Probabilty question

Question

You have a bunch of n keys of which only one one opens the door of a storeroom, You wish to get into the storeroom. You choose one key at random and try it. If it does not work, you discard and try another key at random from the remaining keys. You proceed in this way until you open the door. Find the probability that you will open the door in the (i) 1st attempt, (ii) 2nd attempt, (iii) 3rd attempt. What do you think the probability will be for the kth attempt, where 4 ≤ k ≤ n? Can you justify it?

My answers were: (i) $\frac{1}{n}$

(ii) $\frac{1}{n-1}$

(iii)$\frac{1}{n-2}$

and for the kth attempt: $\frac{1}{n-k-1}$

However, it is given that the probability for the kth attempt is just $\frac{1}{n}$. why is that so?

Appreciate any help!

Answer 1

Before trying them out give each key a different number in $\{1,\dots,n\}$. The key with e.g. number $2$ will be tried out at the second attempt. The probability that the right key gets number $k$ (corresponding with $k$-th attempt) is: $$\frac{1}{n}$$ So this is the probability that door will be opened at the $k$-th attempt.

Answer 2

The probability you open on the second attempt is the probability you fail to open on the first attempt multiplied by the conditional probability you succeed on the second, i.e. $$\frac{n-1}{n} \times \frac{1}{n-1}=\frac{1}{n}$$ and this pattern continues, so by induction the probability you open on the $k$th attempt is the probability you fail to open on the first $k-1$ attempts multiplied by the conditional probability you succeed on the $k$th, i.e. $$\frac{n-k+1}{n} \times \frac{1}{n-k+1}=\frac{1}{n}.$$

A simpler approach is to order the keys at random at the beginning. The key which opens the lock has a probability of $\frac1n$ of being in each position.