The teacher in a certain class room allows you to pass a paper with an assignment around, and whomever it ends up on has to do it. The only two rules are you can't pass it to someone who already has had it and you can only pass it to the person to the left, right, forward, or backward.

In a room of 30 students arranged in a 6 by 5 grid, the teacher starts out with the assignment somewhere on the front row of 6 students. At some point someone is stuck holding the assignment because all his neighbors have had it and passed it on to someone else. If this happens after every student in the room has had it, what is the probablity, for each individual, that he or she turns out to be the lucky winner of the assignment?

If the passing were allowed diagonally, it would be less likely to pass through all the students. In a simulation it took 2,418,340 trials to get to 10,000 in which all students had seen the assignment, so that happened only 1 in 242 trials.  The resulting conditional distribution of last holders is:

  627  297  343  336  312  651
  311  135  140  199  143  298
  406  224  254  256  196  390
  304  155  207  191  143  308
  881  280  402  396  299  916

 1278  609  679
  609  278  339
  796  420  510
  612  298  398
 1797  579  798

Percentages are half of:

13  6  7  7  6 13
 6  3  3  3  3  6
 8  4  5  5  4  8
 6  3  4  4  3  6
18  6  8  8  6 18

And unconditionally (count not only those cases where everyone got the paper):

 1065  340  419  380  327 1083
  292   75  108  114   71  274
  413  109  173  153   92  334
  263   60   96   94   57  249
 1158  212  303  281  228 1177

 2148  667  799
  566  146  222
  747  201  326
  512  117  190
 2335  440  584

21  7  8  8  7 21
 6  1  2  2  1  6
 7  2  3  3  2  7
 5  1  2  2  1  5
23  4  6  6  4 23

Posted by Charlie on 2004-06-25 09:41:07