During my summer hiatus from Who's Counting, a fascinating new paradox, first noticed by puzzle-meister Gary Foshee, became widely known. The following variant of it suggests that summer births may be associated with a much increased likelihood of boy births. Before I get to this paradoxical result, let me put it into context with two simple probability problems.
The first is very straightforward. Assume you know that a woman has two children, the older of whom is a boy. Given this knowledge, what is the probability that she has two boys? Let's count the possibilities. The only two are an older boy and younger girl (B-G) or an older boy and a younger boy (B-B). Since these are equally likely, the probability that the woman has two boys is 1/2.
No problem there, but now consider this second scenario. Assume that you know that a woman has two children, at least one of whom is a boy. You know nothing about this boy except his sex. Given this knowledge, what is the probability that she has two boys?
You might jump to the conclusion that the answer is again 1/2, reasoning that the sex of one child has no bearing on the sex of the other. This conclusion is incorrect, however, since you don't know whether the boy you know about is the older or the younger child.
So let's look at the possibilities. Listing two children in the order in which they might be born, we note four possibilities: B-B, B-G, G-B, G-G. Since you know that at least one of the two children is a boy, the G-G possibility is eliminated. Of the three remaining equally likely possibilities (B-B, B-G, and G-B) only one results in two boys. Therefore the correct conclusion in this case is that the probability the woman has two boys is 1/3, not 1/2.
This much has long been understood, but the paradox mentioned above is considerably less intuitive.
Now for the odd result. Suppose that when children are born in a certain large city, the season of their birth, whether spring, summer, fall, or winter, is noted prominently on their birth certificate. The question is: Assume you know that a lifetime resident of the city has two children, at least one of whom is a boy born in summer. Given this knowledge, what is the probability she has two boys?
On the surface, this appears to be essentially the same as the second scenario -- and so it seems that the probability that the woman has two boys, given that at least one of them is a boy born in summer, should remain at 1/3. After all, the season in which a child is born does not seem relevant and should not affect a sibling's sex. Yet, if demographers were to collect data on women in this city and focus on those with two children, at least one of whom is a boy born in summer, they would find that 7/15 of these women have two boys.
Why should this be? Is there something about those women who give birth to a boy in summer that predisposes them to produce more boys? Is there some previously unknown genetic/climatological link? All sorts of bizarre theories might be constructed to account for this large increase in probability from 1/3 to 7/15.
All of the theories that might be proposed, however, are bound to be wrong. Strange though it may seem, the answer of 7/15 is what we should expect on probabilistic grounds alone.
Let's count the possibilities. Since we're assuming we know a woman with two children, at least one of whom is a boy born in summer, let's list all the possibilities where this condition is met and see how many of them result in the women having two boys.
Abbreviating the four seasons as sp, su, f, and w, and listing the older child first, the 15 equally likely possibilities are: Bsu-Bsp, Bsu-Bsu, Bsu-Bf, Bsu-Bw, Bsu-Gsp, Bsu-Gsu, Bsu-Gf, Bsu-Gw, Bsp-Bsu, Bf-Bsu, Bw-Bsu, Gsp-Bsu, Gsu-Bsu, Gf-Bsu, Gw-Bsu. (Note we don't count Bsu-Bsu twice.)
Of these 15 possibilities, 7 result in two boys: Bsu-Bsp, Bsu-Bsu, Bsu-Bf, Bsu-Bw, Bsp-Bsu, Bf-Bsu, Bw- Bsu.
Thus, knowledge of the summer birth increases our probability estimate from 1/3 to 7/15. If at least one of a woman's two children is a boy, the probability she has two boys is 1/3, but if at least one of a woman's two children is a boy born in summer, the probability she has two boys is 7/15.
I reiterate that this calculation shows that the various theories that might be concocted to explain the change in probability from 1/3 to 7/15 are unnecessary and thus are bound to be bogus. As with many random phenomena (in this case: boys and girls being born in equal numbers and more or less uniformly throughout the year), no explanation other than chance is required.
The puzzle illustrates a deeper truth. Even in clear-cut situations, answers, analyses, and explanations may differ depending on subtle differences in phrasings and assumptions. This is all the more true in political and economic situations, which are nowhere near as clear-cut, but even more sensitive to phrasings and assumptions. Coming up with radically different explanations in these more nebulous contexts is not all that surprising, especially when the various actors don't even attempt to approach the issues carefully and in good faith. The likelihood of consensus in such situations is very low, especially in election season.
John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as, most recently, "Irreligion: A Mathematician Explains Why the Arguments for God Just Don't Add Up."He's on Twitter and his "Who's Counting?" column on ABCNews.com usually appears the first weekend of every month.