Brian Delaney, Sioux City, Iowa writes:Sometimes Marilyn can be made to appear wrong when a problem is construed in a particular way. We favor the construction most favorable to Marilyn.
At work, we had a contest in which the prize was a new car. The six finalists could choose from six keys, only one of which would start the car. In an order chosen at random, each person would select a key and try it. (If the key didn't work, it would be discarded.)Marilyn Vos Savant replied:The second person picked the key that started the car, so the last four people didn't even get an opportunity to choose a key at all. In this kind of contest, would you want to go first? There was much discussion about this but little agreement.
Before I answer your question, let's all try the following one, which is even trickier.As far as we can find out, Marilyn answered only her own question and did not get back to Brian Delaney's question.Say that Brian drops his own car key into that original box of six keys, making the number of keys total seven instead. He's going to randomly remove one key at a time and then try to start his car with it. (If the key doesn't work, he'll discard it.) We know that the chances are one out of seven that he'll retrieve his own car key on the first try. But what are his chances that he'll get back his key on the second try?
ANSWER: Brian's chances of retrieving his own car key on the second try are still one out of seven; in the contest, there is no advantage to any particular position.
Whether or not Marilyn's answer is correct depends on how one construes the phrase "If the key doesn't work, he'll discard it." If it means that he'll throw it back in the box, then Marilyn is right, but if it means that he'll throw it over the fence, then Marilyn is wrong.
Since Marilyn has successfully solved much harder problems than this, we think that the first meaning is the one intended.
One also should also insert the phrase "If he doesn't get it back on the first try" at the end of the problem statement. If the question is construed to mean "If he keeps trying until he gets his key back, what is the probability that he will get it back on the second try?" the answer is 6/49, which is slightly less than 1/7.