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I was playing a game of monopoly the other day, and in the course of strategizing I came up with the idea that how 'safe' you were in the game was a matter of what your expected income/outcome was as you went around the board. I figured it would be a fun project to make a helper program that would compute these values based on a given board configuration, but I found that the statistics got too complicated for me.

At first I was going to just sum up the amount of potential costs you could have on each tile (excluding Chance and Community Chest cards for simplicity), and dividing by 40, the number of tiles on the board, to get the expected value. But I realized that's not actually the expected value because you're not going to land on every tile, and you definitely won't land on 40 tiles in the course of traversing the board. At this point, I figure, there are several assumptions I could make - average die roll is 7, 40 tiles divided by seven steps is ~5.7 turns, so I could divide the total costs by that for an expected value, but that seems like an overly broad assumption in this case - there could be too much variance in the number of turns to get an accurate projection, I think.

I was wondering if anybody more skilled with statistics could give me a hand with this - what is the expected value of your costs in traversing the monopoly board? I intended on ignoring chance and community chest and being sent to jail, as those seemed to overcomplicate the model, but if somebody can incorporate those effects I'd be interested in seeing them.

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Your initial idea was correct: The expected cost upon traversing the board once is the total cost divided by the expected roll. To see this, imagine advancing along a very long line of squares by rolling two six-sided dice per turn. After a large number $n$ of rolls, you'll have advanced by $7n$ squares plus some deviation of the order of $\sqrt n$, so you will have hit a fraction $\frac17$ of the squares with a deviation of the order of $1/\sqrt n$, which goes to $0$ as $n\to\infty$. Thus, in the long run, every square has a probability $1/7$ to be hit on any given traversal. Thus, by linearity of expectation, you just have to add up the costs and divide by $7$ to get the expected costs per traversal.

(Where you say "divide the total costs by that", it sounds as if "that" refers to the $5.7$ – you need to divide by $7$, the expected roll per turn.)

P.S.: Perhaps I understand now why you wanted to divide by $5.7$ – did you want to calculate the expected cost per turn, not per traversal? In that case, you should divide the total cost of the board by $40$, the number of tiles, since that gives you the average cost per tile and you're going to land on exactly one tile (if we ignore the complication that you roll again when you get doubles).

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  • $\begingroup$ this isn't exactly right because of jail and community chest cards, etc that move your place on the board. I think this article is very instructive $\endgroup$ – MichaelChirico Jul 19 '15 at 13:38
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    $\begingroup$ @MichaelChirico: That's right. I followed the "I intended on ignoring chance and community chest and being sent to jail" part, not the "if somebody can incorporate those effects I'd be interested" part. $\endgroup$ – joriki Jul 19 '15 at 13:47
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    $\begingroup$ yep, just leaving this here for reference and is why I am not posting a separate answer :) $\endgroup$ – MichaelChirico Jul 19 '15 at 13:48

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