A Philosophical Essay on Probabilities
Pierre-Simon Laplace, 1814
Why does it seem that after nine heads in a row tails must come up? The great mathematician Pierre-Simon Laplace explains the laws of chance using urns, coins and a lottery, without a single formula. You will learn why a coin has no memory, why even a fair game is unprofitable, how to divide risk, and why an astonishing story needs more proof than an ordinary one.
Who it is for
For those who want to judge chances soberly: with money, in arguments, in rumours and in their own hopes. And for everyone who has never liked mathematics but would like to understand how chance works, without a single formula.
The most important questions of life are, for the most part, nothing but problems of probability.
A Comet People Stopped Fearing
In 1456 a comet with a very long tail appeared over Europe. Europe was already living in fear: the Turks had taken Constantinople not long before and were moving on. People decided that the angry sky was warning of vengeance for the sins of the earth, and they prayed that the disaster would pass them by.
Three centuries later, the same comet caused not terror but impatience. Edmond Halley understood that it had come in 1531, 1607 and 1682, and predicted its return at the end of 1758 or the beginning of 1759. The mathematician Clairaut took into account how Jupiter and Saturn pull it along the way, and refined the date. In 1759 the comet returned, as had been calculated. The sky had stayed the same. What had changed was what people knew about it.
Pierre-Simon Laplace begins his “Philosophical Essay on Probabilities” with this story. The book grew out of a lecture given in 1795, and it contains not a single formula: Laplace wanted to explain his science to those who had not studied it. His central idea sounds bold. The most important questions of life are, for the most part, problems of probability. Should you buy a ticket, believe a story, risk your money? All of these are bets, and the chances in them can be weighed.
Next you will see why a coin has no memory, how nine chances in ten become one in eight, why even a fair game is unprofitable, and why a miraculous story needs more evidence than an ordinary one. There will be urns with balls, coins and a lottery, on which Laplace tested common sense. Let's start with the question of what chance is.
A Mind for Which Nothing Is Random
Laplace offers a thought experiment. Imagine a mind that, in a single moment, knows all the forces of nature and the position of every particle in the world, and is able to calculate all of it. For such a mind, one formula would describe both the motion of the planets and the flight of the lightest atom. Later this imagined mind came to be known by the name “Laplace's demon”.
Laplace is sure that the world is built exactly this way: every event has a cause, even the smallest. The path of a tiny particle of air, in his words, is governed by laws just as strictly as the orbit of a planet. The difference is only that we know the orbit and do not know the particle. That is why for Laplace, chance is the name of our ignorance, not a special force of nature. A caveat is needed here, and it is no longer from the book: the physics of the twentieth century holds that in the world of the smallest particles a complete prediction is impossible even in principle. But for everyday decisions, Laplace's idea works today as well.
If probability depends on knowledge, then different people have different probabilities. Laplace shows this with three urns. One contains only black balls, two others only white ones, and it is not known which is which. A ball is drawn from the third urn. For someone who knows nothing, the chance of drawing black is one in three. Someone who knows that the first urn holds white balls will say: one in two. And someone who knows about both of the first urns is certain the ball will be black. All three reckon correctly. They simply know different things.
From this Laplace draws a human piece of advice. People argue about what is probable most often not because some are foolish and others clever. They stand at different points and see different data. That is why, Laplace writes, one should be indulgent toward other people's opinions, and before enlightening others, weigh one's own opinions impartially.
In your next argument, try asking not “who is right?” but “what does this person know that I do not?” Often the difference of opinion turns out to be the difference of one urn.
About the author
Pierre-Simon Laplace (1749-1827) was a French mathematician, physicist and astronomer, author of “Celestial Mechanics” and “Analytic Theory of Probabilities”. This book grew out of his lecture of 1795 and was first published in 1814.
All books by this author: Pierre-Simon Laplace
Terms from the book
- Bernoulli
- St. Petersburg paradox
- Pascal
- Fermat
- law of large numbers
- Constantinople: The capital of the Byzantine Empire, today's Istanbul. In 1453 the Ottoman Turks took it, and the empire ceased to exist.
- Halley: Edmond Halley (1656-1742), English astronomer. He was the first to understand that the comets of 1531, 1607 and 1682 were one body, and he predicted its return; the comet was named after him.
- Clairaut: Alexis Claude Clairaut (1713-1765), French mathematician and astronomer. He calculated how the attraction of Jupiter and Saturn acts on Halley's comet, and gave the date of its return in 1759.
- Laplace's demon: A later name for the imagined all-knowing mind from this book, which, knowing all the forces and positions of particles, could predict the future exactly. It is a thought experiment, not a scientific term.
- Méré: Antoine Gombaud, Chevalier de Méré, a French writer of the seventeenth century, a worldly gambler and friend of Pascal. His questions about dice games prompted Pascal and Fermat to the first works on probability.
- geometer: In the seventeenth and eighteenth centuries a mathematician in general was called this, not only an expert in geometry.
- mathematical expectation: The sum of all possible gains, each multiplied by its probability, minus the losses counted the same way. It shows how much a bet brings on average; Laplace calls it “mathematical hope”.
- moral expectation: The gain from a bet, taking into account how much this money means to a particular person, not only how much it is. It rests on Daniel Bernoulli's rule, which, in Laplace's words, serves in many cases; Laplace himself calls this quantity “moral hope”.
- votive tablets: Offerings to a temple in gratitude for rescue or a fulfilled prayer, often with the donor's name. In antiquity, for example, people saved from shipwreck left them.
All terms
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