Statistics

   0  1  2  3  4  5  6  7  8  9 10 
  39                            11
  38                            12
  37                            13
  36                            14
  35                            15
  34                            16
  33                            17
  32                            18
  31                            19
  30 29 28 27 26 25 24 23 22 21 20

In a certain board game, there are 40 numbered squares arranged around the board as shown above. The players alternate turns. On each turn a player rolls 2 six-sided dice and moves forward (clockwise) the sum of the two values. If he lands on square 7, 22, or 36 he draws a card which may direct him to move forward to a specified square or to "go to jail". There are 20 cards, and after a card has been drawn it is replaced and the cards are reshuffled:

• 2 cards say "go to jail"
• 1 card says "go to square 11"
• 1 card says "go to square 24"
• 1 card says "go to square 29"
• 1 card says "go to square 39"
• 14 cards don't cause him to move

If he has not gone to jail and has rolled "doubles" (the 2 dice have the same value) he continues his turn by rolling again and behaving as before. However, if he rolls doubles 3 times on one turn, he does not move the amount shown on the third roll -- instead he "goes to jail". The player's turn ends after he moves on a roll that was not doubles, or when he goes to jail.

One additional rule is that if a player lands on square 30, his turn ends after that move and he goes to jail.

We want to analyze the beginning of a 2 player game. Both players start on square 0. We will pretend that a player is removed from the game when he goes to jail or when he gets past square 39 (possibly by being directed to move by a card). So if a card directs a player to a lower numbered square, the player is removed from the game before landing on that square. If a card directs a player to a higher numbered square, he has landed on both the square where he drew the card and on the square he was directed to.

For a particular square we want to know the probability that the 2nd player will land on it without having the first player land on it earlier. Create a class Monopolies that contains a method probability the is given square and that returns the desired probability.