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.
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.
