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.)

See my previous post for set up.

Into L3 I put the probabilty a player being slower than X (i.e. not having won yet) normalcdf(0,100,X*3.5,sqrt(X*35/12))

With two players to have a winner on turn X, there is either a tie, player 1 wins while player 2 is slower, or plare 1 is slower and player 2 wins.  Into L4 goes L2^2+2*L2*L3

The maximum p(28)=.191016
mean = 27.715132
stdev = 2.061973

For three players L5 = L2^3 + 3*L2^2*L3 + 3*L2*L3^2

Maximum p(27)=.219940
mean = 27.010277
stdev = 1.819931

For four players L6 = L2^4 + 4*L2^3*L3 + 6*L2^2*L3^2 + 4*L2*L3^3

Maxmum p(27)=.232752
mean = 26.563970
stdev = 1.678504

The thing I'm not sure of in both posts is whether I should have made the cutoff for winning 99.5 rather than 100 even.  Any thoughts?

Posted by Jer on 2007-06-22 19:52:03