ordinary axis the wild early stretch — the part where the law is visibly working — is squeezed into a sliver at the left edge. Turn it on after students have watched the early tosses arrive one at a time, so the axis change explains itself.

Also show the running total

Draws heads − tails (for dice: the total minus 3.5 per roll) against the same trial count. Expect it to wander, cross zero occasionally, and end somewhere nowhere near zero. Over many runs its typical size grows like √n, which is exactly why the proportion — that quantity divided by n — shrinks toward zero.

If you want the same run twice (lecture, then the same picture in a handout), add ?seed=anything to the URL. The generator is seeded, so the identical sequence of tosses comes back. ?exp=die opens on dice, and ?n=1000 opens with the trials already run.

Common questions

“Is 1000 tosses enough?”

For seeing the settling, yes. For a guarantee, no — the law is a statement about the limit, and at any finite n the proportion can be off by a bit. Press +1000 a few times and watch the distance figure shrink unevenly: it does not decrease every time, it decreases on the whole.

“Why does the line start at 1.0 or 0.0?”

After one toss the proportion of heads is either 1 or 0 — there is nothing else it can be. That is a real feature of the quantity, not a glitch, and it is worth a sentence: an estimate from n = 1 is as extreme as estimates get.

“Does this prove the coin is fair?”

No, and this is a useful place to point forward. The law says if the probability is 0.5 then the proportion settles near 0.5. Asking whether an observed proportion is consistent with 0.5 is the inference question, and that is what a randomization test does.

Where it fits

Natural home is the probability chapter, before sampling distributions. It pairs well with Sampling Distributions (which shows what happens to the spread of a statistic as n grows) and with Stump the Chump, where students discover that real randomness is clumpier than they expect — the same intuition failure that makes a coin feel “due”.

StatLens · Law of Large Numbers · conceptual/law-of-large-numbers/