Rotating train cars into red, green, blue

A train has three cars, coloured red, green and blue, coupled in a uniformly random order. The conductor's only move is to uncouple the last car and attach it at the front, and he may do this any number of times (including zero). What is the probability he can achieve the order red, green, blue from front to back?

$\tfrac12$. Moving the back car to the front is a cyclic rotation, so only the three rotations of red-green-blue (RGB, BRG, GBR) can be fixed. That is 3 of the $3!=6$ equally likely orders.