The idea, in plain English
The law of large numbers says that as you repeat a random process more and more times, the proportion of each outcome gets closer and closer to its true probability. For a fair coin: keep flipping, and the share of heads will home in on 50%.
Two things this does not say:
- It does not say each flip is more or less likely to be heads or tails depending on what came before. Each flip is still 50/50, every time.
- It does not say the absolute number of heads will catch up with the absolute number of tails. After a million flips you might be 1,000 heads "ahead" — and that's perfectly normal. The proportion is what converges.
Why short sequences look wild
If you only flip a few times, expect lopsided results. Variance — the spread of plausible outcomes around the average — is wide when you have few flips, and narrows as the number of flips grows. A useful rough guide:
| Number of flips | Plausible heads-% range (≈ 95% of trials) | What "lopsided" looks like |
|---|---|---|
| 10 | ~ 20% – 80% | 8 of one side |
| 50 | ~ 36% – 64% | 32 of one side |
| 100 | ~ 40% – 60% | 60 of one side |
| 1,000 | ~ 47% – 53% | 530 of one side |
| 10,000 | ~ 49% – 51% | 5,100 of one side |
The ranges are approximate (derived from the standard deviation of a binomial distribution with p = 0.5), but the pattern is the point: variance shrinks as the square root of the number of flips. Quadrupling your flips only halves the spread.
Practical takeaway: if you flip the simulator ten times and get 70% heads, that's well inside normal. To detect a real bias of, say, 1%, you'd need on the order of tens of thousands of flips — not dozens.
Worked example using the on-site stats
The four counters under the coin track total flips, heads, tails, and heads-percentage. Try this:
- Reset the counters.
- Flip 20 times and write down the heads-percentage.
- Flip another 80 times (total 100). Write down the new heads-percentage.
- Flip another 400 (total 500). Write down again.
You'll usually see something like 65% → 54% → 50.6%. Each step has a smaller swing than the last. That trajectory is the law of large numbers — not because the coin is "correcting", but because the few flips at the start matter less and less to the running average as you add more flips on top.
Streaks: how long is too long?
People are bad at intuiting how long a "normal" streak is. A useful rule of thumb: in n independent fair flips, expect the longest run of one outcome to be roughly log₂(n) in length. So:
- In 64 flips, expect a longest run of about 6.
- In 1,024 flips, expect a longest run of about 10.
- In a million flips, expect a longest run of about 20.
If you see a run of seven heads in 100 flips, you have not witnessed an anomaly. You have witnessed a normal Tuesday. We cover the related "the next flip is due" misconception in the probability of a coin flip.
Common mistakes
- Quitting early and declaring the coin biased. Twenty flips is not enough data to detect anything but a wildly broken coin.
- Confusing "approaches 50%" with "equals 50%". The law promises convergence, not equality. Heads counts and tails counts can drift apart in absolute terms while the percentage drifts toward 50.
- Adjusting your "expected" outcome based on streaks. The next flip after a long streak is still 50/50.
- Reading the heads-percentage too often. Watching a counter update each flip exaggerates the feeling of "imbalance". The actual variance is much smaller than visual intuition suggests.
Where this fails: weighted coins
Everything above assumes a fair coin. If a coin is biased — say, it lands heads 55% of the time — the law of large numbers still works, but it converges to the true probability (55%), not to 50%. So if your simulator counter sat at, say, 56% heads after 50,000 flips, the most likely explanation isn't that the law of large numbers has failed; it's that the coin isn't fair. The Coin Toss Simulator uses an unbiased mapping from a uniform random source, so this scenario shouldn't occur — but the same logic applies to any random process you analyse.
Why this matters beyond coin flipping
Coin flipping is a teaching example because it's binary and the maths is clean. The same principle governs casino games, polling, A/B testing, and almost every other situation where you want to estimate a probability from a sample. The lesson is the same: small samples are noisy, big samples are precise, and the noise shrinks slower than you'd guess.