What Is Randomness?

Opening Hook

Write down twenty coin flips. Do it now, before reading further. A sequence of H and T, twenty characters long, completely off the top of your head.

Look at what you wrote. The chances are good that your sequence looks something like this: HTHTHHTHTTHHTTHTTHTH. Alternating fairly regularly, with the occasional short run of two. Tidy. Balanced. Intuitively random.

Now look at a genuine random sequence generated by a computer: HHHHTTHHHTTTHHTTTHHT.

Notice the runs. Four heads in a row at the start. Three tails in a row in the middle. Three more tails before the end. That feels wrong, doesn’t it? It feels like something is off, like the coin is biased or the sequence is fixed. But this is what randomness actually looks like. It clusters. It runs. It does not take turns.

This is the core thing to understand about randomness, and almost nobody grasps it intuitively: random processes do not spread themselves out. They pile up.

The Concept

The problem starts with what you expect randomness to look like. When most people imagine a random sequence, they imagine something even and alternating. They think of randomness as a kind of fairness, as if the universe keeps score and compensates after a long run by switching outcomes. It does not. A fair coin that has just produced five heads in a row has exactly a 50 percent chance of heads on the next flip. The coin has no memory. Each flip is independent of every flip before it.

Independent events are events where the outcome of one has no effect on the probability of another. A fair coin flip is independent of all previous coin flips. A lottery ball drawn (with replacement) is independent of all previous draws. This is not intuitive, but it is true, and a great many statistical errors come from forgetting it.

Because of independence, runs and clusters are not anomalies in random data. They are expected features of it. If you flip a fair coin 100 times, the probability of getting at least one run of six or more heads in a row is over 80 percent. Most people, if shown a sequence with a run of six heads, would suspect the coin was rigged. The coin is doing exactly what a fair coin does.

The technical name for the error is the clustering illusion: the tendency to perceive meaningful patterns in genuinely random data. When you see four heads in a row, your pattern-seeking mind invents a cause. The coin must be weighted. Something must be happening. That instinct is one of the most powerful and one of the most misleading tendencies in human cognition.

The flip side of the clustering illusion is what happens when humans try to generate random sequences themselves. This has been studied extensively. Ask people to write down a random-looking string of coin flips, and they produce sequences with too many alternations and not enough runs. On average, humans alternate about 60 percent of the time when they intend to be random, against the 50 percent an actual coin produces. They over-correct. They avoid placing the same outcome twice in a row because that does not feel random, even though it is just as likely as alternating.

This means a human-generated “random” sequence is actually detectable. It looks too tidy. If you ever need to tell whether a sequence was genuinely random or fabricated by a person pretending to be random, the giveaway is a lack of sufficiently long runs. Real randomness has them. Humans avoid them.

Now here is where the law of large numbers enters, and where people get it backwards.

The law of large numbers says that as you repeat a random process many times, the observed frequency of an outcome converges toward its true probability. If you flip a coin a million times, you will get very close to 50 percent heads. This is a mathematical law, not a tendency or a guideline.

The mistake people make is applying the law of large numbers to small sequences. They reason: “The coin has been landing heads a lot, so it must be due for a tail to balance things out.” This is wrong. The law of large numbers operates across large numbers of trials. It says nothing about the next trial. Randomness does not balance itself out in the short run. It only averages out in the very long run, and only because there are so many trials that early imbalances become statistically insignificant, not because the process corrects itself.

There is no balancing force. There is no memory. There is only each independent event and its probability.

Why It Matters

Once you understand that random processes naturally cluster, you can see how a particular kind of error propagates through public life. The error is this: a cluster appears, someone points to it, and the search begins for a cause. The cluster is real. The cause may not be.

Cancer cluster investigations are the clearest example. When a community notices an elevated rate of a disease, it is natural to ask why. The rate is real. The question is whether it signals something in the environment or whether it is, in full, an expected feature of how random variation works across many communities simultaneously.

Imagine a map of the country divided into thousands of small areas. In each area, a handful of cancer cases occur each year. Even if the underlying risk is identical everywhere and nothing in any environment causes cancer, the cases will not be evenly distributed. Some areas will, by chance alone, have clusters. If you surveyed every area in the country and reported the ones with elevated rates, you would find many. Some would look alarming. Most would be noise.

The hot hand in sport is a closely related case. A basketball player hits four shots in a row. The commentator calls it a hot hand. The coach draws up plays to keep getting that player the ball. Fans feel the momentum. The player is, in the language of sports, unconscious.

Thomas Gilovich, Robert Vallone, and Amos Tversky examined this belief systematically in 1985, analysing the shooting records of the Philadelphia 76ers and the free-throw records of the Boston Celtics. They found no evidence of positive correlation between successive shots. Whether a player had just hit or just missed, their probability of hitting the next shot was not meaningfully different. The hot hand, in the data, was not there. Fans were perceiving streaks in what was, statistically, a sequence of independent events.

The broader stakes are significant. When randomness is misread as pattern, causes get invented. Policies get made. Money gets spent. Occasionally, real causes are missed because the investigation focuses on a spurious cluster rather than a structural one. The misreading of randomness is not a harmless cognitive quirk. It has consequences.

How to Spot It

In 1944, German V-1 flying bombs began falling on London. They were not accurately guided: V-1s were propelled by crude pulse-jet engines and pointed in a general direction, not aimed at specific streets. But as the attacks continued, people began to notice that certain areas seemed to be hit more than others. Clusters were visible on the map. Was the guidance better than the Allies thought? Were some areas being deliberately targeted?

R.D. Clarke, a young actuary working for the Prudential Assurance Company and contributing to wartime analysis, approached this question statistically. He divided a section of south London into 576 equal squares of approximately half a kilometre each, and recorded how many of the 537 V-1 impacts that fell in that area landed in each square. Then he compared the actual distribution of hits per square against what would be expected if the bombs were falling entirely at random, using a mathematical tool called the Poisson distribution.

The Poisson distribution describes how independent random events are distributed across a fixed number of slots when the overall rate is known. It tells you how many squares you would expect to get zero hits, one hit, two hits, three hits, and so on, if the bombs were landing without any guidance at all.

Clarke’s result, published in the Journal of the Institute of Actuaries in 1946, was clear: the observed distribution matched the Poisson expectation almost exactly. The clustering that Londoners had perceived, the areas that seemed cursed, the streets that appeared to be targeted, was precisely the clustering that you would expect from random impacts. There was no guidance. There was no pattern. There was only randomness, which clusters on its own.

The tell in this case was the fit between the actual data and the theoretical random distribution. When that fit is good, the pattern you are seeing is the pattern of noise. Before attributing a cluster to a cause, the question to ask is: how often would a cluster this concentrated appear by chance alone? If the answer is “fairly often, across many areas,” the cluster may require no further explanation.

Your Challenge

A public health officer is investigating a small coastal town where eight people have been diagnosed with the same rare kidney condition over the past five years. The national rate for this condition is roughly one case per 10,000 people per year. The town has a population of 4,000.

At the national rate, you would expect roughly two cases in this town over five years. Eight cases were observed.

The local newspaper runs a story suggesting industrial pollution from a facility on the edge of town may be responsible. Residents demand an investigation. The facility owners commission their own study, which finds no link.

Before deciding what to think about the pollution hypothesis, what statistical question would you ask first? What would you need to calculate? And what would the answer to that calculation tell you about whether this cluster is evidence of anything at all?

There is no answer on this page. That is the point.

References

Gilovich, T., Vallone, R., and Tversky, A. (1985). “The hot hand in basketball: On the misperception of random sequences.” Cognitive Psychology, 17(3), 295–314. Available via: ScienceDirect

Miller, J.B. and Sanjurjo, A. (2018). “Surprised by the Hot Hand Fallacy? A Truth in the Law of Small Numbers.” Econometrica, 86(6), 2019–2047. The reanalysis finding that the original Gilovich et al. result contained a subtle statistical bias; the debate is discussed in: Nautilus, “The Hot Hand Is Not a Myth”

Clarke, R.D. (1946). “An application of the Poisson distribution.” Journal of the Institute of Actuaries, 72, 481. Original paper available at: Berkeley Garcia Lab. Modern treatment in Shaw, M. and Shaw, A., “The flying bomb and the actuary,” Significance (2019): Wiley Online Library

On human generation of random sequences and the alternation bias: Nickerson, R.S. (2002). “The production and perception of randomness.” Psychological Review, 109(2), 330–357. Summary discussion: PMC / NIH — Re-Examination of Bias in Human Randomness Perception

On cancer clusters and the interpretation of random variation: Condon, S., quoted in CBS News, “Cancer clusters: The hunt for a killer.” CBS News. Background on the Woburn, Massachusetts case: SERC, Carleton College — Cancer Clusters: Fact or Fiction?