A man has n keys on a key ring, one of which opens the door to his apartment. Having celebrated a bit too much one evening, he returns home only to find himself unable to distinguish one key from another. Resourceful, he works out a fiendishly clever plan: He will choose a key at random and try it. If it fails to open the door, he will discard it and choose at random one of the remaining n−1 keys, and so on. Clearly, the probability that he gains entrance with the first key he selects is 1/n. Show that the probability the door opens with the third key he tries is also 1/n.