Free Against the Gods: The Remarkable Story of Risk by Peter L. Bernstein

be 1/3 to the fourth power.

    Cardano goes on to figure the probability of throwing a 1 or a 2
with a pair of dice, instead of with a single die. If the probability of
throwing a 1 or a 2 with a single die is one out of three, intuition
would suggest that throwing a 1 or a 2 with two dice would be twice
as great, or 67%. The correct answer is actually five out of nine, or
55.6%. When throwing two dice, there is one chance out of nine that
a 1 or a 2 will come up on both dice on the same throw, but the probability of a 1 or a 2 on either die has already been accounted for; hence,
we must deduct that one-ninth probability from the 67% that intuition
predicts. Thus, 1/3 + 1/3 - 1/9 = 5/9.
    Cardano builds up to games for more dice and more wins more
times in succession. Ultimately, his research leads him to generalizations
about the laws of chance that convert experimentation into theory.
    Cardano took a critical step in his analysis of what happens when
we shift from one die to two. Let us walk again through his line of reasoning, but in more detail. Although two dice will have a total of
twelve sides, Cardano does not define the probability of throwing a 1
or a 2 with two dice as being limited to only twelve possible outcomes.
He recognized that a player might, for example, throw a 3 on one die
and a 4 on the other die, but that the player could equally well throw
a 4 on the first die and a 3 on the second.
    The number of possible combinations that make up the "circuit"the total number of possible outcomes-adds up to a lot more than the
total number of twelve faces found on the two dice. Cardano's recognition of the powerful role of combinations of numbers was the most
important step he took in developing the laws of probability.
    The game of craps provides a useful illustration of the importance
of combinations in figuring probabilities. As Cardano demonstrated,
throwing a pair of six-sided dice will produce, not eleven (from two to
twelve), but thirty-six possible combinations, all the way from snake
eyes (two ones) to box cars (double six).
    Seven, the key number in craps, is the easiest to throw. It is six
times as likely as double-one or double-six and three times as likely as
eleven, the other key number. The six different ways to arrive at seven
are 6 + 1, 5 + 2, 4 + 3, 3 + 4, 2 + 5, and 1 + 6; note that this pattern
is nothing more than the sums of each of three different combinations-5 and 2, 4 and 3, and 1 and 6. Eleven can show up only two
ways, because it is the sum of only one combination: 5 + 6 or 6 + 5. There is only one way for each of double-one and double-six to
appear. Craps enthusiasts would be wise to memorize this table:

    In backgammon, another game in which the players throw two
dice, the numbers on each die may be either added together or considered separately. This means, for example, that, when two dice are
thrown, a 5 can appear in fifteen different ways:

    The probability of a five-throw is 15/36, or about 42% I5

    Semantics are important here. As Cardano put it, the probability of
an outcome is the ratio of favorable outcomes to the total opportunity
set. The odds on an outcome are the ratio of favorable outcomes to
unfavorable outcomes. The odds obviously depend on the probability,
but the odds are what matter when you are placing a bet.
    If the probability of a five-throw in backgammon is 15 five-throws
out of every 36 throws, the odds on a five-throw are 15 to 21. If the
probability of throwing a 7 in craps is one out of six throws, the odds
on throwing a number other than 7 are 5 to 1. This means that you
should bet no more than $1 that 7 will come up on the next throw
when the other fellow bets $5 that it won't. The probability of heads
coming up on a coin toss are 50/50, or one out of two; since the odds
on heads are even, never bet more than your opponent on that game.
If the odds on a long-shot at the track are 20-to-1, the theoretical
probability