Willful Ignorance: The Mismeasure of Uncertainty

APPENDIX: THE PASCAL–FERMAT CORRESPONDENCE OF 1654

The first extant letter in the exchange contains a response by Fermat to some apparent confusion evidenced in Pascal's previous letter.¹ Fermat attempts to clarify a possible misunderstanding regarding a variation on the problem of determining the fair odds for attempting to obtain a 6 with eight potential throws of a fair die. Suppose that “after the stakes have been made,” the players subsequently agree to change the terms.

Instead of winning if a 6 occurs on or before the eighth throw, the player will win only if a 6 occurs for the first time after a specified throw, such as the fourth throw. In other words, the player will win if the first point is made between the fifth and eighth throws, but will lose if the 6 comes up in one of the preceding throws (or not at all). What portion of the total amount staked should be given back to the player in compensation for agreeing to this more stringent condition?

It is important to understand that Fermat was not asking how the player's future expectation would change after four unsuccessful throws had actually occurred. It can be shown that the fair odds in favor of making the point in the remaining four trials would be about 13:12. However, this reduction of the expectation would not be the result of a decision on the player's part but simply to bad luck. So, although his odds against making the point in the four remaining throws would be diminished, the player would deserve no ...