Question

An absent-minded professor has n keys in his pocket of which only one (he does not remember which one) fits his office door. He picks a key at random and tries it on his door. If that does not work, he picks a key again to try, and so on until the door unlocks. Let X denote the number of keys that he tries. Find the pmf of X in the following two cases: (a) A key that does not work is put back in his pocket so that when he picks another key, all n keys are equally likely to be picked (sampling with replacement). (b) A key that does not work is put in his briefcase so that when he picks another key, he picks at random from those remaining in his pocket (sampling without replacement).

Answer 1

(A) since pmf is with replacement then the chance is given by ;

Pmf=X/n

(B) pmf without replacement is given by:

Pmf=X/(n-1)

Explanation:

Where x is the number of keys tried, n is the total number of keys.

Probability is given by : required outcome/total number of possible outcome