Question

Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly, in a ring. How do you determine the probability that no two red marbles are adjacent?

Answer 1

For circular arrangements one blue marble is placed in a fixed position (say-1). Then remaining 7 indistinct blue marbles and 4 indistinct red marbles, totaling 12 marbles can be arranged in a ring in

`((12-1)!)/(7!xx4!)=330` ways.

So this represents the possible number of events.

Now after placing 8 blue marbles there exists 8 gaps (shown in red mark in the fig) where 4 indistinct red marbles can be placed so that no two red marbles are adjacent.
The number arrangements in placing 4 red marbles in 8 places will be

`(""^8P_4)/(4!)=(8!)/(4!xx4!)=70`

This will be the favorable number of events.

Hence the required probability

`P=" the favorable number of events"/"the possible number of events"=70/330=7/33`