514 18. PROBABILITY AND STATISTICSthe game is terminated after three unsuccessful tries. Pascal wrote to Fermat that
the player should be allowed to sell the throws one at a time. If the first throw is
foregone, the player should take one-sixth of the stake, leaving five-sixths. Then if
the second throw is also foregone, the player should take one-sixth of the remaining
five-sixths or ^, and so on. In this way, Pascal argued that the fourth through
eighth throws were worth | [(|)^3 + (§)^4 + (I)^5 + (§)' + (f )^7 ].
This expression is the value of those throws before any throws have been made.
If, after the bets are made but before any throws of the die have been made, the
bet is changed and the players agree that only three throws shall be made, then the
player holding the die should take this amount as compensation for sacrificing the
last five throws. Remember, however, that the net worth of a player is constantly
changing as the game progresses and the probability of winning changes. The value
of the fourth throw, for example, is smaller to begin with, since there is some
chance that the player will win before it arrives, in which case it will not arrive. At
the beginning of the game, the chance of winning on the fourth roll is (|)^3 g, the
factor (|)^3 representing the probability that the player will not have won before
then. After three unsuccesful throws, however, the probability that the player "will
not have" won (because he did not win) on the first three throws is 1, and so the
probability of winning on the fourth throw becomes |.
Fermat expressed the matter as follows:[T]he three first throws having gained nothing for the player who
holds the die, the total sum thus remaining at stake, he who holds
the die and who agrees not to play his fourth throw should take
g as his reward. And if he has played four throws without finding
the desired point and if they agree that he shall not play the fifth
time, he will, nevertheless, have | of the total for his share. Since
the whole sum stays in play it not only follows from the theory,
but it is indeed common sense that each throw should be of equal
value.Pascal wrote back to Fermat, proclaiming himself satisfied with Fermat's anal-
ysis and overjoyed to find that "the truth is the same at Toulouse and at Paris."
1.3. Huygens. Huygens wrote a treatise on probability in 1657. His De ratiociniis
in ludo ale<E (On Reasoning in a Dice Game) consisted of 14 propositions and
contained some of the results of Fermat and Pascal. In addition, Huygens was
able to consider multinomial problems, involving three or more players. Cardano's
idea of an estimate of the expectation was elaborated by Huygens. He asserted, for
example, that if there are ñ (equally likely) ways for a player to gain á and q ways
to gain 6, then the player's expectation is (pa + qb)/(p + q).
Even simple problems involving these notions can be subtle. For example,
Huygens considered two players A and  taking turns rolling the dice, with A
going first. Any time A rolls a 6, A wins; any time  rolls a 7,  wins. What are
the relative chances of winning? (The answer to that question would determine the
fair proportions of the stakes to be borne by the two players.) Huygens concluded
that the odds were 31:30 in favor of B, that is, A's probability of of winning was
|ã and B's probability was |y.