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?

This was tedious and I probably would not attempt it for a larger boards.

I assumed the teacher handed the assignment to a random student (among those possible) and each student passed the assignment to a random student (if possible)  such that the process did teminate with every student having it.

For the 3x3 the teacher must hand it to either the 1st or 3rd student (because of parity).  Probabilities:
1st and 3rd in first row = 5/32
2nd in 2nd row = 1/8
1st and 3rd in last row = 9/32

For the 3x4 the teacher can hand it to any student in the first row (but if he hands it to the second person in the row and they don't pass it to the first person, that first person will end up with it).  Total probabilities by row:

Front row:  91/384, 19/384, 19/384, 91/384
Middle row:  9/256, 5/96, 5/96, 9/256
Back row:  43/384, 11/768, 11/768, 43/384

As the other subject lines note: this is very difficult (and impractical!)

 

Posted by Jer on 2006-02-27 14:51:44