[sparsely]

For the last one, why does the "born on a Tuesday" information change the result? I don't see how it isn't equivalent to "born on a day", since the day of the week has no connection to the rest of the scenario. I understand why "at least one boy" does matter.

[zaik]

If you accept the Bayes theorem, the answer is that the likelihood of "At least one boy is born on a Tuesday" is not the same for different numbers of boys. The more boys the more likely the statement is true. Therefore this information is indicative of how many boys Mrs. Chance has.

[jammaloo]

It seems to me that it comes down to how the day of the week was picked.

If they picked a random day of the week, and there was only one boy, then there is only a 1/7 chance of a boy being born on that day.

If they have one boy, who was born on a Tuesday, and that is why they picked the day, then there is a 100% chance of a boy being born on that day, so no additional information is conferred.

[fenomas]

Puzzles like that one have always seemed dishonest to me. It only makes sense if you start from the conclusion that it's meant to illustrate Bayes rule, and then work backwards to the assumption that the predicate "boy born on a Tuesday" is supposed to be independent of who's being asked about.

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?

[kgwgk]

> If you accept the Bayes theorem

That doesn’t make a lot of sense. A theorem is just a theorem. It’s proved, and in this case the proof is trivial.

The question is whether you accept that the description of the problem in terms of conditional probabilities is adequate, and then whether you accept that the values assigned to those conditional probabilities are appropriate.

[amluto]

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.

[JeffJor]

Mr. Bertrand has (exactly - this needs to be included) two children (not twins, which is not quite the same as different ages). A gender, and a day of the week, that apply to at least one of his children have been written inside a sealed envelope. What is the probability that both children have that gender?

In this problem, we have no gender- or day-specific information. So the answer can only be the probability that he has two of the same gender. Which is 1/2.

Now open the envelope. If the answer changes to P based on what you see written, it has to change to the same P regardless of what you see written. Which means you didn't need to unseal the envelope; the answer was P before, not 1/2.

This is what Joseph Bertrand identified as his Box Paradox in 1889. That word was used to describe an actual contradiction, not a non-intuitive result. It disproves any answer except P=1/2. FOR ANY OF THESE PROBLEMS.

In fact, it is the same reason why the Monty Hall Problem's answer is what it is. Many "explanations" will claim that your original probability can't change, but never justify it. This is the justification - if it changes when one door is opened, it must change the same way when either door is opened.

[knappa]

You could definitely replace "Tuesday" with something like that and part of the pedagogical purpose of the problem is for people to question this. The actual effect comes from not distinguishing the boys. That increases the likelihood that at least one of them will be born on any particular day, upweighing the likelihood that there are larger numbers of boys. i.e. You just get, on average, better coverage of boys-born-on-Tuesday when there are more boys.

[FabHK]

Well, "born on a day" would not convey any information unless it means "during daytime". If that has probability 1/2, the answer would be 3/7. With Tuesday (or, indeed, any other weekday, with probability 1/7), it is 13/27.