r/theydidthemath • u/Confident-Insect-200 • 1d ago
What would be the answer to this interview question? [Request]
[removed] — view removed post
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u/AdAlternative7148 1d ago
The Tuesday thing can matter too. It is an ambiguous question.
There are 14 day and sex combinations per child. 7 days x 2 sexes = 14.
For two children there are 14 x 14 or 196 total combinations.
27 of the 196 involve a boy born on a Tuesday.
13 of 27 are two boys.
13/27 = 48.1%
So it is either 50% if you presume the first child's day and sex doesnt matter, 33% if you presume the sex matters but not the day, or 48.1% if you presume the day and sex matters.
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u/vectavir 1d ago edited 20h ago
So which one is it lol
Edit: to everyone saying "its all of them" no its not. Thats not how math works.
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u/AdAlternative7148 1d ago
Its ambiguous. It depends on how you think the family was selected.
If you look at a random two child family and see one is a boy born on a tuesday then the odds the other is a boy are 50%. The sex and day are irrelevant to how that family was selected.
If you look at a set of two child families, pull out one family that has a boy and then look at them and observe he was born on a Tuesday it is 33%. The sex is relevant to selection but the day was not.
If you look at the set and pull out one family that has a boy born on a Tuesday it is 48.1%. Both factors were relevant.
The question doesnt specify how the family was selected so really all answers should be acceptable imo. But i doubt Citadel feels that way for a job paying so much. Probably they want to see your thought process.
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u/rickrollmops 1d ago
Finally a good answer!
Also see: https://en.wikipedia.org/wiki/Boy_or_girl_paradox - there is a section on "Analysis of the ambiguity"
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u/CordlessOrange 1d ago
Oh, thank god you posted this. This should be the top comment.
I’m guessing the correct answer in the interview would probably be to recognize the ambiguous nature of the question and expand on that.
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u/fynn34 1d ago
How about accounting for the odds of twins? That increases the odds of another Tuesday child, and slightly increases the odds of another male if they are identical
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u/AdAlternative7148 1d ago
Good point and this is the sort of thinking they'd be looking for if they really want to see your thought process instead of just a number.
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u/stewsters 1d ago
Yeah, there are a lot of other things this doesn't consider that you would need to control for:
* Adoptions have a gender preference 75-80% female.
* Child mortality rate is higher for boys than for girls during infancy and childhood.
* Trans people exist, despite what Citadel group wants you to believe.→ More replies (8)4
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u/Grant_S_90 1d ago
I agree it could be worded better but that doesn’t mean all those answers are equally correct.
You sort of have to assume that the people making the statement have been selected for the statement being true, unless you’re told otherwise, else every conditional probability questions is meaningless.
If asked: “if a family have two kids one of whom is a boy, what is the chance they have two boys?”
You could answer: well it could be 0% because the families could’ve been selected for BG families only, in which case there wouldn’t be any BB families at all in the selection. Which would be meaningless.
Similarly: if someone shops at Asda what is the likelihood they also shop at Tesco? It could be 100% because they could’ve selected their sample only by surveying people inside Tesco.
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u/SanityIsOptional 1d ago
Thank you, this makes the most sense, and is definitely important for a high paying statistical job. Selection criteria affect results.
I could see the correct response being: "how was the family selected?"
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u/RainBoxRed 1d ago edited 1d ago
Birth sex is not dependent on sex of other siblings or day of week.
Each child is 50/50 for sex.
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u/Moscato359 1d ago
There is a problem here
The top post of this article added the word "at least"
While the version you are using does not have it
It changes the math
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u/Edward_Durr 1d ago
The 13 permutations he talks about include the Tuesday boy-Tuesday boy pairing, satisfying the “at least” aspect of the problem.
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u/dagreja 1d ago
Isn't the missing 1.9% the odds that both children are boys born on Tuesday? Which should be included, because the wording is "at least one is a boy born on tuesday"?
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u/ManyNeedleworker3693 1d ago
28 of the combinations involve a boy born on a Tuesday. And 14 of them are two boys.
The 27 count excludes counting 2 boys born on a Tuesday twice. But we are counting "Two boys" twice, and "one boy one girl" twice, because we are working in a permutation, not a combination (order matters). So we have to include "2 boys on Tuesday" twice as well.
In combinatorial Math, the MF and FM combinations are the same (order doesn't matter), so there's only 14 combinations where a boy is born on Tuesday, and 7 of them are both boys. Still 50%.
Mixing permutations and combinations is how you get to any other answer - 48, 33, or pretty much any other number depending on what other factor you include.
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u/OpportunityReal2767 1d ago edited 1d ago
The Tuesday is relevant, and the answer is 13/27 (or 48%, which makes sense given the 3% pass rate in this question. You’d think if 50% or 33% were the right answer, you have a much higher success rate.)
Here’s the whole explanation:
Here’s another deconstruction:
https://www.geeksforgeeks.org/aptitude/puzzle-44-girl-or-boy/
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u/Mindless_Insanity 1d ago
I don't like this, and I would fail this question. Because even knowing the answer, I refuse to believe it. It just does not compute.
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u/Moldovah 1d ago edited 1d ago
Yeah, even that Monty Hall problem is easier to comprehend than this.
EDIT: YES GUYS I UNDERSTAND THAT YOU FIND THE MONTY HALL PROBLEM TO BE EASY, WE ARE ALL VERY IMPRESSED WITH YOUR INTELLECT
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u/jeremy1015 1d ago edited 1d ago
I still don’t believe the answer is correct, having read and fully understood the problem. It’s just statisticians fooling themselves into thinking they’ve made a simple problem complicated. The answer is 50% as long as you’re not bringing genetic likelihoods into it.
Edit: For those writing simulation code to prove the answer, how about this simulation code? I don't see how this code violates the constraints of the problem as expressed, and what do you know, the answer is always very close to 50%
import random trials = 1_000_000 third_counts = {0: 0, 1: 0} for _ in range(trials): first = random.randint(0, 1) if first == 0: second = random.randint(0, 7) if second == 0: third = random.randint(0, 1) third_counts[third] += 1 total_third_rolls = sum(third_counts.values()) print(f"Total trials: {trials:,}") print(f"Third number rolled: {total_third_rolls:,} times") print(f"Third number was 0: {third_counts[0]:,} times " f"({third_counts[0] / total_third_rolls:.2%})") print(f"Third number was 1: {third_counts[1]:,} times " f"({third_counts[1] / total_third_rolls:.2%})")385
u/Mindless_Insanity 1d ago edited 1d ago
See I think you're right. What are the odds they have 2 boys is a different problem than what are the odds they have 2 boys and one of them was born on a Tuesday. Knowing one was born on a Tuesday really should be irrelevant. Are all the mathematicians just trolling us? Edit: left a word out
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u/AntonGemini 1d ago
It states at least one child is a boy, so it’s really about the odds the other child is a boy too.
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u/nurseferatou 1d ago
“Doc, what are the odds that I’m going to have a boy”
“Hmm, I’ll need to do a thorough evaluation of your medical history. First: your son, which day of the week was he born on?”
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u/Fraun_Pollen 1d ago
Depends. Do you start your week on Monday or Sunday?
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u/Cronos988 1d ago
That's exactly the misunderstanding.
We're told information about the world we're already in, not trying to reason about the future.
That's how these trick questions work: If you're trying to guess what world you're in, even random tidbits can change the probability, if they're related to how you know in the first place.
Because it's about knowledge, how you know is relevant.
If the family was (self-)selected based on the "boy born on a Tuesday" scenario, that changes the probabilities. Families with two boys have more chances to pass that filter.
If you independently discover that the boy you've been told about happens to be born on a Tuesday, nothing changes because this doesn't apply a filter, and knowing a random fact about this boy doesn't help you.
Interestingly, this is similar to why, in the Monty-hall problem, the host needs to know what's behind the doors for the "always switch" logic to work.
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u/brettersonx 1d ago
This would still be 1/2. In this case you know the older child is a boy born on Tuesday. In order for the question to work, you have to not know which of the two children is a boy born on Tuesday (and the answer could be both but not neither).
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u/fireintheglen 1d ago
Funnily enough, I think this is actually a good illustration of why people are misunderstanding this.
In real life, if you want to know whether a child will be a boy or a girl, you’re probably asking that question already knowing the gender of the previous children. And it is true that if you asked the question “the first child was a boy born on a Tuesday. What’s the chance that the second child is a boy?” then the answer would be 50%.
That is not the question being asked in this post. You don’t know that the first child is a boy born on Tuesday. You just know that if the first child is not a boy born on Tuesday, then the second child must be. This is a fundamentally different problem.
If I’m reading the code in the previous comment correctly, they also make the same mistake. They find the probability that the third roll is 1 given that the first two rolls are 0. They have added information about ordering that didn’t exist in the original problem. To do this correctly you’d have to remove the “if” statements and re-roll every time, but then throw away your answer if you got fewer than two 0s in total. You’d then count how many remaining sets of rolls contained any 1s.
This problem is unintuitive because in real life we almost always have information about ordering, and yet this mathematical problem does not provide any.
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u/Flamecoat_wolf 1d ago
Yes, but I think the mistake is in crossing the two potential datasets and eliminating the option present in each.
The article linked shows the first boy being born on Tuesday and a data set for them, and a second data set for if it's the second boy born on Tuesday.
They're not both going to be in play though. Whichever boy is the boy born on Tuesday he's set in stone. So regardless of whether he's 1st or 2nd, the other boy has all 14 options of being a boy or girl born on any day of the week.
Which means that it does actually come out as 7/14, or 50% chance.
You can also explain it from the point of view that if the first data set is looking at the possibilities for the second child, then the second data set is looking at the possibilities for the first child.
It therefore makes no sense to eliminate one potential option from a parallel universe were you're looking at the possibilities of the first child just because it was already accounted for in the first universe when you were looking at possibilities for the second child.I feel like we don't really have the adequate language to describe this accurately. Hopefully that makes sense though.
We're working with a superposition of both kids potentially being or not being a boy born on Tuesday.15
u/Plane-Scratch8974 1d ago
The reason in the explanation is:
However, the pair (Tuesday, Tuesday) gets counted twice, so we subtract 1. Favorable outcomes: 13.
I think this is wrong, because it's treating (Tuesday_1, Tuesday_2) the same as (Tuesday_2, Tuesday_1) - so eliminating one of the options.
But it does not treat (Monday, Tuesday) the same as (Tuesday, Monday).
If it did, it would come down to 7 favourable outcomes, rather than 13, as would the Boy/Girl options, so the final calculation would become 7/14 = 0.5.
Either you say order matters, and have to treat T1 and T2 separately, or you say order doesn't matter in which case you have to treat all pairs (T, X) the same as (X, T).
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u/Fun-Chemist-3139 1d ago
Birth order isn't really needed, that's just how they're choosing to show it. Here's an explanation without order, if it helps:
Consider the two-boy families on earth where neither son is born on a Tuesday. There's a 6/7 chance that one son isn't born on a Tuesday, so for neither of them to be, that's 6/7 x 6/7 = 36/49 of the two-boy families on earth that have neither son born on Tuesday.
So the proportion of two-boy families with at least one boy born on a Tuesday is the remaining 49/49 - 36/49 = 13/49.
On the other hand, to count the one-girl-one-boy families where the boy was born on a Tuesday, it's obviously just 1/7 = 7/49.
Now a quarter of the two-child families on earth have two boys, so 1/4 x 13/49 =13/196 of the two-child families on earth are two-boy families with a Tuesday boy.
And half of the two-child families on earth have a girl and a boy, so 1/2 x 7/49 = 14/196 of the two-child families on earth are one-girl-one-boy families with a Tuesday boy.
So of the (27/196 in total) two-child families with a Tuesday boy, there are 13 with two boys, and 14 with a girl and a boy.
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u/Plane-Scratch8974 1d ago
I follow this, and I agree with everything you've said.
I think what it comes down to is actually the wording of the questoin. Personally, I take "one is a boy born on Tuesday..." as turning this question into "what is the probability of one child being a boy, given you know the other one is a boy born on Tuesday".
Whereas, I think what you're calculating is the probability of "having" two boys with a Tuesday boy, where you can't say that one has already been born on a Tuesday.
If said to you: Given that I have a boy that was born on a Tuesday, what is the chance that my second child is a boy? Then the answer is obviously 50%.
That's different to: What is the proportion of two-child families, where at least one boy is born on a Tuesday, that have two boys?
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u/engineeringstoned 1d ago
The answers all seem to rule out a 2nd boy, also born on Tuesday.
Nowhere in the text does it do that.
"At least one is a boy born on Tuesday" .. so two boys born on Tuesday is a possibility.
The day does nothing.Another deconstruction seperates the options into 3:
B-B, B-G, and G-B
Which again is misleading, as the age, or "which came first" is not in the text.
The order does nothing.Collapsing this to 50%, as only the gender of the second child is a variable.
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u/jarlscrotus 1d ago
Which is still, basically, 50%, in actuality there is a small variation across populations and environments making it so that in isolated subsets the split actually skews by several points one way or the other, but as a whole, for humanity, it's 50%
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u/draaz_melon 1d ago
Yeah. They are full of shit with this one. At least one being a boy born on Tuesday has nothing to do with any information about the other child. Especially since it could also be a boy born on Tuesday. It could be any gender on any day. Completely independent event.
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u/Vinr_Odinn 1d ago
You are correct. The explanation states that the are three options BB BG and GB; however, if x is a B it can't be a G, so there are only two options.
Below in the article it states that for statistics they would account for all the days of the week, since the factor was provided; however, that ignores the original question which didn't ask for the factor inclusion.
This is just mathematicians making up extra filler that nobody asked for. The result is wrong. The question is simple and the intuitive solution holds true. The rest is filler from people who want to make up alternate probabilities for alternate problems.
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u/PraiseTalos66012 1d ago
I believe the posted one is written wrong. Instead of saying "at least one is a boy born on Tuesday" it should say "only one is a boy born on Tuesday" which then means the other cannot also be a boy born on Tuesday which gives you the fraction they claim.
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u/A7MOSPH3RIC 1d ago
Yeah, If you flip two coins. It's the same probably each time.
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u/TheHrethgir 1d ago
But it doesn't matter what the first child is when it comes to the second. Ea h child is about a 50% chance of being a boy or a girl regar3of the previous child or day of the week.
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u/DrBatman0 1d ago
I thought so too. I wrote simple code to run billions of trials, and it is in fact, just above 48%
Based on the assumption that M/F is 50/50, and each day of the week is 1/7. I don't yet understand it, and I wrote the code intending to prove the 48%ers wrong, but I stand corrected.
(I still don't understand why, but the numbers don't lie)
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u/PB219 1d ago
But the day of the week doesn’t matter, so why is it part of the code?
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u/DanceWonderful3711 1d ago
I also don't get it. They act like boy/girl and girl/boy are two different options, but they're literally the same thing.
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u/rosshm2018 1d ago
The relevance of the day of the week (which is where the supposedly correct answer of 13/27 = 48% comes from) is just utterly lost on me.
If the answer is 48% then the question is poorly written.
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u/djryan13 1d ago
I agree. I also don’t agree with adding the day into the calculation. Obviously the first child was born on A day. He was also born in a month and a year. None of that should matter and be added to the calculation just cause it’s mentioned. It’s added only because it’s mentioned…
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u/rosshm2018 1d ago
I think the question is supposed to be written as something like ~ "If there are two children, and at least one of them is a boy, what are the odds that both children are boys, with at least one of them born on a Tuesday?" But that's not what it says!
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u/stackingnoob 1d ago
Yea the problem was created by someone who understands probability but is not good with language.
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u/karlzhao314 1d ago
This code violates the constraints of the problem because you are completely ignoring the case where the second child may be the boy born on the Tuesday instead of the first. In fact, you didn't simulate at all which day the second child was born on.
Instead of doing [roll first child] -> [if first child is a boy, roll second child] -> [check if second child is a boy], you need to do [roll first child and second child] -> [check if either child is a boy born on Tuesday] -> [if they are, check if other child is a boy].
A simulation set up that way gives you approximately 48%.
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u/Volodya_Soldatenkov 1d ago
If you still think it's 50%, you haven't understood the problem.
You aren't asked whether a specific child is a boy or a girl, you are asked whether both of the random children are boys if we know there's at least one boy among them. These are different questions with different corresponding probability spaces, much like in the Monty-Hall "paradox".
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u/gribson 1d ago
That's because this is bullshit. It's a gambler's fallacy that gets bandied around as esoteric genius-math.
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u/Cavane42 1d ago edited 1d ago
The Monty Hall problem is pretty easy to understand if you make it a large number of doors. Say there are a hundred doors. You pick one. Then they open all of them but yours and one other. If you think it's still a 50-50 chance at that point, that means you think you had a 50-50 chance of picking the right one in the first place.
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u/Donut_Desperado 1d ago
That actually made me immediately understand the Monty Hall Problem, and I've been kicking it around in my head since I first saw it explained in 21.
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u/Embarrassed_Durian17 1d ago
Yeah the way i first learned the monty hall problem was that by switching doors your odds to win are now equal to the original odds to lose
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u/RNG_HatesMe 1d ago
100% correct. Put another way, by switching doors you are *exactly* reversing your original odds. So if the original odds were 1 in 3 (33%), you switch them to 2 in 3 (66%).
In the case of 100 curtains with 99 goats and 1 car, your original odds are 1 in 100. By switching, your odds become 99 in 100.
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u/OpportunityReal2767 1d ago
The only thing that might save me is if that question comes with the pass rate on it and I see that 3%, I know there’s a wrinkle to the question and it can’t be anything easy and obvious. But with a 90 second time limit, I’m not going to come up with that answer unless it’s multiple choice and then I do the game theory thing of eliminating the obvious answers like 50% and 33% because of that pass rate, and hope I make a good educated guess on the last two remaining (I’d probably pick the one that looks closest to 50%.)
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u/kyew 1d ago
This is probably a better interview answer than just getting it right.
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u/aksdb 1d ago
If you want to select for people with quick wits who make good estimations under pressure. But they are searching for a quantitative researcher. So actually selecting people who get these statistical nuances and/or have seen this very problem because they must have encountered it during their education, makes more sense.
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u/doppelbach 1d ago
This is a fairly common statistics 'paradox'. For once, it's actually worded very precisely.
Sometimes you'll see "the older one is a boy born on Tuesday", which completely changes conditional and reverts it to a simple 50/50.
Or sometimes it's "my son was born on a Tuesday" which is deliberately misleading because in any actual conversation that would mean they only have one son.
Sometimes the parent tells you "one of them is a boy born on Tuesday" which is actually kind of ambiguous. It doesn't preclude two boys born on Tuesday but in natural speech that is the obvious implication. And if we rule out two boys on Tuesday then it drops to 46%.
And no one ever considers how the parent decided which kid to tell you about. If it was randomly selected, then we actually have to give extra weight to the two Tuesday boys situation.
Basically the problem is meant to be confusing. This is the most clear I've seen it, where it's gently nudging you to think of conditional probability.
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u/Right_Lengthiness266 1d ago
It's still not a clear question for the same reason the basic boy-girl paradox doesn't have a clear answer.
It depends on how we came to learn Mr. Smith has a boy born on Tuesday. Unless he was specifically selected from the "group of people with two kids, at least one of which is a boy born on Tuesday", then the answer is probably 50/50. If you just picked someone with 2 kids at random, and asked them to randomly select one of their kids and share the gender and the day of the week they were born, it doesn't have any impact on the probability of the gender of the other kid.
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u/doppelbach 1d ago
Yeah great point, I was trying to get at the complexity of how they were selected near the end. but you put it better. The only way to get 13/27 is to make the exact same assumptions the author did. Feels like these questions are popular for the same reason the order of operations questions go viral on facebook or whatever: deliberately ambitious wording so people confidently correct everyone with different assumptions
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u/slartiblartpost 1d ago
Agree. Never liked this problem as it is usually ambiguously stated, leaving away the "at least". Here its clear
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u/Beli_Mawrr 1d ago
The way I have it in my head is "no one ever says 'i have a boy and a boy born on a tuesday'c
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u/deviousbrutus 1d ago
Yeah. That's the part I disagree with as well, but I don't think either reading effects the math?
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u/Beli_Mawrr 1d ago
The article goes into how easily statistics are played with in this regard. It feels like the answer is different depending on how its said and who said it
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u/Interesting_Gap7350 1d ago
But that's the job they're applying for though.
If they are able to tease out a 2% edge just based on the wording or order of a few statements, and ability to isolate prior knowledge vs new evidence.
So if they can see that everyone else thinks it's 50%, but it's really 48%.
Then they get paid $650k and the company makes about $6.5m
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u/andrew_calcs 8✓ 1d ago edited 20h ago
Imagine 50-50 odds of boy-girl, and 7 permutations of days of the week, all with equal probability. Each child has 7*2=14 permutations of biological sex and day of birth. So there are 142 =196 possible permutations.
In 14 of them, the first child is B-Tue. In 14 of them the second child is B-Tue. One of those cases where both are B-Tue overlap, so there are 27 cases of the original 196 permutations that fit the minimum criteria to be considered.
27 is an odd number. From that alone you can already tell that the number can’t be equal. And indeed it isn’t. 13 cases have two boys, and 14 have a boy and a girl.
13/27 = 48%
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u/dkoucky 1d ago
This is where I don't understand. Why can't it be counted twice? Two boys Rob and Joe. Both born on a Tuesday in one scenario Rob is the eldest and in the other Joe is the eldest. Why does this cancel when both scenarios are possible? Which is impossible and cancels out?
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u/SomberDjinn 1d ago
This is also my issue. I don’t ever remember “discarding” equivalent outcomes when creating an outcome matrix when learning probabilities. If there are two ways to get an outcome, both paths are included in calculating the probability.
Also consider this, a child will always be born on some day of the week. Whether that day is named or not is irrelevant. So when the day is not named, is it still 48%? Not according to the author.
I think this problem is wrong.
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u/Charming-Ad6575 1d ago
You would be correct.
It is wrong, or at least the methodology people are using here is.
You're exactly right for exactly the right reasons.
Both children are not the same boy, that's nonsense. If that is nonsense, then both possibilities should be in the set, and for some reason I just cannot fathom, people discard one because they "feel" identical.
You'd think they'd get a clue when you get a different percentile from arbitrarily changing the time set size, eg. day of week to day of year or simplifying day of week to coin flips.
If you calculate the set correctly, regardless of the time input, you ALWAYS get 50%.
That's a huge clue that the operation is correct.
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u/Embarrassed_Durian17 1d ago
When put into algebraic terms I understand why it's 48% but under the original wording i see the fact that one boy being born on a Tuesday as statistically irrelevant information and that the next child born is 50% boy or girl.
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u/ShaggysGTI 1d ago
I’d spend seconds on it making an estimation and move on. Okay, you’re not okay with my guesstimating thats 98% correct? I’ll just go elsewhere.
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u/Graftington 1d ago
The weird part to me is that it becomes a math problem (which like you doesn't make sense to me) but it doesn't take into account biology or genetics which seems like it would take precedent over a hypothetical math outcome.
I'm pretty sure girls are more common than boys? And there is something about subsequent children and affects on gender?
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u/Metals4J 1d ago
If we are doing that, do we need to take into account culture, place, and time in history just in case there was a societal preference for one gender over another?
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u/JustAGuyInTampa 1d ago
Just wait until you find out about the minimum number of people needed in a room for there to be a 50% chance two have the same birthday.
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u/Mindless_Insanity 1d ago
I knew about this one, and it makes sense to me. I believe the other one is really a trick question. They're asking what are the odds they have 2 boys, but they want the answer to "what are the odds they have 2 boys and one of them was born on a Tuesday?". That's the trick. I'm pretty sure.
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u/BitWarrior 1d ago
You know why I hate questions like these? Because some guy opined on the answer for fucking years, and in an interview you have 10 seconds to figure it out.
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u/OpportunityReal2767 1d ago
Well, the thing is, I wonder how many of the 3% who solved this have seen this question before and how many deduced it in the 90 seconds. I’m not a computer guy or quant or statistician, but I’m almost certain I’ve seen this exact question somewhere else before, so I doubt it’s a unique interview question. And it wouldn’t surprise me if some or even most of the 3% who got it right have seen it before.
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u/PopupAdHominem 1d ago
The "right" answer might not even be correct. The way this question is written the answer is 50%. There is nothing in the question that says the other child can't have been born on a Tuesday.
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u/slinkymcman 1d ago
If Tuesday is relevant than the boy is also and the answer is 0%. One is a boy the other is a girl. there’s the “I have 35c in two coins, one isn’t a quarter” riddle which is basically the same but with a different answer.
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u/banananuhhh 1d ago
The trick is that there are only 13 permutations for which days two boys could be born but there are 14 when the other sibling being a girl. That is because 1st child=Girl Tuesday 2nd child=Boy Tuesday and 1st child=Boy Tuesday 2nd child=Girl Tuesday are distinct cases while 1st child=Boy Tuesday 2nd child=Boy Tuesday is only one case. That's why it's 13/27 instead of 14/28
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u/PopupAdHominem 1d ago
Why are there only 13 permutations for which days two boys could be born?
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u/PhilWham 1d ago
Because u already know at least 1 of them was born on a T. B1T/B2T is the same as B2T/B1T so you can't count it twice.
The chart in here (posted above) makes it really easy. https://www.geeksforgeeks.org/aptitude/puzzle-44-girl-or-boy/
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u/pikuray 1d ago
Just another way for HR to feel superior to others once again, not based on any personal merits of their own since their job requires none so they find riddles like this to feel superior to people vastly more qualified than them....
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u/Metabolical 1d ago
It's what I used to call a trivia question. You either know it or you don't. If you don't, your chances of deriving it on the spot are low.
I heard people doing interviews asking how to draw a line on a computer using only integer math. I scolded them because it's not reasonable to ask someone in an interview to derive an algorithm that has a guy's name on it. (Bresenham)
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u/die_lahn 1d ago
Yeah I took a test to work at a water treatment plant after school and about half the questions were questions that you would only know the answer to if you’d already worked in the building you’d be working in, it was wild.
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u/Humg12 1d ago
That probably means they wanted to do an internal promotion but were forced by either law or company policy to do an open interview.
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u/ReversedNovaMatters 1d ago
Those aren't intelligence tests, those are memory tests.
Sounds like they didn't care if you were smart, just if you could remember processes.
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u/tapewizard79 1d ago
I feel like the deconstruction makes assumptions and assertions that aren’t relevant or in the data provided in the question to reach its conclusions.
I’m also an idiot looking at the answer making that confused Jackie Chan face so what do I know?
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u/ringobob 21h ago
The assumptions that it makes that it hasn't stated are:
births are spread equally between boys and girls (mostly true, at global scale, much less true when you get into individual communities, even at the scale of an entire country)
births are spread equally among the days of the week (definitely not true at any scale)
I believe the earliest construction of this question stated those two assumptions explicitly and they've just been lost to time, but the answer based on those assumptions has continued on, so as given, it's no longer accurate. I have seen versions of this question that do state those assumptions. I'm guessing that someone just didn't realize how important they were to getting the specified answer.
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u/PopupAdHominem 1d ago
I feel like the deconstruction makes assumptions and assertions that aren’t relevant
Yep. Nothing in the question says both kids can't have been born on Tuesday. People just add that in somehow and then do math. It is weird.
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u/tofuking 1d ago
The only criticism of this problem is that they don't state "assume independent and uniform pro abilities of birth days and genders", but if that seems unreasonable they probably don't want this person as a candidate anyway
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u/Card-Middle 1d ago
Did you read the linked explanations? The math that results in 13/27 specifically accounts for the possibility that both could be boys born on Tuesday.
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u/GrandmaPoses 21h ago
But were they not born in a certain year as well? Should they not then assign the genders for every day of every year, accounting for leap years as well? I'm sorry but the analysis of the answer is just academics jerking themselves off.
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u/EconJesterNotTroll 1d ago
You say that all over the thread, but that is NOT what people are doing. They are counting two Tuesdays boy births in the same family once, instead of mistakenly double counting them. There are 196 combinations of two children with random gender and day of birth. Of those, one (call them the Jones) has two boys on Tuesday. After we remove all the families with no boys on Tuesdays, how many families have, two boys on Tuesday?? ONE. The Jones family. Eliminating families without Tuesday boys doesn't multiply them or magically double them. They are still one family, now out of 27.
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u/Oh_My_Monster 1d ago
Both of your sources have the same explanation for this but I have a big issue with the explanation. Maybe someone can explain if my thinking is wrong or if it's a legitimate mistake.
They say that that the Tuesday-Tuesday condition shouldn't be counted twice which is what makes it 13/27 instead of 14/28. The problem is they never explain why that condition shouldn't be counted twice even though the Tuesday-Sunday condition is counted twice as is Tuesday-Monday, Tuesday-Wednesday, etc. They ONLY don't count Tuesday-Tuesday twice.
If we count each condition only once we get 7/14 or 50% and if count each condition twice we get 14/28 or 50%. We only don't get 50% by mysteriously not counting just one condition two times.
The second link is better because it has a table.
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u/SivirJungleOnly2 1d ago
I have a PhD in physics and took a Bayesian stats grad course, here is the actual explanation:
The question is written in a VERY misleading manner.
Abstractly, the boy being born on Tuesday doesn't matter, the probability is 1/3.
Where it matters is if the information is given to you as the ANSWER to a question, where you ask "Is there at least one boy born on Tuesday?" and you're told that the answer is yes.
The CRITICALLY important part is that what you asked was a QUESTION, such that if there was no boy born on Tuesday, you could have received the answer "no." The increase in probability comes from the fact that you CORRECTLY guessed a day that at least one boy was born on.
A comparative analogy is to imagine that a couple has 10 children and you are told they either have 1 boy and 9 girls or 9 boys and 1 girl. You then ASK if they have a boy born on Tuesday. If you are then told "yes," you now know it is extremely likely that they have 9 boys and 1 girls, because otherwise it would require you to have guessed exactly the day that the 1 boy was born on.
THIS is where the increase in probability comes from, from the fact that you are much more likely to correctly guess a day when a boy was born if there were two boys than if there was only one boy.
But, if you are just told that there is at least one boy, and then one boy is randomly chosen and you are told their birthday just happens to be Tuesday . . . that changes nothing. You already knew that the boy was born on some day, finding out the specific day they were born doesn't matter.
The entire "trick" of this problem is that the way it is phrased is extremely misleading and quite literally under-defined/underdetermined for what the corresponding statistics are.
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u/Ilovecatsss2026 1d ago edited 1d ago
I remain thoroughly unconvinced by this reasoning. The writer invents a distribution based on spurious semantics that is irrelevant to answering the question as stated.
This is like saying the probability of a coin toss coming up heads is different from 1/2 if I tell you the coin is owned by a man and another coin was flipped by unknown people. It only works if you create a hidden problem other than what you've stated in the question.
I suspect those of you who care about the other coins would make for eager gamblers at the roulette table.
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u/icravedanger 1d ago
If I flip 2 coins and one of them is heads, what’s the probability that the other is also heads?
It depends on how I obtained the information that one of them is heads. If I say “I won’t look, you look and you tell me if at least one of them is heads, and you say “yes, at least one of them is heads”, then the answer is 33%.
If during the flip, one coin rolls under the couch where I can’t see it, but I see the other one is heads, then the answer is 50%.
Same situation here- how did I obtain the information that one of the boys was born on Tuesday? If I put out a survey that says “if you have 2 children and one of them is a boy born on Tuesday, please respond with the sex of your other child.” Then the answer is 13/27. If I’m walking down the street and I see a guy walking with a kid and he says “have you met my son? His favorite day of the week is Tuesday because he was born on Tuesday”. Then the answer is 1/2.
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u/Wulf2k 1d ago
I flipped two coins.
Last Tuesday, I got Heads.
What are the odds that my next coin flip will be Heads?
....why is this a different question?
Also, is it a different question?
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u/Independent-Box-8282 1d ago
It’s different. In your case, you’re specifying that the first coin flip is heads. In the BG case, either one can be the specified boy.
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u/tapewizard79 1d ago
I have flipped a coin twice in my life. Once, on a Tuesday, it landed on heads. What are the odds that the coin landed on heads both times?
How’s that? I feel like this just further highlights the irrelevance of Tuesday. I feel like calculating this with Tuesday involved is just calculating with irrelevant information as if it mattered. Like yes, you’re getting what the correct answer would be if Tuesday made a difference, but it doesn’t and shouldn’t be part of the calculation.
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u/Wjyosn 1d ago
This does actually result in the same outcome as the original question.
Think of it this way:
A trillion people each flipped two coins in their life.
Of all of those flips, some number of them happened on tuesdays (1 in 7), and some number of them were heads (1/2).
Cut the list of outcomes down to only the ones that had at least one heads happen on a Tuesday: that is, only the outcomes that include TuesdayHeads+NotHeads, TuesdayHeads+NotTuesday, NotHeads+TuesdayHeads, NotTuesday+TuesdayHeads, TuesdayHeads+TuesdayHeads are relevant - we use the information given to trim down to only those situations.
Of *those* situations, we need to further differentiate. Any of the "NotTuesday" needs to be split into "NotTuesday but still heads" vs "neither Heads nor Tuesday".
Out of all of those 7 categories, how many had two heads?
So the count of:
([TuesdayHeads+TuesdayHeads] + [TuesdayHeads+NotTuesdayButStillHeads] + [NotTuesdayButStillHeads+TuesdayHeads])
Divided by the count of all 7 categories (and not any of the stuff that was cut out by the original conditions)
The result simplifies down to 13/27, or about 48.1%
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u/Dr_Chickenbutt 21h ago edited 19h ago
the point you're missing is the coin flips (or births) are completely independent events. The probability of tossing a head when you're holding a coin is 50%. The potential outcomes sets are meaningless in this case.
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u/Popetus_Maximus 1d ago
Another way to see why this is non-sense. I throw two dice. At least one of them is a 1. What is the probability that the sum is 5?
The probability is 1/6. Period.
It does not matter that (1,1) appear one in the consideration set and (1,2) and (2,1) are considered two elements in the consideration set. One die is fixed the other has 1/6 probability to land on each number.65
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u/CaptainDogeSparrow 1d ago
That's why I think those kind of questions and answers are from failed stochastic studends.
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u/drumeeney 1d ago
Yep. It's just phisophical mathematical wank.
The real life answer to the question as phrased, with as much info as provided is 50%.
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u/realbobenray 1d ago
Yeah I think the question is meant to learn if the candidate is practical (~50%) or is a wanker who thinks way too much of their own reasoning ability (any other answer)
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u/RoboChrist 1d ago
No, because then it wouldn't be a 3% pass rate.
They're looking for the wankers.
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u/-KFBR392 1d ago
Even a genius wanker couldn’t get this on their own in 90 seconds, this test only shows who has stumbled into this question and answer in the past and remembered it.
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u/Edward_Durr 1d ago
Of course the people looking to pay a statistician $650k are looking for the wankers.
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u/ghunor 1d ago
so, just for funsies I wrote a simple python script to roll a million sets of dice. 2/11 was the ratio fo sets of dice that had at least 1 one die and summed to 5. NOT 1/6.
You can see and run it yourself https://pym.dev/p/2kb8s/
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u/jxf 5✓ 1d ago edited 1d ago
Another way to see why this is non-sense. I throw two dice. At least one of them is a 1. What is the probability that the sum is 5? The probability is 1/6. Period.
Hoo boy. You're not going to like that the correct answer is actually 2/11, are you?
The key difference is that knowing some die is a 1 without knowing which one is a weaker piece of information. This changes the answer.
Here is a (Python) computer program that demonstrates this by rolling 1 million times, if you don't believe me:
``` import random
random.seed(0) N = 1_000_000
at_least_one, at_least_one_5 = 0, 0 first_is_one, first_is_one_5 = 0, 0
for _ in range(N): a, b = random.randint(1, 6), random.randint(1, 6) if a == 1 or b == 1: at_least_one += 1 at_least_one_5 += (a + b == 5) if a == 1: first_is_one += 1 first_is_one_5 += (a + b == 5)
print(f"at least one 1: {at_least_one_5} / {at_least_one} = {at_least_one_5/at_least_one:.4f}") print(f"first die is 1: {first_is_one_5} / {first_is_one} = {first_is_one_5/first_is_one:.4f}") ```
Output:
at least one 1: 55806 / 306306 = 0.1822 (exact 2/11 = 0.1818) first die is 1: 27799 / 166484 = 0.1670 (exact 1/6 = 0.1667)Conditional probability is extremely unintuitive for humans, so it's not surprising that people get this very wrong. It's hard!
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u/Humg12 1d ago
Isn't the difference coming from the assumptions made. You're assuming that the only statement that can be made is "At least one of them is a 1", and all sets that don't include at least one 1 are discarded. The commenter you're replying to is assuming they look at one of the die randomly and say "At least one of them is an X", with X being whatever number they saw.
I think your assumption is more egregious. There's no reason to assume that Mr Smith was selected because one of his children was a boy born on a Tuesday; it's presented as coincidental information.
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u/BlueMacaw 1d ago
You are absolutely correct and this entire thread has me annoyed that the wrong answers are being voted to the top.
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u/PB219 1d ago
How is the Tuesday relevant? The 2nd article made it relevant, but that doesn’t mean it needs to be. It just asks what the probability that both kids are boys, knowing that one of them is. If you know one kid is a boy, why would you not just need to determine the probability of the other kid being a boy?
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u/rhou17 1d ago
I know you didn’t write it, but this is just stupid and pedantic. You either clearly write out the ambiguities and it’s not an interesting question, or it’s a test to see if you’ve seen this exact question before. No sane person is reading that question for the first time and interpreting it that way lmao.
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u/Party-Court185 1d ago
Idk man. Statistically I guess that checks out. But real world, applicable statistics says that this is a 50/50. I understand how adding the Tuesday bit makes you expand the data set for every day of the week, but at the end of the day that is realistically irrelevant.
If you want to argue that women are slightly more common than men, whatever. But for all intents and purposes, I’m going with 50% as the answer.
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u/epicmylife 1d ago
Right, like the math says it works out but one could make the argument from a biological perspective that makes no sense. If you’re a quant trader you want to trade on statistics that matter, not theoretical statistics games that have no reflection on real world processes.
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u/fullsails_openseas 1d ago
Statistics that matter, 100%... https://pubmed.ncbi.nlm.nih.gov/15974501/
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u/jencrs 1d ago
Not really relevant, but there are more boys born than girls in general
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u/skintigh 1d ago
This still makes no physical sense. If I didn't give you Tuesday, you're saying the answer would be different?
In the real world, someone is still born on a day even if that day isn't given. So someone is born on one of the days, and it makes no difference which day when it's not given, so why does the day of the week matter when it is given?
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u/BruceBoyde 1d ago
I kinda track, but why are we taking out the possibility that both children are boys born on Tuesday? It says "at least" one of them is, not that only one is.
I'm surely missing something, but I'm hoping someone can explain.
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u/SignificantLock1037 1d ago
The problem with this is that it ignores the reality that babies are 2.5% more likely to be boys.
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u/Odd_Dragonfruit_2662 1d ago
The math is wrong though because males are born at a slightly higher rate than females, at about 51.2% vs 48.8% in natural human populations not engaging in sex selective abortions (which tend to push males even higher).
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u/zbtiqua 1d ago
The question is worded badly. If you are simply identifying a child being born on tuesday, then the answer is 50/50. The question secretly injects an unstated rule: that you condition uniformally on all two child families with at least one tuesday-born boy. In this case, the correct answer is 13/27.
Child 1 can be a boy or girl born any day of the week. 14 possibilities.
Child 2 is the same. 14 possibilities.
But one case is double counted- the case with one boy born tuesday. 14+14-1=27. So there are 27 possible combinations.
If the first child is born tuesday, there are 7 possibilities for the other to be a boy. Same if the second child is a boy born Tuesday. So 14 chances. But one is double counted again, 14-1=13.
13/27=48.15%. That is the answer the interviewer wants.
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u/dspisayuss 1d ago
God I feel so dumb because I don't understand how any of this works lol.
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u/Current_Swan_2559 1d ago edited 1d ago
While the math is correct, the example it's being used on makes it make no sense. If we don't know the day of the week the boy is born on, they'll argue it's 66%. If you take into account the hour it was born on, we move even closer to 50% than 48.15%. It gets even magically closer if we consider the minute the boy was born on.
The actual correct answer in real life is 50% because the gender of one child and the time of birth is a completely independent event from the gender of the next child. They're not correlated at all. It's like trying to prove flipping heads is more likely because you flipped tails beforehand. The coin flips do not impact each other, neither does the gender of a child.
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u/deadvicariously 1d ago
This is what bothers me about these questions because there's the actual answer which is 50/50, then some math wizardry to conjure up parameters to restrict the real answer.
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u/urbfunsac 1d ago
Fun fact, technically it's genetic I.e. Some couples have a greater probability of having boys and others girls, it's very interesting
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u/Zucc 1d ago
It's a stupid premise. If I changed "Tuesday" to "the 3rd day of the month" or "the 234th day of the year", does that change the odds?
Of course it does, using this logic. Which is why this answer is detached from practical reality.
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u/textbookamerican 1d ago
Yeah great point!! Also if the first child dies does the odds of another boy change? Henry the VIII wants to know
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u/SivirJungleOnly2 1d ago
I have a PhD in physics and took a Bayesian stats grad course, here is the actual explanation:
The question is written in a VERY misleading manner.
Abstractly, the boy being born on Tuesday doesn't matter, the probability is 1/3.
Where it matters is if the information is given to you as the ANSWER to a question, where you ask "Is there at least one boy born on Tuesday?" and you're told that the answer is yes.
The CRITICALLY important part is that what you asked was a QUESTION, such that if there was no boy born on Tuesday, you could have received the answer "no." The increase in probability comes from the fact that you CORRECTLY guessed a day that at least one boy was born on.
A comparative analogy is to imagine that a couple has 10 children and you are told they either have 1 boy and 9 girls or 9 boys and 1 girl. You then ASK if they have a boy born on Tuesday. If you are then told "yes," you now know it is extremely likely that they have 9 boys and 1 girls, because otherwise it would require you to have guessed exactly the day that the 1 boy was born on.
THIS is where the increase in probability comes from, from the fact that you are much more likely to correctly guess a day when a boy was born if there were two boys than if there was only one boy.
But, if you are just told that there is at least one boy, and then one boy is randomly chosen and you are told their birthday just happens to be Tuesday . . . that changes nothing. You already knew that the boy was born on some day, finding out the specific day they were born doesn't matter.
The entire "trick" of this problem is that the way it is phrased is extremely misleading and quite literally under-defined/underdetermined for what the corresponding statistics are.
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u/No_Somewhere_2295 21h ago
Thank you. This is the explanation for which helped me too.
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u/cbbclick 1d ago
That's what I don't understand.
If you say he was born on a day of the year, instead of 27 options, you suddenly have over 700. 364/729. Or wait is it 1400?
The answer approaches 50% depending on what additional information you know. Because the actual odds are 50%, and the extra information is obscuring it.
The Monty Hall problem makes sense. But if sometime got the jackpot last Tuesday, that wouldn't affect you getting it today. The extra info of the empty door would.
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u/CommercialAgency928 1d ago
It says "at least one is a boy born on a Tuesday", so why are you taking out the 1? It doesn't say "just one is a boy born on a Tuesday".
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u/Stampon 1d ago
this is the only correct answer in this thread right now. if you are curious to learn more: https://en.wikipedia.org/wiki/Boy_or_girl_paradox
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u/Xattle 1d ago
Thanks for the link. The more I read the more I need the Patrick and Man-Ray license meme version of this. Mainly what I'm gathering is because of the ambiguity of the question, using it as a good interview question would be more about how they defend/discuss their answer instead of looking for a number.
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u/AdorablePineapple95 1d ago
I dont understand why the tuesday thing is relevant given that sex probability is independent from day of birth…
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u/PopupAdHominem 1d ago
The question secretly injects an unstated rule:
Nope, people are injecting an unstated rule into the question.
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u/trunksshinohara 1d ago edited 21h ago
I'm gonna get downvoted and mobbed. But the Tuesday information is irrelevant. If the question said at least one of the boys likes pineapples. You wouldn't be doing math for every fruit. All the people saying Tuesday is important are working this problem backwards from the stand point that Tuesday is key. One boy is born on Tuesday is a fact not a question. So it's irrelevant. There is a second child. That is a fact not a question.
The question is asking will the second child of a family that has one boy be a girl or a boy?
The same is true if the question was reversed. If it were asking a family has a second child that is a boy. Will the first child be a girl or a boy?
You can replace Tuesday with literally anything it's not important to the question. Feel free to downvote.
Editing to add that the biggest mistake everyone getting the incorrect answer is making is that they are assuming Tuesday is referring a day of the week. (Tuesday could be referring to anything) Which is still irrelevant because the question isn't asking about dates.
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u/Orange_Tang 1d ago
I upvoted you because you're right. Adding irrelevant things to the question doesn't make them relevant. All the calculations getting an answer that isn't 50% are taking into account information that is irrelevant to the gender of the child.
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u/Ectorious 18h ago
I mean removing irrelevant information from word problems was like one of the first things they teach you when you get to word problems in elementary school math classes.
I feel like as people got older and more educated they forgot that basic principle, and are smart enough to argue themselves into believing they have uncovered a secret way of solving the problem. They’ve convinced themselves that they are smarter than… themselves, and everyone else before them. It’s some sort of convoluted logical fallacy taken a numeric form. They’ve created an entirely different question than the one being asked of them.
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u/trunksshinohara 1d ago
Replying to my own comment because it's been bothering me since I posted. This isn't a monty hall problem as in that the answer was between 3 doors and eliminating one. The statistics change when you eliminate one wrong answer.
This is actually a St. Ives riddle (from Die Hard). https://en.wikipedia.org/wiki/As_I_was_going_to_St_Ives
Adding in all this irrelevant information that isn't part of the question.
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u/Boring_Holiday9874 19h ago
People who think Tuesday is relevant are exactly the people the question seeks to filter out. First step of finding a solution is knowing what the problem is and cutting out the noise.
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u/BlueGreenMikey 1d ago
The answer is that people should be more careful when they use words to try to describe a math problem. The English language is often ambiguous and up for interpretation, and asking questions sloppily is bad.
If I were the interviewer, the response that I would hope to see is the applicant asking follow-up questions to ensure the question is being interpreted properly, rather than just giving an answer assuming that my words meant exactly what I meant.
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u/graaahh 1d ago
So I read the article and I believe I understand the confusion (for anyone who's lost).
Three ways of looking at this. One, you could do a punnett square of the four possibilities (2 boys, 2 girls, 1 boy and 1 girl, 1 girl and 1 boy), then discount the one option with no boys. Suddenly, there's a 1/3 chance of 2 boys, or 33%.
Or, you could say it doesn't matter whether the kid born on a Tuesday is a boy, that's a given. What matters is if the other kid is a boy. There's only two options here, so the odds are 50%.
Then, if you're a mathematician who's paid to be pedantic, you say this. Either kid 1 was a boy born on a Tuesday, or kid 2 was a boy born on a Tuesday. That means the other kid was either a boy or a girl, that was born on one of the days of the week. So you have seven possible days, times two genders, times two because one time you count those if you're talking about the first kid and one time you count them if you're talking about the second kid. (If this sounds asinine, just wait, because we're still at 50% odds of two boys here.) THEN, you say "wait, in both sets (first kid vs second kid) I've counted two boys born on Tuesday, so I should scratch one of those out!" And instead of 14/28 of the possibilities having two boys, now only 13/27 have two boys, or around 48%.
So the issue here between the pedantic answer and the obvious answer of 50% is whether you scratch out that extra Tuesday boy or not. Personally, I don't think you should, because it does not impact the gender of the other child in any way. I think it's a cleverly disguised mathematical trick to make it seem like it's relevant but I don't think it's justifiable. And if anyone argued with me that 13/27 was correct, I'd tell them they forgot to take into account the difference in probability between a girl or a boy baby being born.
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u/Honeybadger2198 1d ago
Both children could be born on a Tuesday. You're not double counting shit, they're independant scenarios. You're not drawing 2 cards from a 28 card deck, you're rolling 2 28 sided dice.
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u/RedditSucksMucho 1d ago
But isn’t the third answer wrong. Why can’t both boys be born on a Tuesday?
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u/miggyg6 1d ago
The problem i have with he 13/27 answer is why would you cross out the double counted kid A:tuesday-boy kid B: tuesday-boy? If you assume the question is giving a 50/50 over which kid was the tuesday boy, you would have to consider that the double tuesday-boy outcomes are independent outcomes, one where kid A is the one in the question stem and one where it's kid B.
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u/abbeyadriaan 1d ago
Imagine a stadium with 1000 parents of 2 kids standing up. First, you ask everyone who does NOT have a boy to sit down. 250 people sit down. There are 500 with boy/girl, and 250 with boy/boy.
Then you ask everyone who does NOT have a boy born on Tuesday to sit down. For the ones who have only 1 boy, they have 1 shot on having a boy born on tuesday. So that's 1/7 of them will keep standing. That's 500/7 = about 71 people who will keep standing. The ones with 2 boys have 2 shots, so 7 * 7 = 49 options. Of those 49 options, 13 options have AT LEAST 1 boy born on tuesday. One option even has 2 boys born on Tuesday, lucky parents! So of the 250 * (13/49) = about 66 people keep standing.
In the end, you have 71 + 66 = 137 people standing. Of those, 66 have 2 boys. That means 66/137 = about 48% of the people still standing.
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u/CJFiddler 1d ago edited 1d ago
is the answer not 50%?
You already know that one is a boy, so there is a 50% chance that the other child is a girl and 50% chance it is a boy.
This isn’t like a Monty hall door problem where you are changing your answer after finding new information.
Edit - I’m upset and I don’t like math anymore
Ok so I’m wrong not once but twice.
Assuming I don’t care about Tuesday, like any normal sane person in the world, the answer is 1/3 (NOT 1/2 like I thought). That is because the problem doesn’t specify whether the boy is the first or the second in the pair, and so there is B/G, G/B, or B/B. There are only 2 boys in one of those three options. This is a famous paradox called Bertrand’s box or something similar.
Assuming I DO care about Tuesday like a maniacal statistician, the answer is 13/27 or 48.1% because fuck you that’s why
Edit 2 - somebody wrote a program that brute forced the answer over millions of permutations. Thank you Dr. Batman or whatever your name is you beautiful genius
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u/SivirJungleOnly2 1d ago
I have a PhD in physics and took a Bayesian stats grad course, here is the actual explanation:
The question is written in a VERY misleading manner.
Abstractly, the boy being born on Tuesday doesn't matter, the probability is 1/3.
Where it matters is if the information is given to you as the ANSWER to a question, where you ask "Is there at least one boy born on Tuesday?" and you're told that the answer is yes.
The CRITICALLY important part is that what you asked was a QUESTION, such that if there was no boy born on Tuesday, you could have received the answer "no." The increase in probability comes from the fact that you CORRECTLY guessed a day that at least one boy was born on.
A comparative analogy is to imagine that a couple has 10 children and you are told they either have 1 boy and 9 girls or 9 boys and 1 girl. You then ASK if they have a boy born on Tuesday. If you are then told "yes," you now know it is extremely likely that they have 9 boys and 1 girls, because otherwise it would require you to have guessed exactly the day that the 1 boy was born on.
THIS is where the increase in probability comes from, from the fact that you are much more likely to correctly guess a day when a boy was born if there were two boys than if there was only one boy.
But, if you are just told that there is at least one boy, and then one boy is randomly chosen and you are told their birthday just happens to be Tuesday . . . that changes nothing. You already knew that the boy was born on some day, finding out the specific day they were born doesn't matter.
The entire "trick" of this problem is that the way it is phrased is extremely misleading and quite literally under-defined/underdetermined for what the corresponding statistics are.
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u/conscious-clue-243 1d ago
I will die on this hill: the answer is 50%
Those saying that you are counting ‘Tuesday Boy Tuesday Boy’ twice are just stupid. Give the boy in the question a name, say Rick, then you can extend the problem to ‘Tuesday Boy (Rick) Tuesday Boy (not Rick)’ and ‘Tuesday Boy (not Rick) Tuesday Boy (Rick)’.
Children have names.
Parents never give their kids identical names.
I will die on this hill, come and fight me.
(all of the above assumes that the chances of giving birth to a boy is the same as the chances of giving birth to a girl… in reality, it’s not 50-50)
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u/BlueBod50 1d ago
The day of the week is irrelevant. Nature does not operate on days of the week; it operates on whether an X- or Y-bearing chromosome penetrated an egg cell. You have 2 children, at least 1 is a boy. The second is a coin flip. The answer is 50%.
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u/GalacticDisc 1d ago
Why is no one bringing up the 600k starting compensation? I guarantee this isn’t a real application. Researchers unfortunately are massively underpaid.
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u/Background-Spot-5068 1d ago
Sure, but are they massively underpaid at one of the biggest hedge funds in the world that is willing to pay for even the smallest advantage it can muster, in this case, hiring the 1% of 1%? There are hundreds of thousands of applicants trying to get into the financial companies in NY. Companies can be as picky as they want. They
Reward those who make it.It’s also at the top of the page as a mental distraction. It adds a Level of stress to the interviewee because they don’t want to lose this golden egg of an opportunity. Citadel wants to see you think under extreme stress AND still get the right answer.
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u/OkHelicopter1756 1d ago
this isnt a research role its hiring for the math guy of a top finance firm
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u/voracious-ladder 1d ago
Researchers in academia sure. Researchers in industry are generally compensated very well. At least for STEM, not sure about other areas.
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u/HumbleIntroduction71 1d ago
Quant traders and researchers make 500k+ straight out of college, many make over a million after a few years. It's probably the single highest paying career possible out of college (not including athletics of course), though you kinda need to be a math / computer science genius.
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u/tigerking615 1d ago
Nah, Citadel pays extremely well. This isn’t even a senior position, probably. It’s really boring work though and you have to work a fuckton.
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u/liam_denaldson 1d ago edited 13h ago
Help me understand how it's not 50/50 My thought process goes, we know that one of the kids is a boy so that leave only two options the oldest is the known boy or the youngest is the known boy so lets play out those two senarios.
The oldest is known: Lets say a woman gives birth to a boy and then gets pregnant again. What are the odds this woman ends up with two boys? 50/50. She could have a girl next or a boy next.
The youngest is known: Lets say a woman has a child and it is stolen from the hospital and she never meets this child. Then she has a boy that does not get stolen. Years later she does 23 and me or something and learns the where abouts of her lost child. She finds the address, drives over, and knocks on the door. What are the odds that it will be a boy or a girl that answers. 50/50
In both senarios it strikes me that it can only be 50/50.
This strikes me as the classic, you flip a coin heads five times in a row, what are the odds of flipping heads again? Lets say a woman has 100 kids and 99 we know are boys, what are the odds she has 100 boys?
As for the tuesday thing, I'm completely unconvinced that is relevant. You mean to say the odds change when you add a variable you aren't measuring?
Edit: I get what you guys are saying about the chance being 33% but in the context of this question I don't think you get to count BG and GB as different combinations. The question isn't, of families that have two children and have at least one boy, what are the odds they have two boys? The question is a scenario where Mr Smith states he has two children and one is a boy, what are the odds of the sex of the other kid? That's it. I would guess the low pass rate is from stats people over complicating it. This is like that bell curve meme where the dumbass and the genius reach the same conclusion for different reasons. Tuesday is irrelevant and a red herring and any deviance from a fifty fifty chance because of environmental or biological factors is not what the question is asking about. They are definitely not expecting applicants to cite research about the odds of sex in pregnancy in 90 seconds. With the information given the question is as simple as Mr Smith has two kids, one is a boy and one us unknown, what are the odds of the unknown kid being a boy or a girl? 50/50.
Edit 2: shout out to u/litespeedclassic for being the only one that could actually explain why the statistics are the way they are rather than just say what they are. I'm too cheap for reddit gold but big credit. I get why without the tuesday info it's 33% now, I was divorcing Mr Smith's family from the population at large and treating it like an isolated coin flip. There is no scenario where Mr Smoth is not part of the population at large so he is subject to the same likelyhoods the broader population is. As for the tuesday thing, it absolutely crushes my intuition but it makes sense now that once a new variable is known it changes the shape of all possible subsets.
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u/k37r 1d ago
It feels absurd, but the "Tuesday" detail changes the outcome. This is a classic probability riddle known as the Boy Born on Tuesday Problem, and the additional information restricts the sample space.
To see why the "Tuesday" detail isn't just useless noise, you have to look at how many chances a family gets to satisfy the condition.
Think about the three possible family setups that have boys:
- (Boy, Girl): Has 1 boy (1 chance to hit Tuesday)
- (Girl, Boy): Has 1 boy (1 chance to hit Tuesday)
- (Boy, Boy): Has 2 boys (2 chances to hit Tuesday)
If you ask a parent, "Do you have at least one boy born on Tuesday?", a Boy-Boy family gets twice as many rolls of the dice to say "Yes" compared to a one-boy family.
Because the Boy-Boy family is twice as likely to pass that "Tuesday test," hearing that a family did pass it instantly boosts the likelihood that they are a two-boy family.
- Without the day specified: Knowing "at least one is a boy" gives a 33.3% (1/3) chance of both being boys.
- With Tuesday specified: Knowing "at least one is a boy born on Tuesday" jumps the chance to 48.15% (13/27).
The Math in Brief:
If you map out all 196 possible combinations for 2 kids (14 x 14 gender/day pairs):
- There are 27 total combinations where at least one kid is a Tuesday Boy.
- In 13 of those 27 combinations, both kids are boys.
- 13/27, or approx 48.15%.
The rarer the trait you ask about (e.g., "born on a leap day"), the closer the probability gets to 50%, but it always stays just below it.
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u/wonkim00 1d ago
I love this post and thread and will forever save it as a reminder of how poor reading comprehension, math, and communication skills are in the general (reddit) population.
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u/DrBatman0 1d ago edited 1d ago
I don't get it. It doesn't make sense to me, and I didn't believe it was true, so I wrote some code.
I will include the code below, but here is the english for what the code does.
First, using properly seeded randomness, roll for...
[Child1 Gender](M or F)
[Child1 Day](1-7)
[Child2 Gender](M or F)
[Child2 Day](1-7).
Next, check if Child1 has gender M and day 2 (2 is tuesday here, but it doesn't actually matter)
Then check if Child2 has gender M and day 2.
If neither of those are true (that is, neither child is M and 2), discard this iteration, add 1 to [discarded_results], and begin the next iteration.
If not discarded, continue to check... Are both Child1 and Child2 M?
If yes, add 1 to [Both_M]. If not, add 1 to [Not_Both_M].
Regardless of yes or no, add 1 to [counted_results].
Then, go back and start the next iteration.
So, that's the program I wrote. It then outputs for me, How many iterations have been counted, how many have been discarded, and of the counter ones, what the ratio is of Both_M to Not_Both_M, and then outputs every 10,000 results.
After running for a while (22 million total, 3 million counted (at least one boy born tuesday)), here is the result:
Counted: 3120000 | Discarded: 19511572 | Both M: 48.1756%
And after leaving longer...
Counted: 6010000 | Discarded: 37591807 | Both M: 48.1465%
So clearly, the answer is about 48%.
Here's the code so you can try it yourself, and PLEASE I BEG YOU if there's a mistake in my code, tell me.
import random
discarded_results = 0
Both_M = 0
Not_Both_M = 0
counted_results = 0
print("Running simulation... Press Ctrl+C to stop.")
while True:
c1_gender = random.choice(['M', 'F'])
c1_day = random.randint(1, 7)
c2_gender = random.choice(['M', 'F'])
c2_day = random.randint(1, 7)
c1_match = (c1_gender == 'M' and c1_day == 2)
c2_match = (c2_gender == 'M' and c2_day == 2)
if not (c1_match or c2_match):
discarded_results += 1
continue
if c1_gender == 'M' and c2_gender == 'M':
Both_M += 1
else:
Not_Both_M += 1
counted_results += 1
if counted_results % 10000 == 0:
percentage = (Both_M / counted_results) * 100
print(f"Counted: {counted_results} | Discarded: {discarded_results} | Both M: {percentage:.4f}%")
EDIT: I ran the simulation without caring about weekday at all, and it's 33% (if at least one boy, how likely is BOTH boys?)
I ran the simulation for the same as the original except that at least one is a boy who is NOT born on a tuesday, and that comes out at about 36.35% after about 10 million trials.
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u/Sean9931 1d ago edited 17h ago
Regarding the result (33.3%) if you not care about the weekday.
You'd might expect it to be 1/4 but it's likely that in your code the "at least one boy" condition is still present. Which is not "wrong" but this sort of choice needs to be specified in the math, which may be what caused tte misexpectation of the probability. The code being:
if not (c1_match or c2_match):
discarded_results += 1
continue
Important to note for non-coders, the "continue" is used within a loop (in this case the while = true) to skip the rest of the code after one by "continuing" to the next loop iteration, not that it continues on to the rest of the code (if you want it to carry on the rest of the code, just removing the continue works), it gets rid of the GG case entirely, leaving BB/BB+BG+GB, 1/3, i.e. 33.3%
Edit: Forgive the formatting I'm on mobile
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u/bafben10 1d ago
Why are you checking what day the children are born on and the gender of each? The question is what the probability is that both are male. We already know that one is a male and is born on Tuesday, so the probability that c1_gender==M and c1_day==2 is 100%. There is no requirement for when the unknown child (c2) is born, only that they are male. There is a 50/50 chance they are male or female, so the answer is 50/50.
Given that you reached the same conclusion that is "correct" according to statistical theory, it seems like this is simply case two different ways to interpret a poorly worded question.
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u/Tom-Dibble 23h ago
I think it boils down to exactly that: the question wording, and what the reader infers from it. The key difference is if you ask “do you have a boy born on Tuesday?”, then if yes, “was the other child a boy?” Instead of, “do you have a boy? What day was he born? Was the other child a boy?”
The key case is the one where both are boys born on a Tuesday. This happens just as often as when the first was Tues and the other Wed. However the Tues-Wed combo happens twice as often because it can happen in either order (similar to how heads-tails is twice as common as heads-heads when flipping two coins), and that causes the issue: if the question, as in OP and the above pseudo-code, first filters to cases where one child meets both criteria, then you end up with that Both-Boy-Tuesday case coming up just once even though it fits the criteria twice. But if you first filter for “boy” then ask about the day that specific boy was born (information, doesn’t affect filter), then you get the “expected” result on the “is the other a boy?” question.
TL;DR: if you read “born on a Tuesday” as a fact about the one boy you are introduced to, the answer is the intuitive 50%. If you read “born on a Tuesday” as a selection criterion, then you have reduced the likelihood that the other is also a boy because the both-match situation is only selected once and get the “surprising” 13/27ths (~48%) answer.
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u/clevsv 1d ago edited 1d ago
The problem with the wording of the question becomes simple when you reword it to see clearly what is actually being asked.
Couple things that are assumed to be true:
- The outcome of the roll for sex of child 1 has no bearing on the odds of child 2. Your own code bears this out, correct?
- For the sake of simplicity, we assume there's a 50/50 chance of either M or F.
Another way to reword this problem would be the following:
If you flipped a coin on a Tuesday and it came up heads, and flipped another coin on a random day, what are the odds the randomly flipped coin comes up heads?
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u/Electrochemist_2025 1d ago edited 22h ago
With Bayesian conditional probability the answer is 13/27.
But these 2 events are always mutually exclusive. Additional information in this case has no relevance and the answer is 50%.
If you had 2 kids, the first one has no effect on the sex of the second one.
The question is usually posed to see if candidate is a practical human being or a complicated probability calculating pschyo. 😀
So if you toss a coin a couple of times and one of them turned up heads on a Tuesday, what’s the probability of the other toss being heads?
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u/mp5tyle 1d ago
This is to hire a quant so probability calculating psycho is indeed what they are looking for.
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u/Nice_Actuator1306 1d ago edited 1d ago
Boys birth rate is 51-51,4%, because they died more often, so evolution made the balance. 105 boys for 100 girls.
First children are boy, so we have only one event to count. And take just probability for boys to be born. 51-51,4%
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u/PWannes 1d ago
Reading the explanations for 48.1%. Please help me:
I’m a dad, i already have one son born on a Tuesday, my wife is pregnant and we don’t know the sex of the child yet. If I would ask you to calculate the chance of it being a girl/boy, the outcome (chance) would change depending on me adding details about the birth of my already born son?
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u/thirtychirps 1d ago
Think of it as less predictive and more descriptive. If you have a son born on a Tuesday, and a second child, your info could be pooled together with all others who have two children, including a son born on a Tuesday.
Of that population families, the answer to this question of this thread is basically saying “48.1% of these included families have two sons”
It’s just a math parameters thing though. Independently, it’s still always a 50/50 thing on your next child’s sex. But this is just describing proportions of populations subject to probabilities, really
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u/Btotherianx 18h ago
One of the major problems with the world today is that people that make up stupid ass questions like this are hiring for positions that pay that much
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u/stardust_dog 18h ago
50%. One child already a boy and we know it is so that is actually irrelevant. Because it’s a constant…there isn’t even an option that the boy is sometimes a girl.
It’s really about the other and there is a 50% chance that the other is a boy.
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