2. Here\'s an oversimplified version of a common problem for personnel managers

Question

2.

Here's an oversimplified version of a common problem for personnel managers that nevertheless contains elements of realism. You've advertised an open position in your organization, and n 1 candidates have put their names forward for consideration. You want to hire the best candidate, but before interviewing any of them suppose that their resumes don't provide strong information with which to create a rankingeach of them in your judgment has equal probability of being the best. It would be great if you could just interview all n of them, because you would then know for sure who's best, but (as with the tech sector, for example) this is a fast-moving hiring environment (by the time you get to the end and figure out that (say) candidate 3 is best, that person has probably already taken another job), so you need to be adaptive. Here are the ground rules: ·Once the interviews start, you can rank the candidates you've already seen, but you'll have no information about how the remaining candidates will fit into the ranking; and . After each interview (because of the fast-moving environment), you either immediately hire the candidate you've just seen (and stop the interviewing process) or let that candidate go, with no opportunity to call her or him back. Here's the adaptive strategy you've decided to use:

Explanation / Answer

According to the rule that we are following we have the following guidelines:

Now when we are interviewing i>r candidates. Here total number of candidates that we have interviewed is “i”. Now we are required to find the probability that {the best candidate among the first i people interviewed is one of the first r people}. The number of ways this can happen is “r” (this is intuitive if you think about it, if the best candidate has to be from the group of first r candidates then this can happen only in r ways).

So, going by the definition of probability we have

=Number of favourable outcomes/Number of possible outcomes

=r/i

This is our required probability that the best candidate among the first i people interviewed is one of the first r people.

Now our two events A and B are defined as follows:

A-You hire the best candidate

B-The best candidate is person i in the interviewing sequence

Now, when we consider the probability P(A|Bi) for i<r, it means that we hire the best candidate given the best candidate interviewed was among the first r candidates interviewed in the sequence. But according to the rules that we have decided initially (refer to bullet point 1), it means that we do not hire the first r candidates that were interviewed. So, if the best candidate happens to be interviewed in the lot of first r candidate we do not hire him. In that case event A will not happen, that is we will not be able to hire the best candidate. Thus, P(A|Bi) = 0 for i<r.

If we now consider P(A|Bi) for i>r that is we hire the best candidate given the best candidate was interviewed after the r trial candidates. Now when i>r it means that the best candidate did not happen to be interview till our trial number of candidates r. This means that r candidates have already been interviewed and they are not the best candidate. Now in this situation our best candidate could be (r+1)th person or (r+2)th person …….nth person. Now if we define an event Ci which says that we keep interviewing till we find our candidate i then we have the following

P(Ci) = r/(r+1) x (r+1)/(r+2) x (r+1)/(r+2) x …….x (i-2)/(i-1) = r/(i-1)

Now, if we set r=0, that is we say that we will not interview any candidate without an intention to hire them and our event A is defined as “You hire the best candidate”. If you think logically, there is only one candidate among the n applications who will be the best. So, if we do not reject any candidate callously the probability that “You hire the best candidate” becomes p0=P(A)=1/n.

The second part of this is intuitive as if 0<r<n, then there are some r candidates which will be interviewed on a random basis and will not be selected if event A has to happen that is if we are required to select our best candidate it is possible only if the best candidate i is happened to be interviewed after r candidates. Now going by the even Ci defined previously, we will keep interviewing till we find our candidate i so the summation will be from (r+1) to n for the values of i as we can chance upon our ideal candidate at trial r+1, r+2,…….n. So we consider our event Ci and if r candidates are already interviewed r/n candidates are already ruled out So, P(A)=pr=r/nni=r+11/i-1

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