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How Fair is Monopoly? by Ian Stewart
Reproduced with permission of the author

How Fair Is Monopoly?
Everyone has played Monopoly.
But few, I'd imagine, have ever thought about the math involved. In
fact, the probability of winning at Monopoly
can be described by interesting constructions known as Markov chains.
In the early 1900s the Russian
mathematician Andrey Andreyevich Markov invented a general theory of
probability. I will ignore much of his
work. And I won't review all of Monopoly's rules, but I will convince
you that the game is fair. First, we must
recall how to play it. Players take turns throwing a pair of dice.
The number of dots on the dice determines how
many squares around the board a player may move. A player who throws
a double - say, two l's (snake eyes) - throws
again. All players start from the square labeled GO.
Some rolls, such as 7, naturally happen more often than others. There
are six ways to roll a 7 (1 + 6, 2 + 5, 3 + 4,
4 + 3, 5 + 2, 6 + 1) from 36 possible sums of dots on the dice. So
the probability of a 7 is 6/36, or 1/6. Then come
6 and 8, each having a probability of
5/36; then 5 and 9, having a probability of 1/9. Next, 4 and 10 have
a probability of 1/12; 3 and 11 have a

probability of 1/18; and finally 2 and 12 have a probability of 1/36.
From these values we know that, over the
course of many games, the first player is most likely to land on the
seventh square, a CHANCE square. If he does not
roll a 7, he will probably land on Oriental Avenue or Vermont Avenue,
to either side of CHANCE. Thus, the first
player has an excellent chance of securing one of these properties.
If he does buy one, it lessens the opportunity for
the other players to make a purchase on their first throw.
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