Imagine that you are trying to fill a position (like Secretary of the National B
ID: 3354746 • Letter: I
Question
Imagine that you are trying to fill a position (like Secretary of the National Bank of Blurns) and you know that you have three candidates to interview. Suppose that you know moreover that there is a best candidate (candidate A), a second-best candidate (B), and a worst candidate (C). Your goal, of course, is to hire candidate A. The problem is that you interview the candidates in an unknown (or you might say “random”) order. Here is your strategy: interview and automatically reject some number k of applicants (so k = 0, 1, or 2 in this case) and then pick the first candidate thereafter that is better than all those that came before it.
1. If k = 0, then this strategy boils down to simply picking the first candidate. What is the probability that this strategy picks candidate A? You probably have a good intuitive answer already, but try to write this down carefully. What is the sample space? What is the relevant event in this sample space that you are computing the probability of?
2. Now look at k = 1. What does the strategy mean in this case? What is the sample space and what is the event that corresponds to this strategy picking candidate A? What is the probability of this event?
3. Now do all this for k = 2.
4. In the end, which k yielded the highest probability?
Explanation / Answer
1. For k=0, the interviewer selects the first candidate. Since the candidates get interviewed in a random order, the the probability of A being the first candidate to be interviewed =1/3.
So the probability of A being selected with k=0 is 1/3.
The sample space, S = {A, B, C}
We need to find P(A)
2. This strategy implies rejecting the first candidate and selecting the second one.
Sample space, S = {A, B, C}
The required event is that A gets interviewed at the second place. Since any of A, B or C could be interviewed at the second place, the probability of selecting A = 1/3.
3. In this case, the interviewer rejects the first two candidates and selects the third one.
The sample space, S = {A, B, C}
The probability of A being selected = the probability of A being interviewed at the third place. Since any of the candidates could get interviewed for the 3rd place, the probability of selecting A = 1/3.
4. So we find that all three k-values yield the same probability of selecting A.