
Re: (difficult)Theoretical gambling puzzle
Posted:
Jul 24, 2006 8:27 AM


Hi Earle, thanks for the explanation. I can see the similarities but also the differences:
1. The "Interview" appears to have a infinite number of possible drawings, so I suspect there are some kind of limitation (maybe a flexible budget in proportion to the quality of the one to be hired, or more probable, a fixed maximum limit to the number of interviews). Different from the 52 cards problem, it has a fixed outcome probability for each interview, although we don't know what. I suppose in real life we could assume some kind of statistical distribution to rule the outcomes, normal or bell curve cames to my mind, but this is not sure (if our candidates must know how to do something, then the distribution can be truncated, because that's what schools with admission criteria do when they select their students). In this is also different from the 52 card game, we don't know the distribution, and that's the reason why the ball and urns appears to be a better representative of the interview problem: it doesn't distract us with reasonable assumptions of distributions.
2. In the ball and urns, the number of balls is clearly finite. It is also clear that we can not have any clue about the statistical distribution of his outcomes, only about its order (the "orderable" thing). Each outcome is totally unpredictable but not the order, which is interesting. I think I'm going to attack this problem, next opportunity. I'm not convinced that the best strategy is to define a fixed percentage as a first sampling, but only as first process step. The goal, of course, will be to maximize the expected order position of the value in the paper.
3. In the 52 cards, we know the distribution of its values and we are able to draw the probabilities at each step. The fundamental difference is, we are able to recognize a rare positive result if it happens in the first steps. I think this sums all. "marriage" puzzle appears to be a suitable name, because I don't know any who would do interviews like that (they choose the best from the interviewed pool)... but that is maybe my southEuropean mentality talking (where marriages is supposedly for life). I wish luck to find the results (my firsts has been a disaster).
Cheers, Joao Pedro Afonso
Wrote Earle Jones: > Joao Pedro Afonso wrote: > > Wrote Earle Jones: > > > It has some similarities to the "Interview" > > > puzzle posted here some time back. > > > > Hi Earle, could you provide with some dates for > the appearance of that > > puzzle? I searched the MathForum.org database, > but didn't found anything. > > > > Thanks, > > Joao Pedro Afonso > > * > Joao Pedro: Greetings! Solo se vive una vez! > > The "Interview" puzzle goes like this: (It is > sometimes called the > "Marriage" puzzle, for obvious reasons) > > You are to interview a number of candidates, let's > say a total of n > candidates for a job. You interview each one and > make a decision: > Do you hire them or not? After each interview, you > must decide: Do > I hire this person, or do I go on and interview more > candidates. > After passing on a candidate, you cannot go back and > pick one of the > past candidates. > > What is the best strategy for selecting a candidate? > What strategy > will give the highest probability of finding the best > candidate? > > *** > As you can see, this is sometimes called the > "marriage" problem. > You meet a nice person and then decide: Is this my > life partner, or > should I go on and meet some more candidates? > *** > > Emanuel Parzen, who was my Probability and Statistics > professor at > Stanford, would translate this to a problem of "balls > and urns". > > You have a container with a total of n pieces of > paper. Each paper > has a number on it. The only thing you know about > the numbers is > this: they are "orderable"  that means, they are > all different. > Note: they are not numbered from 1 to n  that > would be too easy. > They are numbered like this: Six, 13.4, 22/7, 12 pi, > 10^19, 43, > 1931 (a prime number and the year I was born!), and > so on. They are > all different. > > Here's a suggested strategy: > > Select a subset of n (say, for example m) and review > them. Determine > the largest number in the subset. Then, continue > selecting until > you come to a number larger than any number in the > first m  the > subset. Select that as your choice. > > How many should you select as the subset m? If you > pick a small > number (say around 0.2 of n), you are likely to find > another number > very quickly  and it is not likely to be the > largest. > > If you select a large m (say equal to 0.8 of n), the > likelihood is > that the largest number is within the selected group > and therefore > cannot be the final choice. > > What is the optimum size of the selected group as a > percent of the > total group. It sounds like about onethird would be > somewhere in > the right ballpark. > > The problem is: What is the optimum m as a percent of > n? > > earle > *

