Answer to Question #373 in Statistics and Probability for Jaan

Question

There are n randomly placed books on a shell with the two-volume edition of Jack London among them. Suppose that the different book arrangements are equiprobable. What is the probability that both volumes are located near each other?

Answer 1

There are n! possible ways to arrange n books on the shell. Two books together (Vol.1, Vol.2) can be placed among these n books in (n-1) different ways (Vol.1 takes places 1..(n-1), Vol.2 takes places 2..n). According to the task, the order doesn’t matter ((Vol.1, Vol.2)~(Vol.2, Vol.1), they just have to be placed together), then there are 2*(n-1) different ways to do that. Thus, the probability that both volumes are located near each other is: P = 2*(n-1)/n!