# What is the probability?

1. Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly, in a ring. Determine the probability that no two red marbles are adjacent. 2. Six people in chemistry class are going to be placed in three groups of two. The teacher asks each student to write down the name of that... show more 1. Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly, in a ring. Determine the probability that no two red marbles are adjacent.

2. Six people in chemistry class are going to be placed in three groups of two. The teacher asks each student to write down the name of that student’s desired partner. If each student picks a favorite partner at random, what is the probability that there is at least one pair of students who have picked each other?
Update: For the first problem, the objects are arranged in a ring, not a linear configuration. I assume the problem does take invariance under rotation into account.
Update 2: I'll throw in my thoughts on the first problem: If there are twelve fixed (and distinct) spots around the ring, we cannot assume invariance under rotation anymore, and 7/33 looks right to me. If we are to put the 12 objects on the continuous circumference, I think the assumption of rotational equivalences is... show more I'll throw in my thoughts on the first problem:
If there are twelve fixed (and distinct) spots around the ring, we cannot assume invariance under rotation anymore, and 7/33 looks right to me.
If we are to put the 12 objects on the continuous circumference, I think the assumption of rotational equivalences is necessary, and Duke's answer (and my original answer) of 10/43 looks correct.

Is my argument sound? I didn't realize whether we are treating the ring as a continuous circumference will make such a difference.

By the way this is problem 6 at
http://web.mit.edu/hmmt/www/datafiles/so...