Mathematical Recreations; April 1996; Scientific American Magazine; by Stewart; 2 Page(s)

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 1¿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.