MONOPOLY -- standard board game. Here are some sci.math posts of interest.
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From: edgar@math.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Monopoly statistic Question
Date: Mon, 16 Jan 1995 09:53:37 -0500

In article <3f9uj0$l3q@odin.community.net>, eldredge@community.net
(Eldredge) wrote:

> : I am doing a report in my calculus clase and I am stumped on this 
> question.What space is landed on the most on a monopoly board? I can't 
> seem to find a stratigy on how to find an logical answer. Any help or an 
> answer would be greatly aprechiated. Thank You.

This is a probability problem.  Moves in Monopoly are (almost) a
Markov chain.  The reference is:
  Ash & Bishop, "Monopoly as a Markov chain", Mathematics Magazine,
   vol 45 (1972) pp. 26--29.

>From their table: assuming you always leave jail immediately, the
most likely square to land on (end your turn on) is Go (probabilily .0368),
the next most likely square is New York Avenue (probability .0366),
the least likely squares are Chance 1 and Chance 3 (probability .0102).

These probabilities can be used to compute other items of interest:
for example, hotels on Boardwalk and Park Place yield
an expected income of $101.40 per opponent's turn, while hotels
on Mediterranean and Baltic yield $17.80 per opponent's turn.
 . . . .
Gerald A. Edgar                  edgar@math.ohio-state.edu
Department of Mathematics
The Ohio State University   telephone: 614-292-0395(Office)
Columbus, OH 43210               614-292-4975 (Math. Dept.) 
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From chrisman@ucdmath.ucdavis.edu Fri Dec 23 14:59:51 CST 1994
Newsgroups: sci.math,rec.games.board
Subject: Re: Puzzler: circular random walk, Monopoly

...
A couple of years ago, I used this technique to analyze the board game
"Monopoly".  A fascinating exercise, which you might give as extra credit to
your honors linear algebra class.  There are a few things to watch out for. 
People spend several turns in jail.  Landing on "chance" often sends you
somewhere else.  If you land on a railroad because of a "chance" card, you
might have to pay double.  Anyway, the results are interesting (and,
amazingly, matched my intuitive "feel" for the game).  

For example, it's not just your imagination that people never land on your
hotels at Park Place and Boardwalk.  These are some of the least-landed-on
properties.  The most-landed-on property (it's one of the reds - Indiana?)
gets landed on almost twice as often as Park Place!  In terms of your
expected income, the oranges are almost as valuable as Park Place/Boardwalk. 
Having all four railroads (if I remember correctly) will bring more income
than hotels on Baltic/Mediterranian.

If anyone is interested, I'll try to find my notes on Monopoly and post them.
 (P.S., if anybody has a new Monopoly game with no missing
Chance/Community-chest cards, could you give me the complete list?  I'm sure
some of mine are missing, which would have perturbed the probabilities
slightly.) 
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From clune@pasta.global Sat Feb  4 03:45:23 CST 1995
Newsgroups: sci.math
Subject: Re: Monopoly statistic Question
...

Actually the problem is a bit more complicated than that.  The cards
that send you to a certain spot also increase the probability of
ending up 7 spaces further on the following turn.
Additionally, these cards are relatively rare events.

At some point a few years ago, a few of my friends had this debate,
so I sat down and computed the probability distribution.  I assumed
that the Chance (and Community Chest) cards came up with probability
1/16 rather than modelling the details of shuffling etc.  I did the
calculation twice once assuming that you paid $50 immediately to get
out of jail and once assuming you would try a few times to get
doubles.  (I even included the fact that 3 doubles send you to jail.)

What I actually computed was the steady state probability
distribution, which can be found as an eigenvector (with eigenvalue 1:
think about it) of the propagation matrix for the game.
I make no claims as to how long one must play to reach this
distribution.  In a real game, the finite play and the starting 
position bias this result and makes the problem much more complicated.

At any rate, the conclusion is that Illinois is the *property* most
often landed on followed by New York  for either assumption about
jail.  Notice that these are not too far after the Jail location.
I unfortunately lost the bet.
Of course my roommate who won the bet had seen the answer in some
book.  At least that gave me some confidence that I had the correct
answer, as the scattering matrix from the cards is a little
tedious.

Unfortunately, the rest of the work is now lost, but I *think* that
jail was indeed the most likely space to end on. 

Another twist:  the probability of ending a turn on a space and of
merely visiting a space on a turn are slightly different.  Not enough
to chang the order among the top spaces however.
...
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