The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern – Keith Devlin

Thoughts: I found The Unfinished Game, a book about the early development of probability theory, browsing the stacks in the Kitchener Public Library. For me, the book mostly underscored things I already knew about probability theory rather than providing many new insights, but I found it to be well written and enjoyable to read. The structure of the book, build around correspondance between Pierre de Fermat and Blaise Pascal, worked better than I initially thought it might.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Devlin, Keith. 2008. The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern. Basic Books.

• 162-163: de Finetti games, named after Bruno de Finetti, are a way of giving meaning to subjective statements of epistemic confidence like I’m 95% sure I remembered to turn off the gas before I left the house: “Suppose you are the person who makes the claim (that you are 95 percent certain you turned off the gas). I now offer you a deal. I present you with a jar containing 100 balls, 95 of them red, 5 black. You have a choice. You can draw one ball from the jar, and if it’s red, you win $1 million. Or we can go back and see if the gas is on, and if it is not, I give you $1 million. ¶ Now, if your ‘95 percent certain’ claim were an accurate assessment of your confidence, it would not make any difference whether you choose to pick a bowl from the jar or to go back with me and check the status of the gas stove. [But if you choose to pick a ball, you thus demonstrate] that what I will call your rational confidence that you have turned off the gas is at most 95 percent.” You can then repeat this game with different ratios of red to black balls, to hone in on how confident a person is about a claim.
  • A de Finetti game thus allows you to convert a statement about epistemic probability into a statement about frequentist probability