A man comes home and he wants to open her door. He has 5 keys, two of which are the door keys but he couldn't remember them. Therefore, he tries the 5 keys independently and at random. If a key is unsuccessful, it will be excluded for further selections. Let N be the total number of trials.

(i) Describe the underlying sample space and the set of possible values for N.

(ii) Find the probability mass function and distribution function of N.

(iii) Compute the expectation of N.

(iv) Consider a simpler situation in which there are 3 keys in total (named a,b,c), and only one of them is the correct key, say key a. We would like to compute the probability that the mane opens the door at the second trial. Xi has a solution to this problem as follows:

When performing the random exprientment, the man keeps trying the keys until he reaches the correct key a. Therefore, the sample space is given by Ω {a, ba, ca, bca, cba,} each outcome recording a possibility of trying the keys until reaching the correct key. Since the problem corresponds to the classical probability model over Ω, we conclude that

P(N = 2) = P{ba, ca} = 2/5

Do you think Xi's solution is correct? If not, is it possible to fix it without changing his sample space Ω?