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.