monopoly

[livn4Him24/7]

I have started a science fair project to find which squares are landed on the most in the game of Monopoly. I have done a lot of research in probabilities and statistics and have decided that they are beyond my understanding right now. I know that others have calculated the probalities of landing on squares using computer models. Since I can't do that yet, is it reasonable to say that I want to find out which squares are landed on the most by rolling the dice lots of times and gathering data on how many times each square is landed on (manually instead of calculated)?
If so, then how do I know how many times I have to roll the dice? My best guess at this point is to say 500 rolls is a "game", and play 100 "games".
I am planning to keep finding the average for each square after every "game", graph the averages for each square, add a trendline on each graph, and stop when the majority of the trendlines flatline.
Am I on the right track?

[ChrisG]

Hi,
Welcome to the Ask an Expert forums. You've come up with an interesting and creative project idea. I do think you are on the right track. This type of analysis is sometimes called a Monte Carlo simulation. Rather than actually rolling dice, it would be much faster to use a spreadsheet program like Excel, or Open Office Calc (which is free). It would be fairly straight forward to set up an equation that would give a random dice roll for 2 six sided dice, and another equation(s) to track the movement of the player piece around the board. I'm sure our experts could help you with that.
If the spreadsheet is not an option, then the dice rolling will work also. It just might take a long time to roll the dice and record the results. If each round of rolling, moving the piece and recording the result takes 10 seconds, then 100 games of 500 rolls would take about 139 hours(!).
I hope that helps,
Chris

[Craig_Bridge]

If there were no "Chance" and "Community Chest" cards that directed the player to physically move and no Go To Jail square, then the probability of landing on any particular square would approach an even distribution. On the first move of each player, the probability of landing on squares a long distance from the starting point is obviously zero so it can't start out as an even distribution. The three doubles in a row that sends you to jail is also something that biases the probability, in this case it decreases the probability suares on the first side of the board.

I don't think you can easily build an accurate simulation without simulating the chance and community chest cards as well.

I'm hard pressed to come up with an opinion on whether you have to simulate the money, property, and property improvement transactions and number of players. Obviously, the end of the game is determined by these. The faster the end game occurs, the more biased the statistics will be to toward the starting probabilities. The number of players will also determine the probability of property improvements occuring quickly which will affect the length of the game.

These are just some my thoughts on this rather complex probability space and its various distribution deforming characteristics. It isn't a simple one!

[livn4Him24/7]

Thanks so much for all your input. I know nothing of computer programming [though at some point I intend to learn], so the computer simulation is a nice idea but not applicable at this point. As far as that's concerned, though, I think one would have to add the chance and community chest cards in somehow. As for game length, I read somewhere (and yes, it's referenced in my bibliography) that going through both decks once consititutes the half-way point, so going through both decks twice is about the length of an average game. 500 rolls per game usually gets me through the decks twice.

I've already started rolling dice. My mom is helping me and we've gotten this thing down to a 3-seconds-a-roll fine art. I'm only rolling the dice and moving around the board, not buying property, etc. I've pretty much finished my background information but I will try to look up Monte Carlo simulations. I know enough Excel to make a chart of my data and I'm graphing the numbers of times landed on per square and the averages for each square.

Thanks!

[ChrisG]

Hi,
To use a computer wouldn't involve any actual programming, just learning a handful of excel functions, such as simple math and logical "if" statements. I put together an example procedure below so you could see what I'm talking about. I just wrote this off the top of my head, so there might be some typos or logic errors, but if you can use this (maybe with our help of with the help of someone who has a little experience with Excel) I think it would save you many many hours of rolling dice, and from ending up with inconclusive results. I implemented the formulas in Excel (without the cards) and ran 50,000 rolls. Even after 50,000 rolls (about 40 hours of your and your mom's work at 3 seconds per roll) the results were fairly noisy and difficult to interpret. It took about 250,000 rolls before the trends in probability among board squares became smooth and convincing (and interesting!)

***********************

This equation:
=INT(RAND()*6)+1
will give you a random number from 1 to 6.

In column A row 3 you would place that equation for the first dice roll. In column B row 3 you would place the same equation for the second dice roll.

Column C row 3 would be the total dice roll, for example:
=A3+B3

Column D row 3 could be a preliminary move, where you first move the piece by the number rolled in C3.
=C3+H2
(Column H will contain the starting square from the previous turn. For row 2 it should be zero, to start the game. Each row will represent a roll of the dice)

Column E row 3 would be an equation to check if you have passed Go (assuming that Go is square 0 and the rest of the squares are numbered 1 to 39).
=IF(D3>39,D3-40,D3)

Column F row 3 would be an equation to check if you have been sent to jail. Square 30 is the go to jail square, and square 10 is the jail cell.
=IF(E3=30,10,E3)

Column G row 3 could be an equation to check for a card draw and to give a result of any place move resulting from the card draw. I don't have the cards handy, but this might involve assigning a movement to each card, shuffling those cards, and then choosing them sequentially. It would require a lookup table function. For a grade 6-8 project, I would see this part of it as optional, though certainly worth doing.

In column H row 3 you could check for the end of the game. There are various ways to do this. I just used your criterion of 500 rolls. If the row number minus 2 is a multiple of 500, the formula sends you back to square zero to start another game.
=IF(MOD(ROW(F3)-2,500)=0,0,F3)


To simulate 100 games of 500 rolls, select row 3 and cut and paste onto rows 4 through 50002 (this is just one operation, you don't have to do it thousands of times).

The final step is to add a separate table 2 columns by 40 rows in dimension. The first column is the numbers 1 to 40. The second column uses the equation
=COUNTIF($h$3:$h$50002,I3)
Where $h$3:$h$5003 are the cells containing your roll values, and H3 is the square number in the other row. This formula adds up all occurrences of your piece landing on that particular square.

By cutting and "paste-values" the second column in the table above into adjacent columns, it quickly simulates 50000, then 100000, then 150000, then 200000 etc rolls.