http://people.howstuffworks.com/trolley-problem.htm

A run-away trolly is traveling down some tracks.

At the end of the tracks there are 5 people tied up. You are standing next to a switch. If you push the switch, the trolly will be moved to a different set of tracks where there is only 1 person tied up.

The only two outcomes are either you do nothing and 5 people die, or you push the switch and 1 person dies. Which one is the morally correct choice?

One common answer is the utilitarian viewpoint: the greatest good for the greatest number of people is what matters. With this thinking, you should take action and save 4 people’s lives by switching the track. An alternate viewpoint suggest that because the run away trolly was already in place (not at the fault of your own), if you push the switch you will be directly hurting another person (the 1 person). But if you do nothing you are not at fault for any moral wrongs. So with this logic you would do nothing.

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat.

He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

A lot of people would say that the choice to switch or stay does not matter.

You should always switch!

The basic form of the problem is the following:

imagine an administrator willing to hire the best secretary out of n rankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator can rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants.

The question is about the optimal strategy (stopping rule) to maximize the probability of selecting the best applicant.

If the decision can be deferred to the end, this can be solved by the simple maximum selection algorithm of tracking the running maximum (and who achieved it), and selecting the overall maximum at the end. The difficulty is that the decision must be made immediately

The problem has an elegant solution.

The optimal stopping rule prescribes always rejecting the first applicants after the interview (where e is the base of the natural logarithm) and then stopping at the first applicant who is better than every applicant interviewed so far (or continuing to the last applicant if this never occurs).

http://en.wikipedia.org/wiki/Secretary_problem

http://en.wikipedia.org/wiki/E_%28mathematical_constant%29