A daughter-son problem (§2.9, §6.1)

You are told that a family, completely unknown to you, has two children and that one of these children is a daughter. Is the chance of the other child also being a daughter equal to ½ or 1/3? Are the chances altered if, aware of the fact that the family has two children only, you ring their doorbell and a daughter opens the door? The psychology of probability intuition is a main feature of some of these problems. Consider the birthday problem: how large must a group of randomly chosen people be such that the probability of two people having birthdays on the same day will be at least 50%? The answer to this question is 23. Almost no one guesses this answer; most people name much larger numbers. The number 183 is very commonly suggested on the grounds that it represents half the number of days in a year. A similar misconception can be seen in the words of a lottery official regarding his lottery, in which a four-digit number was drawn daily from the 10,000 number sequence 0000, 0001,..., 9999. On the second anniversary of the lottery, the official deemed it highly improbable that any of the 10,000 possible numbers had been drawn two or more times in the last 625 drawings. He added that this could only be expected after approximately half of the 10,000 possible numbers had been drawn. The lottery official was wildly off the mark: the probability that some number will not be drawn two or more times in 625 drawings is inconceivably small and is of the order of magnitude of 10-9. This probability can be calculated by looking at the problem as a "birthday problem" with 10,000 possible birthdays and a group of 625 people (see §3.1 in Chapter 3). Canadian lottery officials, likewise, had no knowledge of the birthday problem and its treacherous variants when they put this idea into play: they purchased 500 automobiles from nonclaimed prize monies, to be raffled off as bonus prizes among their 2.4 million registered subscribers. A computer chose the winners by selecting 500 subscriber numbers from a pool of 2.4 million registered numbers without regard for whether or not a given number had already appeared. The unsorted list of the 500 winning numbers was published and to the astonishment of lottery officials, one subscriber put in a claim for two automobiles. Unlike the probability of a given number being chosen two or more times, the probability of some number being chosen two or more times is not negligibly small in this case; it is in the neighborhood of 5%! The Monty Hall dilemma - which made it onto the front page of the New York Times in 1991 - is even more interesting in terms of the reactions it generates. Some people vehemently insist that it does not matter whether a player switches doors at the end of the game, whereas others confidently maintain that the player must switch. We will not give away the answer here, but suffice it to say that many a mathematics professor gets this one wrong. These types of examples demonstrate that, in situations of uncertainty, one needs rational methods in order to avoid mental pitfalls.† Probability theory provides us with these methods. In the chapters that follow, you will journey through the fascinating world of probability theory. This journey will not take you over familiar, well-trodden territory; it will provide you with interesting prospects.