The problem of points. Telis and Wendy play a round of

QUESTION:

The problem of points. Telis and Wendy play a round of golf ( 18 holes) for a $10 stake, and their probabilities of winning on any one hole are P and 1 - p, respectively, independent of their results in other holes. At the end of 10 holes, with the score 4 to 6 in favor of Wendy, Telis receives an urgent call and has to report back to work. They decide to split the stake in proportion to their probabilities of winning had they completed the round, as follows. If PT and pw are the conditional probabilities that Telis and Wendy, respectively, are ahead in the score after 18 holes given the 4-6 score after 10 holes, then Telis should get a fraction PT / (PT + pw) of the stake, and Wendy should get the remaining pw /(PT + pw). How much money should Telis get? Note: This is an example of the, so-called, problem of points, which played an important historical role in the development of probability theory. The problem was posed by Chevalier de Mere in the 17th century to Pascal, who introduced the idea that the stake of an interrupted game should be divided in proportion to the players' conditional probabilities of winning given the state of the game at the time of interruption. Pascal worked out some special cases and through a correspondence with Fermat, stimulated much thinking and several probability-related investigations.