Sunday, 10 March 2013

Musical Chairs

Most of the histories of probability trace their facts back to Hacking and David, and I agree these two books are the best of the bunch I have read.  The Hacking book itself references David.  I love the Bernstein book series but I noticed his page 43-44 has some musings on why the Greeks didn't bother with working out odds behind dice games.  I bet they did.  Anyway, he offers an example of the so-called sloppiness of the Greek observations of dice probabilities by mentioning some facts he gleans from David - namely that when using the astragali they valued the Venus throw (1,3,4,6) higher than (6,6,6,6) or (1,1,1,1). which are, he states "... equally probable".

 No, they are not.  David clearly states that the probability of throwing a six as one in ten; likewise with throwing a one.  And threes and fours are about four in ten events.  This means that even if  order is important, the Venus is indeed more likely.  Second mistake, there's only one permutation of four sixes, and only one permutation of four ones.  But there are many permutations of the four Venus numbers, meaning the probability of (1,3,4,6) in any permutation is even higher again than the strictly ordered (1,3,4,6).

It is this partition/permutation dilemma of probability theory, even today, which is so easy to get wrong.  I just re-read some earlier postings I made on equivalence classes and their information content and key milestones in probability theory, and I still like what I wrote.  Also check out a posting on combinatorics in Cardano and Lull.

It is just a throwaway comment in Bernstein's book and hardly invalidates his wonderful sweeping history of risk but is nicely illustrates the problems of thinking about event space and equivalence class.

Divorce born

I've been thinking about Cardano, Pascal, Fermat and  Huygens a lot recently and hope to make a number of postings.  For now I'd just like to bring some controversy to the usual story found in the literature about these characters and their relative importance.  According to this literature there are three pivotal moments - which I'll call Cardano's circuit, Pascal-Fermat's divorce settlement and Huygen's hope relating to the problems of complex sample space, the arithmetic triangle, and expected value of an uncertain outcome, or to simplify it even further, to factorial, binomial coefficients and the average, all fairly contemporaneous mathematical inventions or discoveries in the Western tradition.

The story usually told is one which lays great praise at the workings of Pascal and Fermat and which makes a big deal of the so-called problem of points.  What I'd like to do during this discussion is show how connected the problem of points is to another famous probability exercise, so-called Gambler's ruin.  I'd like to bring these two problems together and show ways in which they're related to many contemporary decision problems.  I'd also like to claim that the solution to Gambler's ruin is more important than the problem of points, and has more resonance today.  I'd also like to claim that Cardano's discussion of event space has the better claim to being the foundation of probability theory.

In all of the postings to come, I base my readings on the following books, plus free online primary sources, where available in an English translation.

One last introductory point - this thread is clearly a biassed Western history of ideas discussion.  Many of the  commentators below neglect to sufficiently emphasise the great world traditions in mathematics which played into this - especially from the Islamic, Chinese, Indian traditions.  These clearly played in to the so-called canonical view of the birth of probability but that weakness in the line of argument is a weakness for another time and another place.