Another probability question that causes many people difficulty is the children's gender puzzle: I tell you I have two children and that (at least) one of them is a boy, and ask you what you think is the probability that I have two boys. Many people, when they hear this puzzle for the first time, give the answer 1/2, reasoning that there is an equal likelihood that my other child is a boy or a girl. But this is not correct. Based on what you know, you should conclude that I am actually twice as likely to have a boy and a girl as I am to have two boys. So your right answer to my question is not 1/2 but 1/3.
Before I explain the answer, I should clear up a confusion that many people have about problems such as this, which are about what is known as epistemic probability. The probability being discussed here is not some unchangable feature of the world, like the probability of throwing a double six with a pair of honest dice. After all, I have already had my two children, and their genders have long been determined. At issue is what probabilities you attach to your knowledge of my family. As is the case with most applications of probability theory outside the casinos, the probability here is a measure of an individual's knowledge of the world, and different people can, and often do, attach different probabilities to the same event. Moreover, as you acquire additional information about an event, the probability you attach to it can change.
To go back to the original puzzle now, in order of birth, there are four possible gender combinations for my children: BB, GG, BG, GB. Each is equally likely. (To avoid niggling complications, I'm assuming each gender is equally likely at birth, and ignore the possibility of identical twins, etc.) So, if all I told you was that I have two children, you would (if you are acting rationally) say that the probability I have two boys is 1/4. But I tell you something else: that at least one of my children is a boy. That eliminates the GG possibility.
So now you know the possible gender combinations are BB, BG, GB. Of these three possibilities, in two of them I have a boy and a girl, and in only one do I have two boys, so you should calculate the probability of my having two boys to be 1 out of 3, namely 1/3.
If you haven't come across this before, it might take you some time to convince yourself this reasoning is correct. I long ago got past that stage, and hence felt my intuitions would be pretty reliable when I recently came across the following variant of the puzzle.
I tell you I have two children, and (at least) one of them is a boy born on a Tuesday. What probability should you assign to the event that I have two boys?
Before you read further, you should perhaps pause and try to figure this out for yourself.
My initial reaction was that the information about the Tuesday was irrelevant, since at issue was gender, not day of birth. In which case, this was the same problem as the one I just described, and the answer would be 1/3.
But then I began to have second thoughts. I admit my doubts were occasioned by the way I came across the problem: a Twitter feed by the well-known mathematician John Allan Paulos, forwarding a Tweet from the (British) Guardian newspaper science-writer Alex Bellos, who was reporting on the posing of this problem at the recent "Gathering for Gardner" conference in Atlanta by puzzle master Gary Foshee.
Suspecting that there was more to this problem than I initially thought, I set about repeating the same form of reasoning as in the original puzzle, but taking account of days of the week when my children could have been born. As soon as you do that, you realize that Foshee's problem really is different. But how different? My intuition said that, since the original puzzle had the answer 1/3, the new variant would have an answer fairly close to 1/3. After all, knowing the birth day is a Tuesday may (and does) make a difference, but it surely cannot make much of a difference, right?
Wrong. It makes a surprisingly big difference, The correct answer to the new puzzle is 13/27, just slightly less than 1/2, and not at all close to 1/3. This is what really surprised me. To the extent that I checked my solution with the one Bellos published on his blog a few days later.
The crux of the matter is that Foshee's variant seems at first glance to be a minor twist on the original one, but it's actually significantly different. The property it focuses on is not gender, but the combination property gender + day of birth. That makes the mathematics very different, as I'll now show. Instead of just the two genders, B and G, of the original puzzle, there are now 14 possibilities for each child:
B-Mo, B-Tu, B-We, B-Th, B-Fr, B-Sa, B-Su
G-Mo, G-Tu, G-We, G-Th, G-Fr, G-Sa, G-Su
When I tell you that one of my children is a boy born on a Tuesday, I eliminate a number of possible combinations, leaving the following:
First child B-Tu, second child: B-Mo, B-Tu, B-We, B-Th, B-Fr, B-Sa, B-Su, G-Mo, G-Tu, G-We, G-Th, G-Fr, G-Sa, G-Su.
Second child B-Tu, first child: B-Mo, B-We, B-Th, B-Fr, B-Sa, B-Su, G-Mo, G-Tu, G-We, G-Th, G-Fr, G-Sa, G-Su.
Notice that the second row has one fewer members than the first, since the combination B-Tu + B-Tu already appears in the first row.
Altogether, there are 14 + 13 = 27 possibilities. Of these, how many give me two boys? Well, just count them. There are 7 in the first row, 6 in the second row, for a total of 13 in all. So 13 of the 27 possibilities give me two boys, giving that answer of 13/27. (As in the original problem, you have to assume all the combinations are equally likely. In the case of birth days, this is actually not the case, since more babies are born on Fridays, and fewer on weekends, due to the desire of hospital doctors to have weekends as free as possible of duties.)
What misled my intuition (and likely yours as well) was my unfamiliarity with the property gender + day of birth. Fortunately, the math does not lie. Provided you put your intuitions to one side and set up the problem correctly, the math will give you the right answer.
Now that your intuition has been primed, let me leave you with this problem. I tell you I have two children, and (at least) one of them is a boy born on April 1. What probability should you assign to the event that I have two boys? If you think that is going to be too cumbersome, simply tell me whether the probability is close to 1/2 or to 1/3, or to some other simple fraction, and provide an estimate as to how close. (Once more, you should assume all birth possibilities are equally likely, ignoring in particular the well known seasonal variations in actual births.)
If you are still having doubts about all of this, take consolation in the fact that you are not alone. Representing real-world problems correctly to calculate probabilities is notoriously difficult. In my recent book The Unfinished Game, cited below, I describe how no less a mathematician than Blaise Pascal had enormous difficulty understanding an analogous argument by Pierre de Fermat.