![]() Percentages, for landing on any of the squares of the game. One or two in the third decimal place, but it would still beīelow are the two tables of probabilities, expressed as I don'tĮxpect it would change any of the probabilities by more than Time and inclination I may redo this in that way. ![]() The exact probabilities would be generated and there wouldīe no need for doing simulation at all. This much larger matrix would insure that The one before, and the third for having rolled doubles the Would be for having rolled doubles on the last roll but not Not having rolled doubles in the last roll. The first would be for being on that square ![]() Had created a larger Markov matrix with three entries forĮach square. Realized that there is a more efficient way of making the Just as I finished putting together this web page, I Utility and the jail strategy, which affects the rent value. Additionally, the average roll when landing onĪ utility is a bit lower or higher than 7 depending on the Slightly different on certain squares for the two jail The probabilities of two previous rolls being doubles is I simulated 32 billion rolls to make theseĮstimates, so I believe they are reliable and any deviationįrom their exact values is extremely small. Simulation program to find the empirical probability forĮach square and then used these values in my Markov matrix Square and it's not that high for any square. First I used anĮstimate of 1/36, but in practice it's different for each The last two rolls of the dice are doubles (since threeĭoubles in a row sends you to jail). Necessary to estimate the probability-for each square-that Probabilities exactly using the Markov matrix, it is In the process of figuring all of this out I ran into an Me to believe that these results are reliable. The Markov one and the results are very close, which leads I have compared the results from my simulation program and I computed probabilitiesįor both cases and have included them here. The board because the more time you spend in jail, the less Strategy changes the probabilities for all of the squares of Game, it is best to stay in jail as long as possible toĪvoid landing on an opponent's property. To have more opportunities to buy property. Game it is often best to get out as soon as possible so as Jail Free card, but you can also wait until you roll doubles Get out immediately by paying $50 or by using a Get out of I discovered that it is really necessary to model twoĭifferent strategies. I used an extended version of this program to Probabilities using a Markov matrix, which was the methodĭescribed in a simplified form in the Scientific AmericanĪrticle. Although this gave good aproximate answers, I Rules for going to jail and the Chance and Community ChestĬards. Simulates a single person rolling the dice and moving around I'm not much of a Monopoly® player myself, but I'veĪlways enjoyed interesting problems involving probability I ran into some interesting problems but finallyĬame up with the right answers, which you will find hereĪlong with some other useful derived data. The different squares with all of the rules taken intoĪccount. Working on trying to find the probabilities for landing on I was intrigued enough with this problem that I started Without considering the effects of the Chance and CommunityĬhest cards or of the various ways of being sent to jail. Of landing on the various squares in the game of Expected income per roll for other squaresĪpril 1996 issue with additional information in the Augustġ996 and April 1997 issues) that discussed the probabilities.Expected income per roll for each property.Long term probabilities for each square.
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