Three candidates

Probability Level 3

It is time to hire a new ankliopestologist, and you have 3 candidates to choose from. You can only interview them one at a time, and your company has a policy that as soon as an interview is over you must decide on the spot whether you will hire them or not, and can't change your mind after the fact. Only once you have rejected a candidate can you move onto the next one.

Given the optimal strategy, what is the probability you can hire the best one?

Assumption: The relative rankings of the candidates are all equiprobable, they each have a definite ranking from best to worst (no ties) that you can determine from the interview questions.

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Inspiration

5/6 1/3 1/2 1 2/3

2 solutions

Geoff Pilling
Oct 25, 2018

If you pick the first candidate right off the bat, you have a 1/3 chance of picking the best one. However, if you wait for the second one.... If he interviews better than the first, you can make him another offer and you have a 2/3 chance of getting the best candidate. Alternatively, if he interviews worse, then you can pass and you have a 1/3 chance of the last one being the best. So, the probability of finding the best candidate is:

$P = \dfrac{1}{2}\cdot\dfrac{2}{3} + \dfrac{1}{2}\cdot\dfrac{1}{3} = \boxed{\dfrac{1}{2}}$

Fun problem! I'd never heard of the Secretary problem , so thanks (to you and Brian Lie) for introducing me to it. I wouldn't be surprised if this ends up being a POTW at some point.

P.S.. Ankliopestologist? Once again, like with "Frabduzella", absolutely no Google hits; that's quite a feat. :)

Brian Charlesworth - 2 years, 7 months ago

Hahahaha.... Thanks, that would be cool! I thought it was interesting that no matter how many people you have, you can always do better than 1/e...

Geoff Pilling - 2 years, 7 months ago

Brian, an "Ankliopestologist" is an expert on how to rid of a specific pest, which in this case is a "not-klio", or a "not-the-most-amazing-person", usually thought of as a woman modeled after the Greek Goddess and Muse of History, Clio. In other words, an Ankliopestologist helps you rid of certain kinds of annoying people that aren't so selfless and loving. I think in these days, we need more specialists like that.

Michael Mendrin - 2 years, 7 months ago

Yes. That was exactly the meaning I had intended! 😎

Geoff Pilling - 2 years, 7 months ago

Hahahaha I shoulda known it was some sort of Greek god thing. :D

Apparently, but not surprisingly, the Clio Awards are presented for excellence in advertising. :/

Brian Charlesworth - 2 years, 7 months ago

@Brian Charlesworth – Hey, you know, you're right.. I do recall that now. I recall having a friend decades ago who collected Clio publications.

Michael Mendrin - 2 years, 7 months ago

My pleasure, sir. You may enjoy Restaurant Problem , too.

Brian Lie - 2 years, 6 months ago

Jeremy Galvagni
Oct 27, 2018

Adopt the strategy of not picking the first candidate and then picking the second only if they are better than the first. If the second is worse than the first you will end up with the third.

There are $3!=6$ orderings of the candidates and I've ranked them $1<2<3$ . In the chart below I've boxed the orderings where $3$ is picked.

1	2	3
1	$\boxed{3}$	2
2	1	$\boxed{3}$
2	$\boxed{3}$	1
3	1	2
3	2	1

We see we get the best candidate in $3$ out of $6$ orderings so the probability is $\boxed{\frac{1}{2}}$

Incidentally, we end up with 2 in two of the six and 1 in one of the six.

Its funny... I drew this exact same figure on paper when I first solved it... :)

Geoff Pilling - 2 years, 7 months ago

Three candidates

2 solutions

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