This is the crux of the problem. You are treating the cards as separate instances. Most of these probability questions rely on treating the cards/children as a pair, and only as a pair. That way you make sure the two are not independent and it leads to the interesting results.
Well and good. But even in the link nothing in the set up of
Mary has 2 children and at least one is a boy. What’s the probability the other child is a girl?
tells us that such applies.
Yup. Like I said, the real problem is when people insist that the nonintuitive setup must be the right one, even though they were never clear that that was the setup and it would be much more natural to assume the other way.
So what is the way that the problem should be presented to eliminate the intuitive understanding as applying?
If the deck has a total of two cards, then we can be certain the other card is a G.
If the deck has a total of four cards, then there’s a two-thirds chance the other card is a G.
If the deck has a total of 2N cards, then there’s an N/(2N-1) chance the other card is a G.
About as pertinent to the answer as my pointing out that boys and girls aren’t 50 50 odds at birth. So I deserve that. Fresh fully mixed decks each deal. Fair coins.
The easiest way to construct a problem with that interpretation is with a question-and-answer format (with, of course, the right questions). For instance:
Outside of contrivances like that, it’s really difficult to construct real-world problems that lead to the counterintuitive answers. One I came up with a while ago:
This works, because anyone at that dinner had to have at least one son, because my high school was all boys, but either or both of the man’s children could have been alumni. But it’s fragile, because if, for instance, the man had mentioned in conversation “Pardon my tiredness; my toddler kept waking me up last night”, the probability would go back to 1/2, because in that case, we now know that he only has one child who’s an alumnus, and therefore that’s the specific one we have information about.
I submit that the conclusion is erroneous because, for genetic reasons, about 105 boys are born for every 100 girls, and that is being ignored in this problem. There are also genetic propensities in certain families. For example, I have a school bud whose family extending from siblings all the way through grandchildren are about 4 girls to every boy.
Circumstances being what they are, I think this is a very poor problem to give to students if you are an instructor.
I understand that it’s possible to set up a simulation that will prove that the likelihood is 51.8%.
But I still don’t see why Tuesday matters, and why we can’t make up arbitrary criteria that would skew things. He’s human, he breathes oxygen, he has blue eyes, he was born during a snowstorm, etc.
Mary has 2 children. She tells you that one is a boy whose diaper fell off on the first day. What is the probability that the other child is a girl?
Mary has 2 children. She tells you that one is a boy whose left eyebrow has 61 hairs. What is the probability that the other child is a girl?
Mary has 2 children. She tells you that one is a boy whose left eyebrow has 61 hairs AND whose diaper fell off on the first day. What is the probability that the other child is a girl?
Now that I think I get it - it doesn’t work if she just volunteers random information: you had to ask with the possible answers constrained to limited sets.
You have stop framing it as Mary just telling you. You have to ask Mary.
The point is that whatever tiny criteria you choose eliminates the “Mary has both girls that do X” but not the “Mary has both boys that do X” the more finetuned the X is the less times your interviewee will say yes and be part of the statistics. The lower the chance of X the less chance that both boys will be X, and the imbalance works only because of the possibility that both boys are X creating an imbalance. The more difficult the X the closer it will get to 50%, but it’s still just the 2 child problem heavily diluted.
Thing is that is the problem you want presented but not the problem that was presented in the OP or in the link from the OP that reputedly explained the answer.
And sure your initial response “reframed” it “properly” … but that was just presenting a different problem. The correct answer to the OP is that the answer (if boys and girls were evenly distributed) is 50%. Then explaining the difference between the problem presented and the ones that give the non- intuitive answers by contriving a non-intuitive set up.
The question is “why is it 51.8?” Step one of that is explaining how to frame the question correctly so that it is indeed 51.8. Heracles was asking why also.
The answer to the question presented however is actually NOT 51.8. Explaining how 51.8 is the right answer to a different question … confuses.
Your answer (to quote you) “is wrongly structured” because it answers the problem you wanted to answer, not the problem presented. Which is a trap.
To give credit where it is due, following through to the wiki link most of its discussion is exactly about how the precise question presented matters, that such was a major point when the puzzle was first presented.
Mate the OP was asking why Tuesday matters and why it’s 51.8. Tuesday only matters , (and it’s only 51.8) if the question is being tweaked to make Tuesday matter, otherwise the answer is “dunno, as we don’t know Mary’s rationale we can’t answer”. Heracles was also asking why Tuesday, or the exact number of hairs on an eyebrow matter. Again the answer relys on “in order for it to matter, the question needs to be asked like this…”
In both cases they were wondering why there’s a calculable imbalance and the only answer is “firstly we have to make some assumptions about what she means…” otherwise there’s no calculable answer.
But an equally valid answer, given the framing of the question, is “Tuesday doesn’t matter, and the person who said it does was mistaken”.
It actually is THE valid answer.
Now going on to part 2 - explaining why some people mistakenly think that it is the right answer, and how it is the right answer to a different question, which is the one they would have asked if they actually understood the question and answer, is a fair thing to answer as well. But only after it is made clear that as presented the answer is wrong and why.
Subverting the solvers by exploiting their intuitions only works if the puzzle is set to do so. If not there is no gotcha. The setter is the fool instead. A point you chronos have made well.
The problem here is - does the puzzle mean “only one boy born on a Tuesday” or “at least one boy born on a Tuesday”? (I.e. she could have one or more boys born on a Tuesday, she could have a Tuesday boy and an any-day girl)
If the question is - Only one boy on a Tuesday - Teh other chils is not a boy born on a Tueday - Then, if the odds are the same that births happen any day, and boy or girl are equal, then the choices are the combinations:
BTues - BnotTues
BnotTues - BTues
BTues -Gany
Gany-BTues
Out of the set of all possibilities:
(B/Gany and B/G any) minus GGanyany
= = = =
If the question is At least one boy born on a Tuesday then it’s the simple “one is a boy” paradox.
Otherwise, saying “Tuesday” means that you eliminate the “two boys both born on Tuesday” option, which means it BB 1/7 less likely than any BG option. (14 possible ways 2 boys can be born when one is a Tuesday, 2 of which are Tuesday-Tuesday.
7 possible BG, 7 possible GB, and 6 possible BB - 20 options, of which 14 are the other child is a girl. 14/20 is 0.7 odds the other child is a girl.
There are 27 ways
1 B(tue) B(tue)
7 B(tue) G
7 G B(tue)
6 B(tue) B(not tue)
6 B (not tue) b(tue)
Each of these are equally likely given the stated constraints of the intended problem. 14 of them have a girl, 14/27=0.518
I always find these fascinating because they’re not intuitive to me, but I can still follow the math. Two thoughts I’ve had over the years…
- Any kind of addition that changes the sample space will fuck up the math.
- Clearer wording can definitely make it more intuitive. For whatever reason, I struggled with the Monty Hall Problem until I reframed it as “Monty is forced to open a second door”, and then it seemed obvious. Was that already implicit? Probably, but I needed that for the light bulb to go off.
(also, I ran @echoreply’s R code upstream & can confirm it works 
)