Mathematics and the Real World (68 page)

The author's approach to the solution is incorrect, and the paradox, or apparent surprise, does not exist. The writer did not notice that we have insufficient data to reach an unequivocal answer to the question. First, the reason that we cannot accept the proposed solution is that the way the question is formulated, the solutions to the two parts contradict each other. If the answer is
when we know that the child in the house is the youngest, then that would be the answer also if the child in the house is the middle child of the three, and that is also the answer if we know that the oldest of the three is in the house. These are three separate cases, and together they exhaust all the possibilities. And if the probability of each of these cases is
, the probability would also be
if we do not know which of the three children is in the house, that is, the youngest, the middle one, or the oldest. Therefore, from the fact that we get
as the correct answer to the second part of the question, we derive that
is also the correct answer to the first part.

Where, then, is the author's mistake? I could make life easy for myself and claim that the writer was misled by the term
conditional probability
in the title, which is sometimes interpreted in the literature to mean the probability of
A
when it is known that
B
holds (see section 40). This is an incorrect interpretation of the concept of conditional probability, as it ignores the question of how it becomes known that
B
holds. Those who read section 40 will know that to solve problems like these we must use Bayes's thought process. If the author of the article had tried to apply Bayes's scheme, he would have seen that without more information it is impossible to arrive at an unequivocal solution. (We gave an example, the tale of the six competitors in a beauty contest, in section 42. A similar example was also “solved” incorrectly in the article.) As we have stated, if data are lacking, the brain of the person solving the problem supplies what is missing itself, generally unknowingly. The problem is that completing the picture in different ways yields different results.

We will put forward three different versions of how the picture can be completed, that is, how the missing information can be supplied, each of which yields a different answer. In the first, assume that in a family with three children, two, chosen at random, go outside to play. Then the solution to both parts of the question is
(we skip the calculations). In the second version, assume that among the families in that neighborhood children always go out to play with another child of the same sex, meaning boys play with boys, and girls with girls, and if there are three boys or three girls in a family, the two older ones go out to play together. In that case, in the first part of the question the probability that the child who stayed in the house is a boy is
, and the probability in the second part of the question is
, as was claimed in the quoted article (again we skip the calculation, but note that here the calculation quoted in the article is correct). The third possibility is that two children of the same sex always go out to play together, and if there are three of the same sex, then the two who go outside to play are chosen randomly. In that case, in both parts of the question the probability that the child in the house is a boy is
. Thus we see that the information provided in the question can be completed in different ways, which all give different answers to the questions. What is the correct way to fill in the missing data? The formulation of the question does not provide an answer to that. The use that the writer made of the mathematical formula incorporates an unstated assumption about the question itself (for example, one of the assumptions we listed above), an assumption that does not appear in the formulation of the question. That is why I chose my words advisedly and said that the author's approach to the question was wrong, and not that the numerical answer in the article is wrong. (Although, as the writer related only to equations, it is hard to believe that he considered the various possibilities. Herein lies an important lesson for anyone using mathematics: do not use formulae before checking that they are relevant to the situation at hand.)

As we have said, the reason for the error is rooted in the fact that thinking under provisions is not a natural process for the human brain. So much so that the author of the article did not recognize it and thus also did not identify the source of the contradictory results he himself obtained. He preferred to think of the discomfort caused by intuition as a paradox, a paradox that formalism can explain. As we shall see, in this he is not alone.

Possibly the most famous instance of lack of clarity and unknowing completion of information is known from
Let's Make a Deal
, a television game show in which the competitors are given a mental challenge. The following is an exact formulation of the question:

You are a participant in the game. You are shown three closed doors. You are told that behind one door there is a big prize, while behind each of the other two is a goat. You are invited to select a door. Before it is opened to reveal the big prize or a goat, the host, who knows what is behind each door, opens one of the other doors, and reveals a goat. He now offers you a chance to change your mind and select the other door. Is it worth your while to change your selection, that is, to choose the door that the host did not open?