Eddie Thomas, a better philosopher than he is a statistician, poses the following problem. He is interviewing five candidates for two jobs. Each candidate’s chance of receiving a job offer, a priori, is 40%. After interviewing four candidates Eddie wants to offer a job to the best of the four to protect against his taking another job elsewhere. He is puzzled because it now looks like the final candidate has a 25% chance of a job offer, being one in four remaining, while his chances should be unaffected, remaining at 40%.
This is a type of restricted choice problem. The classic illustration of true restricted choice is the old game show Let’s Make a Deal. Monty Hall shows the player three doors, behind one of which is the grand prize, a Hawaiian vacation or a brand new Cadillac Eldorado. The player chooses one door. Monty then reveals another, behind which the prize is not hidden, and asks the player if he wants to switch. The player should always switch. His chance of choosing the grand prize in the first place was 1 in 3. If he switches, it is 2 in 3, because Monty’s choice of which door to open has been restricted by the choice the player already made. Many people don’t believe this even after it has been explained to them, but it’s true, and can be verified easily by experiment if you doubt it.
Here, on the other hand, because of restricted choice, the probabilities only appear to change. To receive a job offer you must be one of the best two of five candidates. Consider it from the point of view of the first four candidates. Each one has a 25% chance of receiving an offer after four interviews, and a 15% chance (.75 X .20) of receiving an offer after five interviews, for a grand total of 40%, as you would expect. So one way to look at it is that the remaining candidate must also have a probability of receiving an offer of 40% for the probabilities to add up to 200% (two job offers).
Since this will satisfy no one, least of all Eddie, I’ll try another approach. To receive an offer he has to be better than the three remaining candidates. However, his three remaining competitors are not randomly chosen; they have already failed to finish first among the first four. Choice has been restricted. Each one of the three, not having received the first offer, has a far less than 40% chance remaining of securing an offer. In fact, they now have half that chance; for they can only finish second, at best, among the five, while the fifth candidate can still finish either second or first. Therefore their remaining chance is half of their original 40%, since one of the two offers has been closed to them, or 20%, rather than 25%. The fifth candidate still has a 40% chance.
God of the Machine: all probability, all the time!