Saturday 8 November 2008

Some Early Probability History Notes

So we make a fair bet, I roll one die, you roll the other, and who ever gets the highest scores a point. If we tie, we just redo the roll, and the first one to five points wins. Easy enough, but then, when the score is three to one my favor, you get an emergency phone call and have to leave. How should we distribute the stakes?

It was just such a problem that formed the foundation of early probability, and when it was solved, it sparked a rapid development of problems, and applications of probability.

I would tell you more, but I just read a neat blog by Keith Devlin that covered just such a development, so here, in part, are the words of a master:

"The Unfinished Game,
The problem of the unfinished game, also known as the problem of the points, was described in a book on arithmetic and geometry written by the Italian mathematician Luca Pacioli in 1494, [The text was Summa de arithmetica, geometrica, proportioni et proportionalita, and you can view it here PAT] though it is known to predate that mention. It asks how the pot should be fairly divided when a multi-round tournament has to be abandoned before it is finished. For instance, suppose two players are rolling a pair of dice and agree to playa best of five rounds tournament. Three rounds are played, leaving one player ahead 2 to 1, at which point they must abandon the game. How should they divide the pot?

Pacioli was unable to solve this problem. So too were a number of other mathematicians (and gamblers) who tried, including Girolamo Cardano, Niccolo Tartaglia, and Lorenzo Forstani. The consensus was that the problem could not be solved.

Then, early in 1654, a gambler by the name of Antoine Gombaud, more often referred to in modern history books by his French nobleman's title of the Chevalier de Mere, asked his friend the mathematician Blaise Pascal. Pascal produced a complicated argument that can be made to work, but was not happy with it, so at a friend's urging he wrote to Fermat about it. Fermat quickly found a simple solution.

There are two rounds left unplayed, argued Fermat. In each round, either player can win, so there are in all four different ways the game could continue to its five-round completion. The player who has won one round to the other's two must win both those final rounds in order to win the contest; in the other three possible endings, the player who is ahead after three rounds will win. Therefore, said Fermat, the player who is ahead when the game is abandoned should take 3/4 of the pot, with the other player taking 1/4.

To anyone who sees this solution today, it seems simple enough. (The solution assumes the tournament is thought of as a "best-of-five" rounds, as opposed to a "first-to-three". You need a slightly more complicated argument in the latter case, but the answer is the same, a 3 to 1 division of the pot.) But no one before Fermat saw it, including Cardano who did work out all of the basic rules we use today to combine probabilities. Moreover, when he did see Fermat's solution, Pascal could not accept it, and nor could various of his colleagues he showed it to. What was their problem?

Since the computation is trivial, indeed no different from the calculation of the odds in any game of chance (and actually much simpler than many), the only thing that could be holding everyone back was the fact that what Fermat was counting were "possible futures." Something that two thousand years of received wisdom said was not possible.

Once word got out about Fermat's breakthrough, however - presumably through the highly mobile network of gambling European noblemen - it did not take long for others to jump into the "future prediction" act. Within a single lifespan, modern future prediction and risk management were in place.

The speed of developments that followed the solution to the problem of the unfinished game is staggering.

1657. Christian Huyghens writes a 16-page paper that lays out pretty well all of modern probability theory, including the notion of expectation, which he introduces.[This one is LIBELLUS DE RATIOCINIIS IN LUDO ALEAE and can be found here

1662. John Graunt, an English haberdasher, publishes an analysis of the London mortality tables, and in so doing establishes the beginnings of modern statistical inference.

1669. Huyghens uses his new probability theory to re-compute Graunt's mortality tables with greater precision.

1709. Nikolas Bernoulli writes a book describing applications of the new methods in the law. One problem he shows how to solve is how long must elapse after an individual goes missing before the court can declare him dead and allow his estate to be divided among his heirs.

1713. Jakob Bernoulli writes a book showing how the new probability theory can be used to predict the future in the everyday world. This is the first time the word "probability" is used in the precise, mathematical sense we use it today. He also proves the law of large numbers, of which more in a moment.

1732. The first American insurance company begins in Charleston, S.C., restricted to fire insurance.

1732. Edward Lloyd starts the precursor of what in 1734 becomes Lloyd's List, and eventually gives birth to the insurance company Lloyds of London.
1733. Abraham de Moivre discovers the bell curve, the icon of modern data collection.

1738. Daniel Bernoulli introduces the concept of utility to try to get a better handle on human decision making under uncertainty.

1760s. The first life insurance companies begin.


A pretty concise History for one blog... If you have additional notes to offer, please do.

1 comment:

SarahL said...

I wonder if you know of any modern games which depend on the division of stakes theorem. I've already found tennis and I think Texas Hold 'Em. Thanks.