Chance, data and large numbers

Two round diagrams cut into slices of very unequal sizes, in blue, red and black, with explanatory text. — Florence Nightingale’s “Diagram of the causes of mortality in the army in the East”, sent to Queen Victoria in 1858: each slice is a month; in blue, deaths from preventable diseases, in red those from wounds, in black other causes. Florence Nightingale, Public domain, source

When you throw a die, there is no knowing what will come up. Over a thousand throws, the outcome can be predicted with astonishing precision.

This whole page lies in that gap.

Repeated chance becomes regular

A single event is unpredictable. An event repeated a great many times becomes, on the contrary, very regular.

Over six throws, it is entirely possible to get no ace at all. Over six thousand throws, you will get about a thousand, since on average one throw in six gives an ace. Not exactly a thousand, but very rarely off by more than sixty.

Interactive

Six dice, or six thousand?

When a die is thrown, there is no knowing what will come up. Over thousands of throws, the share of aces becomes almost predictable.

The question

A die lands on the ace one time in six: is that true over six throws?

Without JavaScript, here are the animation’s data.

What the calculation predicts (one ace in six on average) (calculated values)
Throws	Aces expected on average	In 95% of series, number of aces	That is, a share of aces of
6	1	0 to 3	0 to 50%
60	10	5 to 16	8.3 to 26.7%
600	100	82 to 118	13.7 to 19.7%
6,000	1,000	944 to 1,057	15.7 to 17.6%

Over 6 throws, there is no ace about one time in three (33%). Over 6,000 throws, the number of aces is more than 60 away from 1,000 in only 3.6% of series: “very rarely”, as the article says.

These figures are calculated exactly (binomial distribution), for a fair die.

The takeaway A small number of throws can give almost anything. The more you throw, the more tightly the share of aces gathers around one in six: this is the law of large numbers.

Sources 1

Bernoulli, Jacques, Ars Conjectandi, Bâle, 1713, public domain ; partie IV traduite en anglais : Sheynin, Oscar, On the Law of Large Numbers, Berlin, 2005, part IV, chapter 4.
even the most stupid person, all by himself and without any preliminary instruction, being guided by some natural instinct […] feels sure that the more such observations are taken into account, the less is the danger of straying from the goal
Public domain and open access open the source · checked on 24 October 2026

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The day people decided to calculate chance

For a very long time, chance seemed impossible to measure. What happens by luck happens, and that is all — some even saw fate in it. The idea that it could be calculated is a break, and it can be dated.

In 1654, a Frenchman, the Chevalier de Méré, put a question to a mathematician he knew, Blaise Pascal. Here it is, and it is easy to grasp.¹

Two players stake the same sum. The first to win three rounds takes everything. But the game is interrupted at two rounds to one. How should the money be shared?²

Splitting it down the middle is unfair: the one who was leading had a better chance of winning. Giving everything to the one who was leading is unfair too: he had not yet won. So one has to calculate what each would probably have obtained. In other words, to put a number on something that did not happen.

Pascal wrote to another mathematician, Pierre de Fermat, and the two men exchanged letters all summer. This is usually taken as the starting point of the calculus of probabilities.³

A caveat is needed, however. This story is often told as the moment probability was born. But the problem of points has a history before Pascal: Italian authors discussed it throughout the sixteenth century, and one of them, Tartaglia, judged it “more judicial than rational in nature”.⁴

The regularity has a name

This regularity has a name: the law of large numbers.

Everyone had long suspected it: if you keep throwing a die, the 5 eventually comes up roughly once in six throws. What was difficult was not noticing it, but proving it. A Swiss mathematician, Jacob Bernoulli, managed it, and he said he had kept the solution to himself for twenty years. His book would not appear until 1713, eight years after his death.⁵

This law explains why insurance exists, why opinion polls more or less work, and why casinos win in the long run without needing to cheat.

It also sheds light on an important point: no one can say what will happen to a particular person. One can only say what happens to many people in their situation. It is not the same thing, and confusing the two is a very common mistake.

Counting letters to break a code

Here is a surprising use of counting, long predating the mathematics of chance.⁶

In a given language, some letters occur much more often than others. This regularity is invisible in a short sentence, and perfectly stable over a long text.

In the ninth century, the scholar al-Kindi wrote a treatise explaining how to use it to read a coded message. His method: take a long ordinary text in the same language, count its letters, rank them from the most frequent to the rarest; do the same with the enciphered message; then set the two lists side by side.⁷

The most frequent sign in the message very probably corresponds to the most frequent letter in the language. Step by step, the message is deciphered.

This is already statistical reasoning in embryo, applied long before statistics became a discipline.

Counting people

Counting objects is useful. Counting people is something else.

Very different empires organised censuses: in Babylon, in China, in Egypt, and as far as Peru, where the Incas counted their population in the fourteenth century. In the Andes, numerical information was recorded on knotted cords.⁸

Two examples show clearly what this was for. In Rome, every five years, citizens were registered to assess their property and the tax they owed. In 1086, the King of England had almost his entire kingdom surveyed, plot by plot, to find out who owned what and how much tax he could draw from it.⁹

None of these counts was made out of curiosity. People are counted to know how much tax can be raised, how many soldiers can be mobilised, how many hands can be requisitioned.¹⁰

Modern censuses appeared only quite late: in Iceland in 1703, in Sweden in 1749, for example. Counting everyone, at regular intervals, is a recent invention.¹¹

In France, the first census of the modern era took place in 1801, the same year as in England and Wales; it counted thirty-three million inhabitants. Since 2004, it has been based on annual data collection.¹²

Making a table of figures speak

A column of numbers says nothing. The human eye cannot read three hundred numbers in a row.

Giving these numbers a shape — a drawing, a curve, a map — changes everything. All at once you see a trend, a bump, an anomaly.

The most famous example deserves a closer look at the way it is usually told.

In 1854, cholera struck a district of London. A doctor, John Snow, rejected the authorities’ explanation, which blamed poisoned air: for him, cholera travelled through water. He investigated: he compiled the list of the dead, noted where these people lived, and at which pump they fetched their water. He questioned their families. In the end he pointed to one particular water pump, in Broad Street.¹³

What followed is often told like this: Snow marked the houses of the dead on a map, saw from it where the disease was coming from, removed the handle of the pump, and the epidemic stopped. It is a fine story. It is wrong on two counts.¹⁴

First, the map was not used to find the cause: it served above all to show an idea the investigators already had. Second, when the handle of the pump was removed, the epidemic was already on the wane.¹⁵

What made the discovery possible was the investigation. The map served to win acceptance for it.

Another famous drawing was put to exactly the same use. During the Crimean War, in the middle of the nineteenth century, Florence Nightingale compiled the figures for the deaths of British soldiers, month by month, separating deaths from wounds received in battle from those due to diseases that could have been prevented.¹⁶

Then she drew them. Each month occupies a slice of a disc, and the area of each colour gives the number of deaths. The blue, for preventable diseases, outweighs the red, for wounds.¹⁷

Her drawing shows two things at once, and generally only one is remembered.

The first: the soldiers died less from the fighting than from the filth of the hospitals. The second, just as visible in the figure: after the arrival of a sanitary commission, in the middle of the war, the blue collapses. The graph does not merely point to the culprit — it shows that the remedy worked.¹⁸

Nightingale did not draw to decorate a report. She drew to persuade the government to improve army hygiene. She had two thousand copies of the report containing these figures printed.¹⁹

Two details that show her rigour

The first concerns the drawing itself. Her first version represented the deaths by the length of the radii. She realised that this misled the eye, which looks at areas: if you double the radius of a disc, its area is multiplied by four.²⁰

She first had an erratum slip inserted into the already printed work, then redid the figure entirely, this time calculating the areas.²¹

She acknowledged the error in her own graph, even though that graph served the cause she was fighting for.²²

Interactive

Length or area? The error Nightingale corrected

During the Crimean War, Florence Nightingale gathered the deaths of British soldiers, month by month, and drew each month as a slice of a disc. The choice remains of how the slice grows with the number.

The question

Should a number twice as large get a radius twice as long?

Without JavaScript, here are the animation’s data.

Two slices of a disc: what the eye sees (calculated values)
The second number is	Area seen, if the radius follows the number	Area seen, if the area follows the number
the same as the first	×1	×1
twice the first	×4	×2
three times the first	×9	×3
four times the first	×16	×4

The area of a slice of a disc is proportional to the square of its radius: doubling the radius multiplies the area by four. The numbers in the animation are examples, not Nightingale’s figures.

The takeaway The eye compares areas. If the radius follows the number, a number twice as large looks four times as large. To avoid misleading, the area must follow the number.

Sources 2

Small, Hugh, “Florence Nightingale’s Statistical Diagrams”, University of York, Department of Mathematics, description of the diagrams (“bat’s wing”, then “wedges”).
The length of the radial line in each month is proportional to the death rate, […] Florence recognised this error and inserted an erratum slip, but then replaced this diagram in later documents (nos. 3, 4, and 5 listed above) with what I will call the "wedges" diagram. […] In this diagram, Nightingale resolved the problem of the "bat's wing" by using areas to represent the variation in the death rate, instead of the length of radial lines.
Open access open the source · checked on 9 September 2026
Science Museum (Londres), “Florence Nightingale: The Pioneer Statistician”, section “The Crimean war”, then “An analytical approach to medical care”, then “Nightingale’s polar area diagram”.
The Crimean War (1853–6) was the first high-technology conflict of the modern age […] Nightingale set about collecting statistics in Crimea. She treated this activity—counting the number of soldiers killed, injured or diseased […] In her diagram, each wedge represented a month, and the area of the wedge showed the number of soldiers who had died that month.
Open access open the source · checked on 25 October 2026

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The second concerns her reply to her opponents. A pamphlet accused her of having exaggerated the number of deaths. In 1859, she published a reply — anonymously — based on the figures the army itself had just made public.²³

These figures showed that she had not exaggerated them: she had underestimated them.

So the two must be remembered together. A well-made graph is not an instrument of discovery: it is an argument. You discover by investigating. You draw to be believed.

The flip side must be added at once. A badly made graph — or one made that way on purpose — can make accurate figures say more or less anything. It is enough to choose where the axis starts, which years are shown, which colours are used. A drawing is never neutral.

Counting gives power

One last point remains, and it is a serious one.

Being able to count people means having a hold over them. It has made it possible to organise schools, to detect epidemics, to understand why people were dying. It has also made it possible to tax, and to conscript people by force.

It has served far worse ends.

During the Second World War, in France, the State forced Jewish people to register. Then their names were filed on index cards.

None of this was an accident. In July 1942, the police took out 27,000 of these cards to hand them over to the teams in charge of the arrests. Here, counting served to locate people.²⁴

No computer was needed for that work. The lists that mattered were simple index cards, filed in the card index of the Paris police prefecture.

This is something to remember, and it still holds today: the danger is not in the machine. It is in the question one decides to ask people, and in the act of writing down the answer.

The tools used to measure a society are also used to control it. They are the same tools.

We must go one step further, because the history of these tools is troubling.

Some very useful calculations are used every day by scientists in their research. Two of the best known, correlation and regression, were invented more than a hundred years ago by two scholars, Francis Galton and Karl Pearson.²⁵

But they did not invent them for the sake of mathematics. They were among the most enthusiastic proponents of an idea called eugenics: in their view, certain people should be encouraged to have children, and others discouraged, or even prevented — rather as a breeder chooses his best animals.²⁶

Yet human beings are not livestock, and no one has the right to decide such things for others.

This was not an idea dreamt up on the back of an envelope. In 1911, in London, Pearson became the first holder of a chair in eugenics, funded by Galton’s will — and, the same year, he founded the world’s first department of statistics at the university. He headed it until 1933.²⁷

Today, this idea of sorting human beings is seen for what it is: deeply unjust.

It would be wrong, however, to think that they had invented some neutral tool that was later used for a bad idea. These calculations were shaped for their own question: how far does a child resemble its parents? Measuring the strength of the link between two things is exactly what a correlation coefficient does. The tool was born of that very question.²⁸

It works none the worse for that — indeed, it works very well. Today it is used to treat people and to understand the world. But knowing where the tools we use come from is part of the job.

This does not mean we should stop counting. It means we should always ask who is counting, and what for.

Three different things called “the figures”

What it is	What it is for	The trap
A probability	predicting what happens often	it says nothing about a particular case
A collected datum	describing what exists	it depends on who decided what to count
A graph	making a trend visible	it can be built to convince

Note 1

En 1654, le problème des partis resurgit dans des conditions assez obscures. Le chevalier de Méré le propose à Pascal

In 1654, the problem of points resurfaced in rather obscure circumstances. The Chevalier de Méré put it to Pascal

Trotignon, Nicolas, Pascal, Fermat et la géométrie du hasard, mémoire, IUFM de Créteil, 1998, preprint arXiv:1309.2824