Heads or Tails: the real probabilities across 10,000 tosses

In April 1940, the German army invaded Denmark. A South-African mathematician, John Edmund Kerrich, was visiting his in-laws in Copenhagen — he was arrested as a British subject and interned until the end of the war in the Hald camp, in Jutland. To pass the time, Kerrich decided to run an experiment none of his university colleagues had ever had the patience to attempt: he flipped a coin. Again. And again. Ten thousand times exactly, by hand, recording every result. When he tallied the books, he had 5,067 heads out of 10,000. Not 5,000 flat, no wild outlier — just 5,067. That figure, published in his book An Experimental Introduction to the Theory of Probability in 1946, remains one of the most cited experimental verifications of the law of large numbers.

This article extends Kerrich’s experiment with today’s tools. What really happens when you flip a coin 10,000 times? The spontaneous mental image — heads-tails-heads-tails in tidy alternation, around a clean 50/50 — is almost always wrong. The reality is more instructive: the probability of any single toss stays at 50%, but what you see on screen is organised by rules our intuition does not expect. Here are those rules, in numbers.

What intuition predicts, and what actually happens

Ask anyone how many heads they expect across 10,000 tosses of a fair coin. The spontaneous answer is almost always: ‘about 5,000’. The ‘about’ is the interesting part.

For a sequence of independent two-outcome trials — what is called a binomial distribution — the uncertainty around the mean is measured by a simple formula: the standard deviation, equal to the square root of the number of tosses multiplied by each face’s probability and its complement. For 10,000 tosses at 50/50, the calculation gives √(10,000 × 0.5 × 0.5) = 50 tosses.

Concretely: roughly two times in three, your final tally will land between 4,950 and 5,050 heads. Around nineteen times in twenty, it will land between 4,900 and 5,100. And the probability of landing exactly on 5,000 heads is about 0.8% — statistically one of the least frequent outcomes, even though it is the one everyone expects to get. Kerrich, with his 5,067 heads, sits 1.34 standard deviations above the mean: a perfectly mundane result, among the most likely.

Intuition makes two mistakes at once: it forgets that hitting the mean exactly is almost impossible, and it underestimates the expected gap. Across 10,000 tosses, drifting by 50 or 80 heads is not a sign of a rigged coin — it is exactly what an honest coin produces.

Surprising streaks: why 13 heads in a row are nothing special

Here is a question rarely asked before playing, and the answer reshapes how we see randomness. Across 10,000 tosses, what is the longest unbroken run of the same face?

The theoretical answer is calculated with the log₂(N) formula, the base-2 logarithm of the number of trials. For 10,000 tosses, log₂(10,000) ≈ 13.29. In other words, you should expect to see, somewhere in the sequence, a run of about 13 to 14 consecutive heads — or 13 to 14 consecutive tails. Not as a stroke of luck or a remarkable event, but as a normal one.

This figure crashes head-on into intuition. Seven heads in a row already triggers in most observers a sense of anomaly, even of a fix. Yet across 10,000 tosses, seven consecutive heads are essentially guaranteed — you’ll get them several times. The fact that our brain reads them as suspicious comes from a well-identified mechanism that we unpack in our article on the gambler’s fallacy: we expect a short sequence to already look like ‘well-shuffled’ randomness, when authentic randomness naturally produces clusters.

If you want to see for yourself, run the sequence at scale on Heads or Tails and note the longest streak you get after a few hundred tosses. You’ll see runs of five, six, sometimes eight consecutive faces appear — without any rule being broken. That is precisely what we call the mechanics of randomness: not a smooth distribution, but an irregular one whose regularity only shows up at very large scale.

Probability, proportion, variance: the distinction that changes everything

If there is one thing to take away from this article, it is this. Three words look alike and refer to different things; confusing them is the root of most false intuitions about randomness.

Probability is what applies to one trial. When you click Heads or Tails once, the probability of getting heads is 50%. That probability never changes, depends on nothing, doesn’t accumulate. It stays at 50% on the first toss and on the ten-thousandth.

The observed proportion is what we measure after a certain number of trials. Across 10 tosses, you might get 7 heads; the proportion will be 70%. Across 1,000 tosses you’ll be closer to 50% — perhaps 51%. Across 100,000 tosses you’ll be very close to 50.0%. That convergence movement is what we call the law of large numbers.

Variance — or its cousin the standard deviation — measures the uncertainty around that proportion at a given moment. And its behaviour is the great surprise: variance does not shrink in proportion to the number of trials, it shrinks as the square root of the number of trials. Across 100 tosses, the standard deviation is 5 (a 5% relative gap). Across 10,000 tosses it climbs to 50 — but proportionally it falls to 0.5%. That explains the central paradox: the more you toss, the larger the absolute gap can grow, and yet the closer the result gets to 50% in proportion.

Grasping this triple distinction defuses, in one go, most of the illusions. ‘50% chance’ does not mean ‘exactly 50% of results’. Those two phrases speak about different things.

And what about a ‘real’ coin?

All the previous calculations rest on the assumption of a perfectly balanced coin — 50% heads, 50% tails. Is a real physical coin really that?

The answer, surprisingly, is: not quite. In 2007, the mathematician Persi Diaconis, professor at Stanford and incidentally a former professional magician, published with Susan Holmes and Richard Montgomery a paper titled Dynamical Bias in the Coin Toss in SIAM Review. Their demonstration combines physical modelling and slow-motion observation: a hand-tossed coin doesn’t spin perfectly on its axis; it undergoes a slight precession — a gyroscope-like effect that means the face initially pointing up has roughly 51% chance of landing in the same orientation. An empirical study released in 2023 across more than 350,000 hand tosses confirmed that bias with striking precision: 50.78% landings on the same side as the start.

For everyday life, that’s negligible. For a digital random-draw site, it’s an interesting point: a coin simulated by algorithm, which has no initial orientation and no physical precession, is in fact fairer than a real coin. Our article on how our draws work details the exact mechanism — Math.random(), uniform distribution, no memory between draws. The digital coin on Heads or Tails does not cheat, and it doesn’t precess either.

Check it yourself, in 30 seconds

The advantage of mathematical randomness is that it’s entirely reproducible. You can replicate Kerrich at home, without 75 years of internment.

Method 1 — With Heads or Tails. Toss the coin on Heads or Tails about a hundred times and note the results. You should land around 50/50, with a gap that can reach 10 or 12 heads one way. Note the longest streak observed too: for 100 tosses, expect 6 or 7 consecutive flips of the same face.

Method 2 — In the browser console. On any page, open the console (F12 key, ‘Console’ tab) and type: let p = 0; for (let i = 0; i < 10000; i++) if (Math.random() < 0.5) p++; p. In a fraction of a second you’ll get your number of heads across 10,000 tosses — a complete Kerrich in less than a blink. Run the command several times: you’ll get a different number each time, almost always between 4,900 and 5,100.

That’s all the law of large numbers says: not a promise of 5,000 exactly, but a stable bracket around 50%, with a predictable variance. Randomness, in this pure form, is neither mysterious nor hostile — it follows rules, simply different from the ones our intuition projects onto it.

To dig deeper, you can revisit our article on the gambler’s fallacy — which explains why seven heads in a row feel suspicious when they aren’t — or lift the hood on our draws to see, line by line, the code that produces these probabilities.