You just rented a large house and the realtor gave you 5 keys, one for each of the doors. They keys look all alike, so to open the front door, you try them at random. What is the PMF of the number of trials you will need to open the door, if after each unsuccessful try, you mark the key so you don

Step-by-step explanation:

The simplest way to look at the problem is the correct key is equally likely to be the first key you try, the second, and so on. So if NN is the number of trials needed, we have Pr (N=1) = Pr (N=2) = Pr (N=3) = Pr (N=4) = Pr (N=5) = 1/5

Perhaps you are using the notation fN (k) for the probability that N=k. Then you could write fN (k) = 1/5 if k=1,2,3,4 or 5, and fN (k) = 0 otherwise.

One can also do the problem more slowly. The probability that the first key opens the door is fairly clearly 1/5, for there is only 1 good key, and all keys are equally likely to be the good one.

In order to open the door with the second key, two things must happen:

(i) The first key you try doesn't work and

(ii) the second one does.

The probability the first doesn't work is 4/5. If the first doesn't work, then the probability the second does is 1/4, since you have marked the first key. So the probability you get a failure and then success is (4/5*1/4). This simplifies to 1/5.

A similar argument shows that the probability of failure on the first two, followed by success, is (4/5*3/4*1/3), which again simplifies to 1/5.

And so on! We get the same answer as with the first solution, but with more work needed.