Against the Gods: The Remarkable Story of Risk (14 page)

All sorts of patterns greet the eye when we first glance at Pascal's
Triangle, but the underlying structure is simple enough: each number is
the sum of the two numbers to the right and to the left on the row above.

Probability analysis begins with enumerating the number of different ways a particular event can come about-Cardano's "circuit." That
is what the sequence of numbers in each of these expanding rows is
designed to provide. The top row shows the probability of an event
that cannot fail to happen. Here there is only one possible outcome,
with zero uncertainty; it is irrelevant to probability analysis. The next
row is the first row that matters. It shows a 50-50 situation: the probability of outcomes like having a boy-or a girl-in a family that is
planning to have only one child, or like flipping a head on just one toss
of a coin. Add across. With a total of only two possibilities, the result is
either one way or the other, a boy or a girl, a head or a tail; the prob ability of having a boy instead of a girl or of flipping a head instead of
a tail is 50%.

The same process applies as we move down the triangle. The third
row shows the possible combinations of boys and girls in a family that
produces two children. Adding across shows that there are four possible results: one chance of two boys, one chance of two girls, and two
chances of one each-a boy followed by a girl or a girl followed by a
boy. Now at least one boy (or one girl) appears in three of the four outcomes, setting the probability of at least one boy (or one girl) in a twochild family at 75%; the probability of one boy plus one girl is 50%.
The process obviously depends on combinations of numbers in a manner that Cardano had recognized but that had not been published when
Pascal took up the subject.

The same line of analysis will produce a solution for the problem of
the points. Let us change the setting from Paccioli's game of balla to the
game of baseball. What is the probability that your team will win the
World Series after it has lost the first game? If we assume, as in a game
of chance, that the two teams are evenly matched, this problem is identical to the problem of the points tackled by Fermat and Pascal.'s

As the other team has already won a game, the Series will now be
determined by the best of four out of six games instead of four out of
seven. How many different sequences of six games are possible, and
how many of those victories and losses would result in your team winning the four games it needs for victory? Your team might win the
second game, lose the third, and then go on to win the last three. It
might lose two in a row and win the next four. Or it might win the
necessary four right away, leaving the opponents with only one game
to their credit.

Out of six games, how many such combinations of wins and losses
are there? The triangle will tell us. All we have to do is find the appropriate row.

Note that the second row of the triangle, the 50-50 row, concerns
a family with an only child or a single toss of a coin and adds up to a
total of two possible outcomes. The next row shows the distribution of
outcomes for a two-child family, or two coin tosses, and adds up to
four outcomes, or 22. The next row adds up to eight outcomes, or 23,
and shows what could happen with a three-child family. With six
games remaining to settle the outcome of the World Series, we would want to look at the row whose total is 26-or two multiplied by itself six times, where there will be 64 possible sequences of wins and losses.*
The sequence of numbers in that row reads:

Remember that your team still needs four games to win the Series, while the opposing team needs only three. There is just one way your team can win all the games-by winning all the games while the opponents win none; the number 1 at the beginning of the row refers to that possibility. Reading across, the next number is 6. There are six different sequences in which your team (Y) would gain the Series while your opponents (0) win only one more game:

And there are fifteen different sequences in which your team would win four games while your opponents win two.

All the other combinations would produce, at least three games for the opposing team and less than the necessary four for yours. This means that there are 1 + 6 + 15 = 22 combinations in which your team would come out on top after losing the first game, and 42 combinations in which the opposing team would become the champions. As a result, the probability is 22/64-or a tad better than one out of three-that your team will come from behind to win four games before the other team has won three.

The examples reveal something odd. Why would your team play out all six remaining games in sequences where they would have won the Series before playing six games? Or why would they play out all four games when they could win in fewer games?

Although no team in real life would extend play beyond the minimum necessary to determine the championship, a logically complete .rolution to the problem would be impossible without all of the math ematical possibilities. As Pascal remarked in his correspondence with
Fermat, the mathematical laws must dominate the wishes of the players
themselves, who are only abstractions of a general principle. He declares
that "it is absolutely equal and immaterial to them both whether they let
the [match] take its natural course."

The correspondence between Pascal and Fermat must have been an
exciting exploration of new intellectual territory for both men. Fermat
wrote to Carcavi about Pascal that "I believe him to be capable of solving any problem that he undertakes." In one letter to Fermat, Pascal
admitted that "your numerical arrangements ... are far beyond my
comprehension." Elsewhere, he also described Fermat as "a man so outstanding in intellect ... in the highest degree of excellence .... [that his
works] will make him supreme among the geomasters of Europe."

More than mathematics was involved here for Pascal, who was so
deeply involved with religion and morality, and for Fermat the jurist.
According to their solutions, there is a matter of moral right involved
in the division of the stakes in Paccioli's unfinished game of balla.
Although the players could just as easily split the stakes evenly, that
solution would be unacceptable to Pascal and Fermat because it would
be unfair to the player who was lucky enough to be ahead when playing ceased.16

Pascal is explicit about the moral issues involved and chooses his
words with care. In his comments about this work, he points out that
"the first thing which we must consider is that the money the players
have put into the game no longer belongs to them ... but they have
received in return the right to expect that which luck will bring them,
according to the rules upon which they agreed at the outset." In the
event that they decide to stop playing before the game is over, they will
reenter into their original ownership rights of the money they have put
into the pot. At that point, "the rule determining that which will
belong to them will be proportional to that which they had the right to
expect from fortune .... [T]his just distribution is known as the division." The principles of probability theory determine the division,
because they determine the just distribution of the stakes.

Seen in these terms, the Pascal-Fermat solution is clearly colored by
the notion of risk management, even though they were not thinking
explicitly in those terms. Only the foolhardy take risks when the rules
are unclear, whether it be balla, buying IBM stock, building a factory,
or submitting to an appendectomy.

But beyond the moral question, the solutions proposed by Pascal
and Fermat lead to precise generalizations and rules for calculating
probabilities, including cases involving more than two players, two
teams, two genders, two dice, or coins with two sides. Their achievement enabled them to push the limits of theoretical analysis far beyond
Cardano's demonstration that two dice of six sides each (or two throws
of one die) would produce 62 combinations or that three dice would
produce 63 combinations.

The last letter of the series is dated October 27, 1654. Less than a
month later, Pascal underwent some kind of mystical experience. He
sewed a description of the event into his coat so that he could wear it
next to his heart, claiming "Renunciation, total and sweet." He abandoned mathematics and physics, swore off high living, dropped his old
friends, sold all his possessions except for his religious books, and, a short
while later, took up residence in the monastery of Port-Royal in Paris.

Yet traces of the old Blaise Pascal lingered on. He established the
first commercial bus line in Paris, with all the profits going to the
monastery of Port-Royal.

In July 1660, Pascal took a trip to Clermont-Ferrand, not far from
Fermat's residence in Toulouse. Fermat proposed a meeting "to embrace you and talk to you for a few days," suggesting a location halfway
between the two cities; he claimed bad health as an excuse for not
wanting to travel the full distance. Pascal wrote back in August:

I can scarcely remember that there is such a thing as Geometry [i.e.,
mathematics]. I recognize Geometry to be so useless that I can find
little difference between a man who is a geometrician and a clever
craftsman. Although I call it the best craft in the world it is, after all,
nothing else but a craft .... It is quite possible I shall never think of
it again. 17