Hi! I have one rather unusual question. Couple days ago, me and my friends played a game of monopoly. And something happened that you don’t see every day: my friend landed on the same jail box every single time, he just couldn’t go past jail tile 12 times in a row. I want to ask, whats the mathematical probabilities of this event happening to someone again?
We're looking for the probability of starting in Jail, moving exactly 20 spaces around the board to land on Go To Jail, and repeating 12 times.
To approximate the likelihood of moving exactly 20 spaces over an indeterminate number of rolls, I simulated 10 dice rolls (the max it would take to move 20 spaces), and checked whether the cumulative sum was 20 spaces at any point. I repeated this 1 million times, and found that a person rolling two dice will move exactly 20 spaces about 14.1% of the time. That is, when starting from Jail and rolling normally, you stand a 14.1% chance of landing on Go To Jail before making it any further.
The likelihood of doing this 12 times is simply 0.141 ^ 12, or 6.17 x 10^-11, somewhat less likely than one in ten billion. It's extremely unlikely that this would happen to you, although given that there are a quarter billion Monopoly sets, each of which is possibly played multiple times by multiple people with multiple trips around the board, it's reasonably likely that this has happened to someone in the history of Monopoly.
These probabilities assume that the player pays to get out of jail and is rolling normally, and does not require doubles to escape jail. The distribution of likely spaces will change somewhat if we require that the first roll must be doubles. I also don't account for any other means of going to jail, like rolling doubles three times in a row, or getting a Chance card that sends you there. It also does not account for any Chance/Community Chest cards that would move you elsewhere on the board, as it assumes that all movement is done as a result of the dice roll.