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.)

Over the course of 266,899 trials, for some reason, it took on average 29.04 rolls to get over 100, rather than the 100/3.5 = 28.57 I had expected.  Maybe it has to do with 29.04*3.5 = 101.64, while on average the final toss takes the total to 103.5.

The other statistics are:

   avg # of tosses                  avg positions of
1playr 2playrs 3playrs 4playrs    2nd pl 3rd pl 4th pl  trials



29.04  27.58   26.87   26.43      94.84  89.80  83.43   266899

If two players reached the finish in the same turn, 2nd place was considered to be in position 100 regardless of how much the roll was in excess of the number needed.

DECLARE FUNCTION min# (x#, y#)
RANDOMIZE TIMER
DEFDBL A-Z
CLS
DO
 trial = trial + 1
 REDIM p(4)
 nOver = 0: turn = 0
 DO
   turn = turn + 1
   FOR i = 1 TO 4
    r(i) = INT(RND(1) * 6 + 1)
   NEXT
   FOR i = 1 TO 4
    IF p(i) < 100 THEN
      p(i) = p(i) + r(i)
      IF p(i) >= 100 THEN
       nOver = nOver + 1
       over(nOver) = turn
       overWho(nOver) = i
       IF nOver = 1 THEN
         REDIM pList(3): plp = 0
         FOR j = 1 TO 4
           IF j < i THEN
            FOR k = 1 TO plp
             IF pList(k) < p(j) THEN EXIT FOR
            NEXT
            FOR l = plp TO k STEP -1
              pList(l + 1) = pList(l)
            NEXT
            pList(k) = p(j)
            plp = plp + 1
           ELSEIF j > i THEN
            FOR k = 1 TO plp
             IF pList(k) < min(100, p(j) + r(j)) THEN EXIT FOR
            NEXT
            FOR l = plp TO k STEP -1
              pList(l + 1) = pList(l)
            NEXT
            pList(k) = min(100, p(j) + r(j))
            plp = plp + 1
           END IF
         NEXT
       END IF
      END IF
    END IF
   NEXT
 LOOP UNTIL p(1) >= 100 AND p(2) >= 100 AND p(3) >= 100 AND p(4) >= 100
 REDIM pers(4)
 FOR numb = 1 TO 4
   FOR j = overWho(numb) TO 4
    IF pers(j) = 0 THEN pers(j) = over(numb): tot(j) = tot(j) + over(numb)
   NEXT
 NEXT
 FOR pl = 1 TO 3
   totProg(pl + 1) = totProg(pl + 1) + pList(pl)
 NEXT

 'FOR i = 1 TO 4
 '  PRINT over(i); overWho(i)
 'NEXT

 PRINT USING "####.##"; tot(1) / trial; tot(2) / trial; tot(3) / trial; tot(4) / trial;
 PRINT USING "####.##"; totProg(2) / trial; totProg(3) / trial; totProg(4) / trial;
 PRINT trial
 REM
LOOP

FUNCTION min (x, y)
 IF x < y THEN min = x ELSE min = y
END FUNCTION

 

Posted by Charlie on 2007-06-19 12:03:28