I have encountered some issues with the following question and have also seen the solution to it, but yet don't seem to understand why.
An employer is about to hire one new employee from a group of N candidates, whose future potential can be rated on a scale from 1 to N. The employer proceeds according to the following rules: (a) Each candidate is seen in succession (in random order) and a decision is made whether to hire the candidate. (b) Having rejected m-1 candidates (m>1), the employer can hire the mth candidate only if the mth candidate is better than the previous m-1. Suppose a candidate is hired on the ith trial. What is the probability that the best candidate was hired?
I am attaching the solution provided in the solutions manual. I will share why I don't understand this solution
If we suppose that there are three candidates A, B, and C who are ranked as 1, 2, and 3 respectively. Then there are 6 ways (3!) for them to be interviewed which are:
$1. A,B,C$
$2. A,C,B$
$3. B,A,C$
$4. B,C,A$
$5. C,A,B$
$6. C,B,A$
I may be wrong in setting it up this way already, but my thought process was the following.
If we assume that the second candidate was hired and we want to find the probability that they were the best of the $3$ (in this example candidate C), then what I did was select the options that were still "Valid". For example, we cannot say that options $3, 5$ and $6$ are valid because the second candidate interviewed was worse than the first one. So we only have three valid opitons remaining ($1, 2$ and $4$). Since they are all equiprobable (if they are not please let me know why not, need to have hope in humanity restored) then the probability of having the selected the best candidate would be $2/3$.
If we were to follow the answer given in the solution it would be $1/2$.
Thanks for any help a good fellow citizen can provide, I believe that my interpretation may be incorrect and don't know what other approach to consider since it seems that there is not much to grab on to.
It is in the first chapter of the book, page 41 (exercise 1.32) of Statistical Inference (2nd edition) by Geroge Casella and Roger L. Berger
I believe the answer in general is i/N. Consider these events:
A: The event that the best of candidates 1 ... i is at location i.
B: The event that the best of candidates 1 ... N is at location i.
We are asked to find P(B|A). This is P(A|B)P(B)/P(A). However, B=>A, therefore P(A|B)=1. Now plug in P(A)=1/i, P(B)=1/N.