p(2s) + (1–p)(–2s)
Looks like just another routine algebraic equation, doesn’t
it? Perhaps, but like every other routine algebraic equation this
one has a meaning. It tries to solve a problem.
What’s this particular problem?
Deciding whether to accept a double of the bet in a game of backgammon.
That’s just one of the lessons Ronald Gould professes to 15
freshmen in his seminar “Mathematics in Sports, Games and
“The point of this course is to take mathematical concepts
and apply them in ways you wouldn’t normally think about,”
said Gould, Goodrich C. White Professor of Mathe-matics.
Sure enough. Class assignments mix terms like “mean,”
median” and “standard deviation” with “home
run.” Card tricks and games in which the results can be predetermined
through mathematics are outlined.
And algebraic equations, all based on the laws of probability, can
be attached to parlor games many people learned how to play with
their grandparents—games like backgammon.
For instance, the sum of that equation at the top of this story
must be greater than the result of not accepting the bet. That sum
is –s, s being the amount of the bet. The above equation is
the probability of winning the bet (p) multiplied by the amount
to be won (2s), plus the probability of losing the bet (1–p)
multiplied by the amount to be lost (–2s).
Once the math is computed out, the sum is p > 1/4. So, in a game
of backgammon and someone doubles you, if you have better than a
25 percent chance of winning, you should accept the bet.
Gould goes on to give examples of when a player himself should double
(50 percent chance of winning) or when he wants an opponent to accept
a double (with a 75 percent chance of winning).
The course is not all fun and games; the material has a deeper meaning.
By uncovering the mathematical principles underlying games of chance
and fun, the students’ strategic-thinking muscles are flexed—and
those skills are applicable anywhere.
This is the fourth time Gould has taught the class, and he is somewhat
minimalist about it. The syllabus is a single page, and the class
utilizes just one textbook (Edward Packel’s The Mathematics
of Games and Gambling) and a variety of handouts. And he tends to
focus on material the class appears more interested in.
While previous classes have eaten up sections on baseball statistics,
the fall 2002 seminar wasn’t as enamored, so Gould backed
off a bit on the subject.
Still, for the backgammon segment, Gould had to do a little preparation.
He brought in a laptop computer so he could teach the game to the
students; most were unfamiliar with it. They played against the
Mathematics in sports and games is all well and good, but what about
gambling? After all, the entire backgammon segment was about betting.
Is it a very good idea to discuss gambling techniques with freshman?
Sure, Gould said. And with good reason.
“If anything, this class is teaching them that they shouldn’t
gamble,” Gould said. The only casino game the class has investigated
in depth so far is roulette, which is set up to favor the house.
Players, in fact, through class computation, were found to be at
a 5 percent disadvantage.
“So, for every dollar you bet, you’re donating a nickel,”
Gould said. “No matter how you bet. That game is set up to
have a fixed probability against you.”