A man has n keys on a key ring, one of which opens the door to his apartment. Ha

Question

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. (Hint: what has to happen before he even gets to the third key?) 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. (Hint: what has to happen before he even gets to the third key?)

Explanation / Answer

First key:

Total Number of keys available = n

Probability of opening = 1/n

Second key:

Total number of keys available = n-1

Probability of opening with second key = 1/(n-1).

But, since the first attempt is a failure with probability = (n-1)/n,

Probability of opening = 1/(n-1) X (n-1)/n = 1/n

Third key:

Total number of keys available = n -2

Probability of opening with third key = 1/(n-2)

But, since the first ttempt is a failure with probability = (n-1)/n

and the second attempt is a failure with probability = (n-2)/(n-1),

Probability of openng = 1/(n-2) X (n-1)/n X (n-2)/(n-1) =1/n

Thus, the probability the door opens with the third key he tries is also 1/n.

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