I think this particular question illustrates a major oversimplification in the entire premise of the webpage. If you have a probability problem that isn't well-specified, no amount of "mechanical" magic, Bayesian or otherwise, will give you a fully correct answer, since you are missing relevant details.

Let's consider this particular question:

"Mrs. Chance has two children of different ages. At least one of them is a boy born on Tuesday. What is the probability that both of them are boys?" [Source: https://news.ycombinator.com/item?id=45052502]

The question is bizarre and there are planty of ways to interpret it.

Here's how I guessed it was intended to be interpreted: I'm a person who just met a stranger, and the stranger told me they had two children of different ages (i.e. not twins). I did a tiny bit of investigation and found an undated article stating that they had a baby boy born on an unstated Tuesday in the past. The article gave no indication as to whether any other children had been born yet. I believe, a priori, that each of the strangers' children is either a boy or a girl with 50% probability each, i.i.d. for both children. I believe that there are no further biases in the article (e.g. if the child in question was a second child, then the article would have been equally likely to be published and found by me regardless of the gender of the first child).

The only relevant thing I learn is that the children were not both girls. Then the problem is essentially identical to the sisters problem higher up on the webpage, and there is a 1/3 a posteriori probability that both children are boys.

Now let's interpret the same question differently. I meet a stranger, and the stranger mentions that they have two children, and I determine, a priori, that each child is a boy or a girl, with 50% probability for each child, i.i.d. For some bizarre reason, I decide to ask the stranger "Do you have a son who was born on a Tuesday. Answer yes or no, and do not give any other information!" and, for some bizarre reason the stranger actually remembers or calculcates the answer and answers honestly, and the answer is yes. And the probability that the stranger gives a correct, honest answer is independent of the birth dates and genders of both children, which is a very strange assumption indeed. Now you get the scenario in the webpage: it is dramatically more likely that there was a boy born on a Tuesday if there were two boys than if there were only one.

The older HN thread that the article links has some fun comments giving even more differing interpretations (e.g. that "born on a Tuesday" refers to the most recent Tuesday, in which case, if the children are not twins, one might reasonably conclude that the younger child is a newborn boy and that absolutely no information is gained about the elder child.

This whole situation illustrates one of my major pet peeves about the way that statistics is often done. The real world in complex, and there are many reasonable experiments that one might do, and there are many reasonable questions one might ask about what was learned from the experiment. Nonetheless, it's very very common to see a conclusion that consists almost exclusively of something to the effect of "X significantly improved Y", and, while this might be mechanically correct in the sense that you could shove the numbers into your favorite statistics software and get that answer, you don't know enough details about the study to translate that result into any useful answer to any clearly stated question about the world.

Use Bayes rule to mechanically solve probability riddles