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Marble-Go-Round (Posted on 2004-12-28) Difficulty: 4 of 5
Consider N holes arranged in a circle on a wooden board. A marble is placed in one of them. You toss a fair coin to determine if you should move the marble one hole clockwise or one hole counterclockwise. You keep doing this until the marble has been in each hole at least once.

What is the probability that each of the N holes turns out to be the last hole visited by the marble? Number the holes 1 through N, clockwise starting with the hole in which the marble starts. Obviously the probability for hole 1 is zero, since it already has the marble and there are other holes to visit still.

No Solution Yet Submitted by neshal    
Rating: 3.2000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): solution (spoiler) | Comment 3 of 4 |
(In reply to re: solution (spoiler) by Hugo)

The results I got for 100,000 trials (I didn't count the tosses) are:

 1             0
 2             11144
 3             11049
 4             11198
 5             11073
 6             11020
 7             11089
 8             11141
 9             11115
 10            11171

with the program:

DIM answer(10)
FOR trial = 1 TO 100000
  tr = tr + 1
  REDIM visited(10)
  visited(1) = 1
  visitCt = 1
  posn = 1
  DO
   dir = (INT(2 * RND(1)) - .5) * 2
   posn = posn + dir
   IF posn < 1 THEN posn = posn + 10
   IF posn > 10 THEN posn = posn - 10
   IF visited(posn) = 0 THEN
     visited(posn) = 1
     visitCt = visitCt + 1
   END IF
  LOOP UNTIL visitCt = 10
  answer(posn) = answer(posn) + 1
NEXT
FOR i = 1 TO 10
   PRINT i, answer(i)
NEXT
PRINT

 


  Posted by Charlie on 2005-01-02 17:33:23
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