Use Bayes rule to mechanically solve probability riddles

But in plain English, ".. At least one of them is a boy born on Tuesday" suggests the speaker is giving a fact that was chosen because it's true of the person spoken about - like if the kids were both chosen on Thursday then that would be the day named. And read that way, the Bayes illustration doesn't stand and the "correct" answer makes no sense.

To make it honest, it should really be worded like: "Mrs. Chance has two children of different ages. You ask whether at least one of them is a boy born on Tuesday, and you are told yes. What is the probability that both of them are boys?" Or am I missing something?

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.

As far as I’ve been able to find the text below would be the original question (“the host, who knows what’s behind the doors, opens”) and answer (“the host, who knows what’s behind the doors and will always avoid the one with the prize, opens”).

Are you referring to that?

https://web.archive.org/web/20130121183432/http://marilynvos...

Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? [Craig F. Whitaker - Columbia, Maryland]

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it.

Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly opens a door [0], this specific day it happens to contain a goat, should you switch?

And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory).

[0] Note that specifying that the host knows what's behind what door doesn't help. They could still pick a door randomly.