Wednesday, 26 April 2017

D'Alembert loses at roulette...

Our experience and understanding of the laws of nature teach us that the same event never happens many times in a row, and it is by virtue of this acquired knowledge that we dismiss the repetition of "heads" or "tails" many times consecutively.
(d'Alembert, Opuscules mathématiques, 1780, p.48)


If Google hits are anything to go by, d'Alembert is remembered chiefly today as the author of a codified gambling staking plan.  Although there is no documentation, the d'Alembert System probably dates from the late 18th or early 19th century and was not "invented" by d'Alembert at all, though it is loosely based on his speculations on the mathematics of probability.

The system is associated particularly with outside bets in Roulette, but applicable to any game of chance based on 50-50 probability. It relies on the idea that prolonged runs of a single outcome are less likely than an even distribution of outcomes. The strategy is simple: the player always bets on the same outcome; if he loses he increases the stake by one chip, if he wins he take away one chip. As  the table shows, if he succeeds in returning to his original bet, he will have made one chip for every winning coup.  Provided the game lasts long enough to approximate to mathematical probability, the result is will be a modest gain. (The Martingale system is similar but the number of chips are doubled for each loss making the risk much higher).

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See "d'Alembert system", BettingSites.org.uk
http://www.bettingsites.org.uk/guides/betting-systems/d-alembert-system/


So where does d'Alembert fit in?


 Catherine Lusurier, Portrait of d'Alembert, 
1777,   musée Carnavalet, Paris
The betting plan is named for d'Alembert since his work on probability supported the theory that a consecutive run of similar events (for example repeated tosses of the same side of a coin) was inherently unlikely.

D'Alembert's position is based on  a distinction between what is "metaphysically possible" (the mathematical laws of probability)  and the "physical possible" (what can actually occur),   No event can be precisely duplicated in nature; indeed, observation shows that long sequences of identical events can never happen purely by chance:
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...In the ordinary course of nature, the same event (whatever it might be) happens only rarely twice in succession, more rarely three or four times, and never 100 consecutive times.(Opuscules mathématiques1761, p.10)

 D’Alembert famously considered the problem: “When a fair coin is tossed, given that heads have occurred three times in a row, what is the probability that the next toss is a tail? He  insisted  the probability of a fourth tail must “obviously” be greater than 1/2: 

However I ask if ... if the number of times that heads has already successively occurred by the hypothesis, does not make it more likely the occurrence of tails on the fourth time? Because after all it is not possible, it is even physically impossible that tails never occurs. Therefore the more heads occurs successively, the more it is likely tail will occur the next time.  If this is the case, as it seems to me one will not disagree, the rule of combination of possible events is thus still deficient in this respect. (p.14)

Despite the elegance of the formulation - and sadly for exponents of the d'Alembert system - this is really just the classic "gambler's fallacy" recast.  There is no mysterious "law of nature" governing repeated events in games of chance and a series of identical outcomes is as mathematically probable as any other pattern. Here is the verdict of the Merriam-Webster's Guide to Everyday Math:


Perhaps fortunately, d'Alembert himself was not optimistic that mathematicians made effective gamblers. In the Encyclopédie article "Géomètre" he takes issue with Diderot's dismissive comparison of the two: granted, both compute and calculate; but a good gambler operates by instinct and  seizes his opportunities without lengthy reflection.  If the evidence of experience is anything to go by, great geometers are poor gamblers - a truth which d'Alembert claims to have learned from bitter experience.
See Hankins, Jean d'Alembert (1990), p.93: https://books.google.co.uk/books?id=gwjc3vGW9-MC&pg=PA94#v=onepage&q&f=false


References

Prakash Gorroochurn, "Errors of probability in historical context", The American Statistician, 65(1), 2011 p. 246-54  [reprint]
http://www.columbia.edu/~pg2113/index_files/Gorroochurn-Errors%20of%20Probability.pdf

On the early history of Roulette, The online Guide to traditional games
http://www.tradgames.org.uk/games/roulette.htm

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