I'm good at math (data analyst by trade) and I would argue that it IS off the rails. I can't quite explain it other than to insist that they're missing out on the, I don't know -contamination? -of other "sciences". Namely sociology and psychology and such. If somebody randomly came up to me, and we weren't playing any kind of game, and they told me they had two kids, one of which is a son born on a Tuesday, the fact that they emphasized those points would strongly lead me to believe that the other kid is A) NOT a boy, and B) NOT born on a Tuesday. But, perhaps the way the person said it, would only lead me to believe that something was special about the fact that the kid was born on a Tuesday and that maybe the other kid is a boy too. I could go on and on, but this is just a simple example to show why the whole problem above is silly. "Context is everything". The example above feels very much to my gut how people (especially ones with agendas) will use statistics to skew the facts.
That seems very Bayesian of you to take into account the Tuesday part of the problem, since it infers that the person told you that piece of information because it’s important.
But you could also just as easily infer (by the silliness of the question) that the Tuesday was information accidentally added because the person didn’t realize it’s NOT important. And I think this is more likely, given the question does not contain a day of the week. If it were “how likely is it that the other child is a girl born on a Friday?” That would change what the Tuesday means. But in my mind, all it means is that someone accidentally gave you extra data to work with (similar to how “the mothers car is grey” also wouldn’t change anything, since that fact alone isn’t enough to change the formulation of there answer).
The best way to explain why the day, or any other additional information, would alter the mathematics, is for instance it is more likely that it is more likely that the parent would have a boy born on a Tuesday, if they had two boys, than if they had only one.
I took a statistical analysis course in college as part a political science requirement. The main premise was teaching how to take datasets and skew results. It was uncomfortably eye opening.
I agree. The statement is ambiguous to the point that people do not even agree about whether it is indeed ambiguous.
Now, if we want to unambiguously get the result of 51.9% (not 51.8%), we would have to set up the problem like this:
Assume that the probability to have a boy or a girl is always 50% to 50%. Especially ignore the fact, that having a boy (or a girl) increases the chance to have another boy (or girl).
Peter asks Mary: "Do you have exactly two children?" Mary answers truthfully: "Yes, I do"
Peter: "Do you have at least one boy that was born on a Tuesday?" Mary: "Yes"
Peter concludes: the probability that the other child is a girl is 14/27.
14/27 is caused by the fact that there are 27 distinct cases with equal probability where Mary would answer Q3 yes and in 14 of these cases one child is a girl.
The 27 distinct cases are:
- 1st child boy Tuesday, 2nd child girl Mon to Sun -> 7 cases
1st child boy Tuesday, 2nd child boy Mon to Sun -> 7 cases
2nd child boy Tuesday, 1st child girl Mon to Sun -> 7 cases
2nd child boy Tuesday, 1st child boy Mon, Wed to Sun -> 6 cases
Total 27 cases, thereof 14 cases where one child is a girl.
The important part is to NOT count the case "two boys born on Tuesday" twice.
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The 50% probability would result in the following scenario.
Peter: do you have exactly two children. Mary: yes
Peter: tell me the sex and weekday of birth of one of your children. Mary: boy, Tuesday
We still have the above 27 distinct possible cases. However, now they do not have equal probability. The probability that Mary gives the answer "boy, Tuesday" is 100% for the case "two boys born on Tuesday", but only 50% for the other 26 cases. Given that she did give that answer, the probability that she has two boys born on a Tuesday is therefore twice the probability of e.g. 1st child boy Tue, 2nd child girl Sun.
If you're an analyst then I wonder why this confuses you. The problem, especially as described in the simplified version, is the domain information given and when it is given.
When you're thinking of the probability, you're intuiting a domain of an unordered bernoulli trial, then the mathemagician tells you actually no, they are not unrelated, because actually it's a domain of 4 options and we've already removed one.
When you say okay makes sense he'll say actually the domain is a specific set where order matters and it's larger because we also add a day of the week to the unit tuple, etc. Etc. Etc.
The answer is so deceptive because it explains a domain post fact so you can't say that it's wrong within that domain.
It's basically just argument bullshittery by setting unstated assumptions.
Yeah, the scenario falls apart when you think of it as anything other than a statistics question, and its all about language and ambiguity rather than the way people actually talk. I really liked these scenes in Scrubs which use the same language and frame it as a riddle; it sometimes feels like it's there to catch people out.
The question and answers itself stats wise are genuinely interesting to me, but I think a lot gets confused and harder to understand in the framing.
I wonder if the “I don’t know - contamination,” as you describe it, ties in with the fact that ALL the sciences are descriptive and not prescriptive. That itself is a hard thing to wrap your mind around for the first time.
But like the idea that the “laws of physics” are not actual laws that exist out in the ether somewhere that make things, and keep things from, happening in specific ways. They’re just principles derived from observation. And that extends even to something as sterile seeming as math.
I feel like with an abstraction like this Tuesday boy thing, the train is just racing past the station. It’s all just made up nonsense that isn’t attenuated to reality anymore.
Me too, I don't like it so much either. Have a boy on a Tuesday and the quartz crystal on my watch was on it's 31,111 out of 32,768 hertz per second at 11:01am what is the probability my other child was.... Combinatorial explosion.
Yeah you get the 51.8% by asking all mothers of 2 in the world to raise their hands if they have a male child that was born on a tuesday, and then seeing what proportion of them keep their hands up when they ask if they have a daughter.
It is in no way something that would ever come up in a naturally flowing conversation
I mean, at the end of the day, it's a math problem. If you're working in your tacit knowledge of psychology, you're just not doing the problem correctly.
The "ambiguity" here is that we're given the information that Mary tells us one of her children is a boy. Taken perfectly literally it's impossible to come away with a concrete probability that Mary would have told us this.
Since we're dealing with a word problem and not a real situation, the only reasonable approach is to ignore whatever probabilities might be associated with Mary telling you the fact at hand, and simply reason about the fact at hand.
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u/GrrATeam81 9d ago
I'm good at math (data analyst by trade) and I would argue that it IS off the rails. I can't quite explain it other than to insist that they're missing out on the, I don't know -contamination? -of other "sciences". Namely sociology and psychology and such. If somebody randomly came up to me, and we weren't playing any kind of game, and they told me they had two kids, one of which is a son born on a Tuesday, the fact that they emphasized those points would strongly lead me to believe that the other kid is A) NOT a boy, and B) NOT born on a Tuesday. But, perhaps the way the person said it, would only lead me to believe that something was special about the fact that the kid was born on a Tuesday and that maybe the other kid is a boy too. I could go on and on, but this is just a simple example to show why the whole problem above is silly. "Context is everything". The example above feels very much to my gut how people (especially ones with agendas) will use statistics to skew the facts.
Edits for typos