Question

The mathematics main office has backup keys to the offices in Kiely Hall. One day Dr. Unlucky locked himself out of his office. So he borrowed the keys from the main office. Unfortunately, these keys have no room number on them and thus he has to try one by one. Suppose there are n keys and only one key will open Dr. Unlucky’s office.

(a) If he tries the keys at random and discards those that do not work, what is the probability that he will open the door on his k-th (k=1,2,...,n) try?

(b) What if he does not discard previously tried keys?

I need an answer asap please. Thank you.

Answer 1

a) The probability that the door opens in 1-st try = 1/n.

The probability that the door opens in 2-nd try =

Similarly, the probability that the door opens in k-th try is

Thus the probability that he will open the door on his k-th (k=1,2,...,n) try is 1/n.

b) Now, here he does not discard previously tried keys then in each try the probability that the key is right is 1/n. Thus the probability that the door opens in 1-st try = 1/n

The probability that the door opens in 2-nd try

Similarly, the probability that the door opens in k-th try is

Thus the probability that he will open the door on his k-th (k=1,2,...,n) try is