 Pascal’s assumptions about utilities and probabilities are now in place. In another landmark moment in this passage, he next presents a formulation of expected utility theory. When gambling, “every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason”. How much, then, should a player be prepared to stake without transgressing against reason? Here is Pascal’s answer: “… the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss …” It takes some work to show that this yields expected utility theory’s answer exactly, but it is work well worth doing for its historical importance and is included in  a footnote.[5].)