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 might see something like 65% → 54% → 50.6%. The exact numbers will differ every time. 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.
Law of large numbers vs. the "law of averages"
These two get mixed up all the time, including in exam questions:
- Law of large numbers (real): the long-run proportion of heads approaches the true probability. It says nothing about any individual flip and applies only in the long run.
- "Law of averages" (folk belief): after a run of heads, tails becomes more likely so that things "even out" soon. That is false for independent events, and it's the gambler's fallacy.
A quick test: if a statement predicts the next flip, it isn't the law of large numbers. The law can't be used to predict the next outcome. The chance the next flip is heads stays 50% because each flip is independent, not because of the law.
The weak and strong laws, briefly
Mathematicians state two versions. If p̂n is the proportion of heads after n flips:
- Weak law (Jacob Bernoulli, published 1713): for any small margin, the chance that p̂n is further than that margin from 0.5 shrinks to zero as n grows. This is convergence in probability.
- Strong law (Émile Borel and later Kolmogorov): with probability 1, the sequence p̂n eventually settles at 0.5 and stays arbitrarily close to it. This is almost sure convergence.
For flipping coins in practice the difference doesn't matter. Both say the proportion converges and neither says the counts become equal. The typical gap between the heads percentage and 50% shrinks in proportion to 1/√n.
Proportion converges, the count difference doesn't
This is the part that trips people up. As you flip more, the heads percentage gets closer to 50%, but the typical gap between the heads count and the tails count actually grows, roughly as √n:
| Flips | Typical heads − tails gap | Typical distance from 50% |
|---|---|---|
| 100 | ~ 8 | ~ 4 points |
| 10,000 | ~ 80 | ~ 0.4 points |
| 1,000,000 | ~ 800 | ~ 0.04 points |
(“Typical” here is the average absolute gap, about 0.8 standard deviations.) Try it with the bulk flipper: a million flips will almost never be a perfect split, yet the percentage will read 50.0-something nearly every time.
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.
Frequently asked questions
What does the law of large numbers say about flipping a coin?
As you flip a fair coin more and more times, the proportion of heads gets closer to 0.5. It doesn't say each flip changes its odds, and it doesn't say the number of heads will ever equal the number of tails. It's a statement about the long run only.
According to the law of large numbers, what happens when you flip a fair coin a great number of times?
The proportion of heads approaches 0.5. It isn't always exactly 0.5, the per-flip probability doesn't drift, and the counts of heads and tails don't have to be equal. If you see this as a multiple-choice question, "the proportion of heads will approach 0.5" is the right answer.
Is the law of large numbers the same as the law of averages?
No. The "law of averages" usually means the belief that outcomes even out in the short run, so a tail is "due" after a run of heads. That's the gambler's fallacy. The law of large numbers makes no prediction about the next flip; it only describes the long-run proportion.
Why does the percentage even out if the coin has no memory?
Because early results get diluted, not corrected. If you start with 7 heads in 10 flips and then flip 990 more at an even 50/50, you'll have about 502 heads in 1,000, or 50.2%. The early surplus of 2 extra heads hasn't gone away; it just matters less and less as the total grows.
After 100 coin flips, the proportion of heads gets closer to 0.5. What does this represent?
That's the law of large numbers: the sample proportion (an estimate) converging toward the true probability as the sample size grows. In statistics this is called convergence in probability, and it's the weak law of large numbers.
Is flipping one coin many times the same as flipping many coins once?
Yes, for fair, independent coins. A thousand flips of one coin and one flip each of a thousand coins have the same distribution of heads. You can test it with the bulk flipper on the simulator.
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.