Imagine that you are trying to fill a position (like Secretary of the National B
ID: 3355009 • 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
Different strategies involve different values of k: {0,1,2}
---- Part (1)
For k = 0, the probability that the first candidate is the best candidate i.e. candidate A = (1C1)/(3C1) = 1/3 [Answer]
The event is the first candidate and the sample space is the following: {A,B,C) and favourable conditions: {A}
---- Part (2)
For k = 1, the event is the first two candidates and the sample space is the following: {AB,AC,BA,BC,CA,CB} and favourable conditions: {BA,CA} => P(k=1 works) = 2/6 = 1/3 [Answer]
---- Part (3)
For k = 2, the event is the all 3 candidates and the sample space is the following: {ABC,ACB,BAC,BCA,CAB,CBA} and favourable conditions: {BCA,CBA} => P(k=2 works) = 2/6 = 1/3 [Answer]
---- Part (4)
What this means is that all 3 strategies work equally well to get the best candidate to be chosen for the role.