In a simplified version of a child’s board game, players "race" around the board going from Start to Finish, moving their tokens based on the roll of a six sided die. The board has 101 spaces where the Start space represents "0" and the Finish space represents "100". An exact roll is not required to cross the finish line. To ensure that one child does not have an advantage, all the children roll the die separately, but move simultaneously.

Part I. With one child playing, how many turns (on average) will it take to have a winner?
With 2, 3, or 4 children playing, how many turns will it take (on average) to have a winner?

Part II. With 4 children playing, when one of the children crosses the finish line, how far along should the child in 2nd place, 3rd place, and 4th place be? (Again, looking for the statistical average position.)

Part III. What would be the solution to the previous questions if an exact roll is required to cross the Finish line? (If the roll is too high, the player loses a turn and does not move.)

(In reply to monte carlo --parts I and II by Charlie)

The same stats for part III come out:

 

   avg # of tosses (rolls)          avg positions of
1playr 2playrs 3playrs 4playrs    2nd pl 3rd pl 4th pl  trials
33.31  30.16   28.92   28.21      97.45  94.07  88.65   267917

The changed line of the program is:

   IF p(i) < 100 and p(i) + r(i) <= 100 THEN

as r(i) is the amount to be added this turn.

Posted by Charlie on 2007-06-19 12:20:07