Question

3. (12pts) 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) (6pts) 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) (6pts) What if he does not discard previously tried keys?

Answer 1

3.

a.

in first k-1 tries he should not get the key

P(in first k-1 tries he should not get the key)

= (n-1)P(k-1) / (n)P(k-1)

P(get key in k th try)

= 1 / remaining keys

= 1 / (n-k+1)

P(get key on kth try) = ((n-1)P(k-1) / (n)P(k-1))*(1 / (n-k+1))

= (n-1)P(k-1) / (n)P(k)

= [ (n-1)! / (n-k+1)! ] / [ n! / (n-k)! ]

= (n-k) / (n)

b.

if doesn't discard :

P(correct key) = 1/n

P(incorrect key) = (n-1)/n

P(get correct key on kth key) = ((n-1)/n)^(k-1) * (1/n)

= (n-1)^(k-1) / n^k

(please UPVOTE)