彩虹列车
Rainbow Trains
题目详情
火车一共有三节车厢：蓝色、红色、绿色。这列火车有一名售票员，他的工作是断开后面的火车车厢并将它们放在前面。他们可以根据需要多次或多次执行此操作。如果三节车厢随机排列，售票员能将其重新排列成彩虹顺序的概率是多少？ （注：此处，彩虹顺序表示先红，然后绿，然后蓝）
There are three train cars: a blue one, a red one, and a green one. This train has a conductor whose job it is to disconnect the train cars in the back and put them in the front. They can do this as many or as few times as they want. If the three train cars are randomly ordered, what is the probability that the conductor can rearrange it into rainbow order? (Note: Here, rainbow order means red, then green, then blue)
解析
把红、绿、蓝分别记为 1、2、3，则正确的彩虹顺序是 123。售票员每次操作“把尾部若干节接到前面”相当于对顺序做循环移位。
三节车厢共有 种排列，而与 123 同一个循环类的排列只有 3 个：123、231、312。故所求概率为
Original Explanation
Let us denote the colors 1, 2, and 3 such that a correct rainbow ordering of the cars is 123. There are ways to order the three cars, and of those orders, only three preserve the rainbow order of the cars: 123, 231, and 312. Thus, the probability that the conductor can rearrange the order of the cars into rainbow order is: