Warm Seat

I am really rather pleased with my reading of the history of the theory of probability.  Four points struck me about it, firstly that Cardano has a much stronger claim than the authors of histories of probability give him credit for.  Second that Pascal was wrong in criticising Fermat's combinatorial approach in the case of more than two players in the problem of points and that his mistake was an equivalence class / ordering misunderstanding about the reading of three thrown dice.  Third, that Pascal's solution is a bit like using dynamic hedging for an exotic option (one which doesn't exist yet, but which I'll call a one-touch upswing option).  And fourth, that Huygens's gambler's ruin can be made into a problem of points by using participant stakes and separately some tokens which are transferred from the loser to the winner after each throw.  On the last three of these points Todhunter and the authors Shafer and Vovk agree with me, variously.

A better name for the problem of points is the warm seat price.  And the original first-to-six game, and also Gambler's ruin with plastic tokens and stakes can both be seen as specific games for which there's a warm seat price - the fair value of the game for a participant if he wanted to get out of the game immediately.  Gambler's ruin doesn't have a definite time in the future at which point it will with certainty be known who the winner is.

It is also amusingly my warm seat moment since I didn't discover anything myself, but followed in other peoples' footsteps, and have experienced the warm seat experience of discovery others had made before me.

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.