That Tuesday Boy problem

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

Much effort has been expended on this one. The initial presumption is that the answer must be 1/2 or 1/3, depending on how the set up of the question is interpreted.

Actual answer: 13/27. Seeming implication: having a boy born on a Tuesday makes it less likely to have another one. As one would suspect, it's not true. You can do this kind of trick with lots of ancillary events (for boy born night or day, the answer is 3/7).

Sorely lacking in the hours of reading this blogger has done on the topic is any intuition for the 13/27 answer, and indeed the attempted explanations seem to veer ever deeper into the philosophy of probability.

So here's our attempt. By specifying not just that one child is a boy but is one born on a Tuesday, the question has done two things. (1) It has introduced a 2nd characteristic into the event i.e. day of week as well as sex and (2) it has ruled out some events that would otherwise count. Example: 2 boys born on Saturday don't count as positive events. 2 girls born on Saturday already didn't count because of the specification that one child was a boy. In other words, the day-of-week seems irrelevant, but it's still grabbing a portion of the outcome space and putting other parts of it off limits.

The implications of this logic for an opinion poll asking whether the US president is a socialist are left to the reader.