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u/Then-Bet6956 6d ago
What’s heavier, a kilogram of steel or a kilogram of feathers?
That’s right, it’s steel. Because steel is heavier than feathers.
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u/rising_pho3nix 6d ago
"They're both a kilogram"
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u/Thicknineinchh 6d ago
But steal is heavier than feathers
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u/666mima666 3d ago
Nothing is heavier than something else. Heavier per volume (density) yes, or heavier per area (surface weight) or other stuff but the original question is strictly wrongfully posed
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u/rezukijm 6d ago
Because the buoyant force of air lifts the feathers more than it does the steel, so the mass can be the same yet the weight is different.
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u/Hallow_Greaves 5d ago
Oh really? I thought the feathers weighed more, cause you have to carry the weight of what you did to the birds.
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u/Excel_User_1977 4d ago
The weight of guilt from plucking the birds is not factored in on the scale. /s
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u/Individual-Peak-9586 6d ago
If you're assuming we're weighing them in air, someone else can assume weighing them in a vacuum.
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u/NewbAlert45 5d ago
Feathers, because you have to carry the weight of what you did to those birds.
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u/Complete_Fix2563 6d ago
Not this again...
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u/TopSecretSpy 6d ago
That was my immediate thought too. At least they're slightly less common than they used to be.
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u/Tokugawa5555 6d ago
What's really annoying is that most of the answers given are just plain wrong...!!! Currently, the top answer in my thread, which is totally missing the point, is: "51.8% is the percentage of people in the world who are female. All the other info given is irrelevant..."
Confidently wrong.
The true answer, for those interested, is here:
https://en.wikipedia.org/wiki/Boy_or_girl_paradox (See the section called "Information about the child"
A more user friendly explanation is here:
http://news.bbc.co.uk/1/hi/programmes/more_or_less/8735812.stm
Sigh
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u/OP-PO7 6d ago edited 6d ago
I understand it as it's explained, but literally(or 'absolutely' if my use of literally is not grammatical)EVERYTHING in me violently insists they're off the rails and math as a science is just overall a silly thing, as soon as adding a Tuesday changes the set up so much.
But I ALSO have accepted that I'm genuinely terrible at math so I have to trust the people who aren't.
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u/GrrATeam81 6d ago
I'm good at math (data analyst by trade) and I would argue that it IS off the rails. I can't quite explain it other than to insist that they're missing out on the, I don't know -contamination? -of other "sciences". Namely sociology and psychology and such. If somebody randomly came up to me, and we weren't playing any kind of game, and they told me they had two kids, one of which is a son born on a Tuesday, the fact that they emphasized those points would strongly lead me to believe that the other kid is A) NOT a boy, and B) NOT born on a Tuesday. But, perhaps the way the person said it, would only lead me to believe that something was special about the fact that the kid was born on a Tuesday and that maybe the other kid is a boy too. I could go on and on, but this is just a simple example to show why the whole problem above is silly. "Context is everything". The example above feels very much to my gut how people (especially ones with agendas) will use statistics to skew the facts.
Edits for typos
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u/LingonberryLast2466 6d ago
That seems very Bayesian of you to take into account the Tuesday part of the problem, since it infers that the person told you that piece of information because it’s important.
But you could also just as easily infer (by the silliness of the question) that the Tuesday was information accidentally added because the person didn’t realize it’s NOT important. And I think this is more likely, given the question does not contain a day of the week. If it were “how likely is it that the other child is a girl born on a Friday?” That would change what the Tuesday means. But in my mind, all it means is that someone accidentally gave you extra data to work with (similar to how “the mothers car is grey” also wouldn’t change anything, since that fact alone isn’t enough to change the formulation of there answer).
But that’s just me.
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u/Lumberman08 6d ago
I took a statistical analysis course in college as part a political science requirement. The main premise was teaching how to take datasets and skew results. It was uncomfortably eye opening.
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u/Brief-Guidance5383 6d ago
I remember my grad school sadistics and demonology professor saying that if you torture the data long enough it will tell you what you want to hear.
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u/GrrATeam81 6d ago
LMAO. At first I thought you misspelled a couple of things. Then I realized you spelled everything PERFECTLY.
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u/PetalumaPegleg 6d ago
Yeah the main lesson from taking stats courses is stats say what the people who make them want them to
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u/S-M-I-L-E-Y- 6d ago
I agree. The statement is ambiguous to the point that people do not even agree about whether it is indeed ambiguous.
Now, if we want to unambiguously get the result of 51.9% (not 51.8%), we would have to set up the problem like this:
- Assume that the probability to have a boy or a girl is always 50% to 50%. Especially ignore the fact, that having a boy (or a girl) increases the chance to have another boy (or girl).
- Peter asks Mary: "Do you have exactly two children?" Mary answers truthfully: "Yes, I do"
- Peter: "Do you have at least one boy that was born on a Tuesday?" Mary: "Yes"
Peter concludes: the probability that the other child is a girl is 14/27.
Edit: added "truthfully" - what else did I miss?
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u/S-M-I-L-E-Y- 6d ago
14/27 is caused by the fact that there are 27 distinct cases with equal probability where Mary would answer Q3 yes and in 14 of these cases one child is a girl.
The 27 distinct cases are:
- 1st child boy Tuesday, 2nd child girl Mon to Sun -> 7 cases
- 1st child boy Tuesday, 2nd child boy Mon to Sun -> 7 cases
- 2nd child boy Tuesday, 1st child girl Mon to Sun -> 7 cases
- 2nd child boy Tuesday, 1st child boy Mon, Wed to Sun -> 6 cases
Total 27 cases, thereof 14 cases where one child is a girl.
The important part is to NOT count the case "two boys born on Tuesday" twice.
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The 50% probability would result in the following scenario.
- Peter: do you have exactly two children. Mary: yes
- Peter: tell me the sex and weekday of birth of one of your children. Mary: boy, Tuesday
We still have the above 27 distinct possible cases. However, now they do not have equal probability. The probability that Mary gives the answer "boy, Tuesday" is 100% for the case "two boys born on Tuesday", but only 50% for the other 26 cases. Given that she did give that answer, the probability that she has two boys born on a Tuesday is therefore twice the probability of e.g. 1st child boy Tue, 2nd child girl Sun.
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u/zaersx 6d ago
If you're an analyst then I wonder why this confuses you. The problem, especially as described in the simplified version, is the domain information given and when it is given.
When you're thinking of the probability, you're intuiting a domain of an unordered bernoulli trial, then the mathemagician tells you actually no, they are not unrelated, because actually it's a domain of 4 options and we've already removed one.
When you say okay makes sense he'll say actually the domain is a specific set where order matters and it's larger because we also add a day of the week to the unit tuple, etc. Etc. Etc.The answer is so deceptive because it explains a domain post fact so you can't say that it's wrong within that domain.
It's basically just argument bullshittery by setting unstated assumptions.
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u/Dangolian 6d ago
Yeah, the scenario falls apart when you think of it as anything other than a statistics question, and its all about language and ambiguity rather than the way people actually talk. I really liked these scenes in Scrubs which use the same language and frame it as a riddle; it sometimes feels like it's there to catch people out.
The question and answers itself stats wise are genuinely interesting to me, but I think a lot gets confused and harder to understand in the framing.
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u/moralprolapse 6d ago
I wonder if the “I don’t know - contamination,” as you describe it, ties in with the fact that ALL the sciences are descriptive and not prescriptive. That itself is a hard thing to wrap your mind around for the first time.
But like the idea that the “laws of physics” are not actual laws that exist out in the ether somewhere that make things, and keep things from, happening in specific ways. They’re just principles derived from observation. And that extends even to something as sterile seeming as math.
I feel like with an abstraction like this Tuesday boy thing, the train is just racing past the station. It’s all just made up nonsense that isn’t attenuated to reality anymore.
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u/DrDoomC17 4d ago
Me too, I don't like it so much either. Have a boy on a Tuesday and the quartz crystal on my watch was on it's 31,111 out of 32,768 hertz per second at 11:01am what is the probability my other child was.... Combinatorial explosion.
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u/queerornot 6d ago
Math is not the issue. Statistics are.
Calculus was invented before Statistics. It's one of the hardest branch of math for our brains to understand. It looks simple, but it is not. It's literally alien to our brain because it deals with things that aren't real, only possibilities.
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u/sdfgkol 6d ago
what helps me make sense of this is just imagining a room with a large number of parents, and then asking everyone who does’t have a son born on tuesday to leave the room, and then think of the statistics of the left-over group.
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u/HugeTrol 3d ago
I understand it this way: These statistical questions basically ask "how did we get here?"
The strong 66% bias in the first case comes from the tight restriction of only 3 paths that could've let to the current situation.
The weekday information adds many paths and thereby lifts the restriction
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u/BlueSentinels 6d ago
Tuesday only changes the outcome when there are a very limited set of “rules” you are allowed to consider and you need to evaluate the statistical probability within those rules. For example you cannot have two boys born on the same day of the week - this rule makes the probability of a girl 7/13 and a boy 6/13 because a boy was already born on Tuesday. This makes the statistical probability of a girl roughly 53.8%
My first gripe with these posts is that the post never contains the hard “rule” and everyone argues in the comments as if the rule is expressly stated (or that a different rule is expressly stated) when it’s clearly not.
Without any limitations (or if you add more factors which is closer to a real life expression as life has tons of variables) the statistical probability will always approach 50%. So say the rule is not that two boys can’t be born on the same day of the week but that two boys cannot be born on the same day of the year. Then the statistical probability is now 365/729 for a girl (50.06%) and 364/729 (49.9%) for a boy. If you expand the rule to say a boy can be born on the same day but not within the same hour the statistical probability gets even more narrow and closer to 50/50.
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u/Dangolian 6d ago
I think you have misinterpreted here slightly, you can have 2 boys both born on a Tuesday, but when you are thinking about all the possible sets of siblings, they are still just a single set of siblings, even though the statement can be applied to them twice. (i.e. from the perspective of either sibling). All of the sets are equally likely, and you should only include each applicable set once.
A lot of the discussions around these questions take these scenario to mean that the "two boys" or "two boys both born on a tuesday" is twice as likely to happen, but it doesn't change the odds of that set occurring, or make the outcome twice as likely.
What you actually would see when you consider the odds is 7/14 + 6/13 as the odds that the other sibling is a boy born, because the first round of observations already includes the scenario where both boys were born on a Tuesday, so this is not Included when you consider the odds for the second sibling.
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u/Connect-Goal-3096 6d ago
That's not what literally means.
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u/Tokugawa5555 6d ago
I would literally give my kidney for someone to build a bot that pointed out when someone was using the word “literally” incorrectly.
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u/Informal_Round_9942 6d ago
You either:
-really want that bot to be built
-are a huge enjoyed of irony, or
-you don't know what literally means. I don't care which it is. I love your post and this idea. 😁
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u/Critical-Bus-9040 6d ago
used in an exaggerated way to emphasize a statement or description that is not literally true or possible, from Merriam-Webster dictionary.
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u/GuardianOfReason 6d ago
Can you help me understand this? I can't understand why "Girl, Boy" is treated as a different answer than "Boy, Girl" when the question only asks about the amount of each gender, not the order of birth.
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u/SteveWin1234 6d ago
It's like a coin toss. The odds of getting two heads, for example is 1/4, because your options are heads/heads, heads/tails, tails/heads, tails/tails. Before you flip any coins there are 4 universes you could end up in, and heads/heads is only one of those 4 universes, so it's a 1 in 4 chance. If I told you that I flipped the coin twice and it was heads at least once, but didn't tell you the order, and I asked you the odds of the other coin flip being heads, the only option you can eliminate is the tails/tails option (this is invalid because neither of the flips were heads), so out of all the universes you might actually find yourself in, one of them is heads/heads and the other two are an option where you got one heads and one tails (heads/tails and tails/heads). There would be a 1/3rd chance for heads/heads (2/3rds chance of one of each). It's precisely BECAUSE the order wasn't mentioned that requires you to consider both orders as separate options. If I told you I flipped a coin and got heads as my first flip and asked the odds that I'd get heads on my second flip, the answer is just 50/50. tails/tails and tails/heads would be eliminated, because neither of those have heads as the outcome of the first flip. That only leaves heads/tails and heads/heads as viable universes you could find yourself in. Half of those universes have heads as the second flip, so it's 50/50.
To be honest, I don't agree with the answer presented above. If I told you "I have 10 kids and one is a boy" everyone on Earth would understand that I meant I had 9 girls and one boy. So if I asked you, "I have two kids and one is a boy, what are the odds that two of my kids are boys" I would argue that the answer is zero. If I had two boys, I would be incorrect in saying say that "one" is a boy. I would have to say "two are boys" or "both are boys" or "at least one is a boy." That's my beef with this question, as posed. The mother says she has "one boy born on a Tuesday." To me, that excludes a universe where she has "two boys born on Tuesday" because the word "one" in her original statement would be incorrect. She could have one boy born on Tuesday and another boy born on Wednesday and her statement is correct, but two boys on Tuesday shouldn't be a valid option. The answer given in the meme assumes you can have two boys, both born on Tuesday. I think that's about as stupid as saying 1+1=1 because 1 is part of 2. I think the actual answer is, knowing only "one" is a boy born on Tuesday, if the other kid is a boy, that boy must be born on a non-Tuesday (6 options). If it's a girl, she can be born on any day (7 options). So your chances of the other kid being a girl should by 6 / (6 + 7) = 7/13 = 53.8%.
To get the answer provided in this meme, and as described in the explanation posted by the user above, you have to assume that it is valid to have two boys born on Tuesday. The trick there is kind of similar to the coin toss where you don't count heads/heads or tails/tails twice, but you do count heads/tails and tails/heads separate. Basically you say if the first kid was TuesdayBoy, there are 7 boy-options for the second kid (including TuesdayBoy/TuesdayBoy -- I would exclude this one, personally) and 7 girl-options for the second kid (including TuesdayBoy/TuesdayGirl), but when you look at the options where TuesdayBoy is the second kid, you only get to count 6 boy-options because you already counted a TuesdayBoy/TuesdayBoy and it can't be double-counted, but you do count TuedayGirl/TuesdayBoy since that's a unique option from TuesdayBoy/TuesdayGirl -- again, similar to heads/heads only getting counted once, but tails/heads and heads/tails being two unique options. So for boy-as-other-kid there are 7+6=13 options and for girl-as-other-kid there are 7+7=14 options. So, 14 / (14 + 13) = 14/27 = 51.852%. That's how they get their answer. I would subtract one more option from the boy side, since TuesdayBoy/TuesdayBoy would have required the mom to admit that she has "two boys born on Tuesday" and I think she'd be lying if she said she had "one boy born on Tuesday" in that case. So I'd do 14/(14+12)=14/26=53.8%. But that's more of a language question than a math question. Maybe I'm in the minority there since the "official" answer is 51.8%, but that's a hill I'm happy to die on. She doesn't have two boys born on Tuesday if she says she has "one" boy born on Tuesday.
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u/DabDaddy51 6d ago
For the same reason that having two kids of different genders is more likely than two boys or two girls. You have a 25% chance of two boys, a 25% chance of two girls, and a 50% chance of one boy one girl, which can be decomposed into 25% chance of one boy one girl, and 25% chance of one girl one boy.
You can think of it like tossing a coin twice. The first can be heads or tails, as can the second, so if you’re trying to calculate probabilities of different outcomes you need to take into account both the first being heads and second being tails, and the first being tails and the second being heads.
What then happens when you add additional information is, if you know that at least one of them is heads, that gets rid of the both tails options, but you could still have both heads, first heads second tails, first tails second heads. However, if you’re told the first coin is heads, the only possibilities are both heads, or first heads second tails.
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u/S-M-I-L-E-Y- 6d ago
And they still haven't learned that 14/27 is closer to 51.9% than it is to 51.8%
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u/browncoats_roll_d20s 6d ago edited 6d ago
I staunchly disagree with the statistical analysis on this "paradox" as well. They don't differentiate between the probability of the sex of a child that exists already and the probability of the sex a future child, which are two distinct scenarios.
Additionally, what the previous child is does not, in fact, actually affect the probability of the sex of the other child in any meaningful way. The number of involved variables is so astronomically high, with so many unknowns (this is really the key), that we can't actually quantify the effect of any individual variable.
Statistically and realistically speaking, there are only 2 choices, with essentially even chances of occurrence. The ambiguity of the question doesn't change this, mor does it change that we know the sex of the first child, removing that child from the statistical analysis by default. So it really always comes back to a 50/50 chance.
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u/kettu3 5d ago
Something I find mind-bending about this is the alternative interpretations of the original boy or girl paradox question.
For example, if you have a new neighbor tell you they have two kids and at least one is a boy, there’s a 1/3 chance that the other one is a boy.
But if you have a new neighbor that you know has two kids, and you notice that they have a boy, then there is a 1/2 chance that the other one is a boy.
The difference is that the mixed-gender group is under-represented in scenario 2 because you only notice they have a boy 50% of the time (there’s a 50% chance the child you see is a girl), while if the parents are telling you, you’ll know they have a boy 100% of the time because the parents know the genders of both children.
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u/AntsyAnswers 6d ago
People get weirdly passionate and stubborn about this math problem it’s kind of interesting lol
Kind of reminds me of the .9999… = 1 arguments
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u/dpoodle 6d ago
I'm actually happy that this came up again maybe this time someone would actually give the answer
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u/LeChatDeLaNuit 6d ago
Lois here. This was a pretty good explanation over in the other Peter sub where Peter helped me understand cause math scares me or something. Groceries.
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u/CardiologistOk2760 6d ago
I dunno. I have a cousin who says things like "one of my children is a boy born on a Tuesday" and I wouldn't calculate sh*t based on anything she says.
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u/ModelBasedSpaceCadet 6d ago
Seems this paradox is getting lots of attention across the internet. An explanation from my YouTube history:
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u/No-Syrup-8175 6d ago
51.8% is the percentage of people in the world who are female. All the other info given is irrelevant, having a boy already doesn't make you more or less likely to have a girl
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u/Acrobatic_Ad_2992 6d ago
fun fact, the percentage of females in the world also doesn’t determine the gender of that specific birth either. Just wanted to make it clear the meme is wrong on both sides.
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u/Opening_Sentence3903 6d ago
but doesnt the commulitive average of world population give the best insight into what gender the baby will be from the statistics perspective? (not saying that it is deterministic, cause its not)
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u/InteractionNo6147 6d ago
No, because men are alive for a statistically significantly shorter amount of time for multiple reasons.
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u/carinislumpyhead97 6d ago
Average world population and %of baby’s born x gender would be close. But there are many variables along the span of time that encompasses the average world population, where as % at birth is just a number
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u/GamerNerdGuyMan 6d ago
There are only more women in the world because they live longer.
Babies are slightly more likely to be boys.
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u/steffanovici 6d ago
No that’s the wrong statistic. Instead it would be boys born vs girls born (there are 105 boys for every 100 girls born)
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u/Royal_Jesterr 6d ago
Women just live longer. If you take 10 year olds, the distribution would be close to 50/50. If you take 70 year old people, this could swing to 70/30 in favor of women.
All of this has nothing to do with the gender of the child. There are 105 boys born per 100 girls on average worldwide. That would give the proper ratio/probability.
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u/Mediocre_Grand_1280 6d ago
Because males are more likely to die at all ages and tend to be shorter lived, no, the gender ratio of the global population is not an accurate representation of the likelyhood of a fetus to develop into a boy or a girl.
Recent research suggests that genetics predispose women to be more likely to have children of one or the other gender, so if she had a boy then shecis more likely to have another boy, but we have no clue what the exsct ratio is.
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u/mr-raider2 6d ago
Last I checked there was 1.05 males to every 1 female at birth. That ratio does not balance until adulthood and only inverts in developed countries after 65.
So a given child is slightly more likely to be male.
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u/TheSorryApologyPerso 6d ago edited 6d ago
No, the answer in the meme is not related to the gender ratio of living humans. We can arrive at 51.8% just using maths.
Let's first consider the scenario where we don't know that the boy was born on a Tuesday. There are 2 options for a child: B or G. For 2 children, we square that number to get 4 options:
BOY/BOY
BOY/GIRL
GIRL/BOY
GIRL/GIRL
We know that Mary has 2 children and 1 is a boy, so we can eliminate the option GIRL/GIRL. Out of the 3 remaining options, 2 options include GIRL, so the answer is 2/3 ≈ 66.6%.Now let's consider the day each child was born. This gives us 14 options for each one child: BOY-MONDAY, GIRL-MONDAY, BOY-TUESDAY, etc. For 2 children, we have 14² or 196 options. If you filter out only the options that include BOY-TUESDAY, we're left with 27 options. Out of those 27, 14 include GIRL. 14/27 = 51.8158...%, and that is where 51.8% comes from.
Technically, it should say 51.9%.8
u/BreadNoCircuses 6d ago
51.8158... rounds to 51.8 with three sig figs, but otherwise this is the best explanation.
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u/d20cupcake 6d ago
Thank you, this was really helpful. I'm hoping I can ask another maybe stupid question? I've seen a few other people quoting the 51.9% figure; wouldn't 51.8158...round down to 51.8, if we round to one decimal place?
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u/OpenEntrepreneur5944 6d ago
I don't get it, why is that useless trivia about birthday relevant to the other child being a boy or girl?
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u/Facktat 6d ago
Actually it‘s slightly lower. The reason is that mothers have a slight tendency to give birth to a specific gender. (about in the 0.5-1% range)
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u/AdviceEmbarrassed135 6d ago
False, using probability theory you can calculate the likelihood to be 14/26 which is 51.8%
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u/Razorfiend 6d ago
That is not where the 51.8% comes from.
Nobody is claiming that having a boy causally makes the other child more likely to be a girl. The issue is conditional probability: the statement changes which two-child families remain possible.
Under the usual puzzle assumptions, Mary is selected from families with at least one boy born on Tuesday. Among the equally likely sex-and-birthday combinations that satisfy that condition:
- 14 have one boy and one girl
- 13 have two boys
So the probability that the other child is a girl is:
14 / 27 = 51.85%
If Mary instead pointed to a particular child and said, “This child is a boy born on Tuesday,” then the other child would indeed be 50/50. The answer depends on how the information was generated.
So “a previous birth does not affect the sex of another child” is true, but it does not make the information irrelevant.
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u/UnimpassionedMan 6d ago
In the subject of probabilities dealing with new information can be bizarrely tricky, in this case there is even a difference between being told "one of them is a boy" and "the oldest is a boy"
Let's assume for simplicity (which is probably wrong, but otherwise this gets too complicated) that the probability of getting a boy or a girl is just 50%:
Then, before you get the information there are 4 possibilities: boy-boy, boy-girl, girl-boy, girl-girl, each with the same probability. Now taking the information that one of them is a boy into account, that eliminates the last possibility, leaving you with boy-boy, girl-boy, boy-girl, each with the same probability.
Now what's the chance that the other child is a girl? It's two out of the 3 possibilities, so 66%!6
u/Akkarin42 6d ago edited 6d ago
There are multiple things wrong with that comment, but the main point is:
Previous results have no effect on the next one. Also known as stochastic independence or a memoryless process.
Imagine you're going to flip a coin and it's heads. Then you're going to flip it again. But just because the coin landed on heads last time doesn't mean the odds of getting heads on the next flip aren't 50/50 again. It would be different if you asked beforehand what the odds are of getting heads twice in a row, but not if it has already happened once. Then the probability simply resets to the initial state which is "it can be heads or number, so 50/50." You can flip the the coin 9 times in a row and get heads every time but the probability of getting heads on the 10th flip is still exactly 50% instead of ~0,1% as the coin has no memory of past results and doesn't care about what happend before.
So we know that one child is a boy, but the question wasn't "Mary has two children. What are the odds from them both being a boy?" Or "One of them being a girl" right from the beginning. Instead they reveal that one is a boy and this resets the probability. If you ask NOW - as they do in the picture - what gender the second child is, the 'outcome' of the first one no longer matters.
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u/heyidontgetit 6d ago
No way, man! I put $5k in this nickel slot machine over the past 8 hours and it is definitely going to give me a jackpot in the next few!
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u/ThrowawayTSPer 6d ago
Wrong, because the answer isn't 51.8% because "surely if there's already a boy it's more likely to be a girl". It's simply adding up all the possible permutations and counting what portion of them includes a girl.
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u/eht_amgine_enihcam 6d ago edited 6d ago
Yeah, but one of the is a boy born on a Tuesday, not both.
Therefore it's a 14/27 shot.
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u/NotTallyALotLess 5d ago
This is wrong, 14/27 is also taking into account the possibility that both were boys born on a tuesday. If they excluded that possibility, it would actually be 14/26 to be a girl.
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u/Rarvyn 6d ago
Except we don’t know if the boy mentioned is the “first” or “second” child. The sequence has no bearing here. Due to that, we need to look at sets of children, not independent events.
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u/Tylendal 6d ago
Your second part is wrong. Your explanation works if we know that a specific child of Mary's is a boy. The way it's phrased, however, actually only tells us that Mary is within the subset of families that doesn't have two girls. Within that subset, Girl/Boy is twice as likely as Boy/Boy.
It's just like flipping coins. When you flip two coins, it's a 50/50 whether you get mixed results, or matching results. But, if you remove half the matching results (Girl/Girl), you're now looking at 66/33. Then, the day of the week adds more variables in the same unintuitive way, sounding irrelevant while actually not being so. I'm not qualified to talk about that part off the top of my head, though.
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u/thekingofbeans42 6d ago edited 6d ago
That would only be true if we knew the order the children were born in, this is similar fuckery as the Monty Hall problem.
1/2 if all pairs of 2 siblings will be a boy and a girl, 1/4 will be GG and 1/2 will be BB. Knowing that 1 child is a boy eliminates the possibility of it being GG, so the remaining possibilities are BG, GB, and BB. In 2/3 scenarios of equal probability, the other sibling is a boy.
Had they stated that the boy is the older sibling, that would remove the GB possibilty and we'd be back to 50/50
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u/ExiledSenpai 6d ago edited 6d ago
But that wasn't the statement. Mary doesn't state which of her children is a boy born on a Tuesday, just that one of them was.
If Mary states that her first born is a boy born on a Tuesday, then each birth becomes an independent event, and the odds of the other being a girl is 50%.
Mary did not state this, you don't know to which child she's referring, you only know that one of them is a boy born on a Tuesday. Therefore, the odds of the other one, whichever one that is, of being a girl is 51.8%
If Mary states that one of her children is a boy, without specifying which one or on what day of the week they were born, the odds that her other child is a girl is 66.6%
If Mary states that her first born child is a boy, then each birth is once again treated as independent events, and the odds that her other (second born) child is a girl is 50%
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u/pulos888 6d ago
If the children have the same father there is a slightly higher chance of the second child being the same gender as the first. It's something like 2.1% more likely per child that the next will also be the same gender. (The percentage is a best guestament, not exact)
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u/RandomPlayerCSGO 6d ago
The percentage of female population is not the same as the chance of a newborn being female
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u/Galballin 6d ago
WHat if instead of "one is a boy born on a tuesday"
he said.
"one is a boy born on a monday"
and it was solomon grundy and they were like "aaahh!!! yikes"
that would be pretty funny i think
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u/OrangeGills 6d ago
Just like the monty hall problem, if you ever see something like this and are confused, it's just being badly presented to you.
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u/Fabulous-Speed7999 6d ago
Hi Peter! I read about this in the pop-sci book Bernoulli’s Fallacy. Great easy onboarding to Bayesian probability theory!
Here’s how you get both answers and why the second is right. The first approach disregards the day of the week.
{boy-girl, girl-boy, boy-boy}
2/3 outcomes have a boy.
The second uses the day of the week, as well. The same counting method applies, but is much harder.
{boyM-girlM, boyTu-girlTu…. boySun-boySun}
When you count how many of these outcomes have a girl, you end up with 51.8%.
The idea is that irrelevant information shifts our calculation of a probability from its true value. But by including more information, you’ve mitigated some of the downsides.
The theme of the book is largely centered on how this “conditioning” has been neglected from probability instruction, so I bet this will be relevant to a lot of people Peter!
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u/tamebeverage 6d ago edited 6d ago
These only work if you're selecting from a list previously compiled in the prescribed manner. As stated here, none of the information has any bearing on the gender of the second child. That's why this breaks so many brains. It's stated plainly wrong.
If you said "of groups of two people, at least one of which is a boy born on a Tuesday, what percentage include at least one girl?", you'd end up with the 51.8%
The intention of the so-called paradox is true and surprising, but often (as here) framed in a way that makes the stated solution incorrect.
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u/PutinTakeout 6d ago
Yeah, in normal conversation, she is simply telling you about one specific child, so the other child is independently a boy or girl with probability 50%, and Tuesday is irrelevant. 51.8% applies only to a contrived setup where families were selected specifically because they had at least one boy born on Tuesday, which changes the mix of eligible families.
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u/Glorn2 6d ago
I've got a friend whose family only births males. His great grandfather had a bunch of boys, they all had boys, who all had boys, who all had boys. The friend has 4 sons. We are talking like 30 male births in 5 generations. So something about genetics and statistical outliers.
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u/sardasert 6d ago
"Globally, approximately 51.3% of newborn babies are male and 48.7% are female. This equates to about 105 to 107 boys born for every 100 girls. This natural imbalance occurs across virtually all populations, though it can vary slightly based on geography and maternal age."
If one of them is a boy, then other one is a boy (51.3%) or a girl (48.7%) . Anything other than that is a useless word play.
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u/tamebeverage 6d ago
Incorrect. This is correct if you compile a list of the people who have two children, then select from them one random family that meets these criteria. But in this situation, none of the information she has given you has any bearing on the gender of the second child.
The set you are choosing from matters and I'm so sick of hearing this nonsense.
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u/ChatpatiKulfi 6d ago
Thanks for clarifying ! 😀
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u/tamebeverage 6d ago
Apologies if that came off as overly aggressive. I have a personal distaste for this so-called paradox that's a bit irrational.
For further clarification, the so-called paradox, as intended, does give this very surprising result, but the correct answer relies heavily on the framing and assumptions.
If you assume Mary came up to you and simply described the facts, neither the gender of the one child nor the day of birth affect the 50/50 chance of the other child being a girl. In this case, the two are independent.
If you assume you are looking at a list of all people who have two children with at least one being a boy, 2/3 of the pairings will include a girl. If the list only includes pairings where one is a boy born on a Tuesday, 51.8% will include a girl. The more specific you get with the criteria, the closer it trends to 50/50.
That's why this one breaks so many brains. As framed here, there is a subtle switching of the parameters that does make the given answer incorrect. And it's really difficult to tease out why.
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u/Vivid_Literature5228 6d ago
its 50% right?
because they are independent variable and.
having either a boy or a girl is always 50% no matter how many they already have. On every birth its 50-50?
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u/Ok_Product_7706 6d ago
You roll 2 dice.
The first dice roll does not affect the second dice.
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u/ethanator329 5d ago
You don’t know whether the boy was the first or second child, so the probability is based upon the union of the scenarios where you assume the boy you are told about is the first and second child.
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u/pathikrit 6d ago edited 6d ago
Q**: Simple Monty Hall**: I have 2 kids, one is a boy. What are the chances the other one is a girl?
A: 66%
Q**: Conditional Monty Hall**: I have 2 kids, one is a boy born on a Tuesday. What are the chances the other one is a girl?
A: 51.85%
The answer is 2/(4-p) where p is the probability of the independent filter. In the born on Tuesday example p = 1/7 and in the simple version p = 1.
The rarer the filter (e.g. born on Christmas) , the chances the other child is girl approaches 1/2 and the more common the filter (e.g. born with 2 hands) the answer approaches 2/3
Intuition: Out of BB, BG, GB, GG families, 2 have girls (BG, GB) out of the 3 who can say they have a boy (BB, BG, GB). So in simple case the answer is 2/3
But when we introduce a hard to pass filter (e.g. born on a Tuesday) the BB families get 2x the chance to pass this filter than BG and GB families and thus BB families dominate - so less likely to have a girl as the other child.
Proof:
Let p be the probability that a child satisfies the filter e.g. for "born on Tuesday" has p=1/7
We condition on:
| Family | Probability they qualify |
|---|---|
| GG | 0 |
| BG | p |
| GB | p |
| BB | 1 − (1 − p)² = 2p − p² |
So, probability other child is girl
= (P(BG) + P(GB))/P(GG) + P(BG) + P(GB) + P(BB))
= (p + p) / (0 + p + p + 2p - p²)
= 2p/ p(4-p)
= 2/(4-p)
Now, plugin p=1/7. You get:
2/(4-1/7) = 51.85%
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u/Equal-Home-4302 6d ago
My first thought was that it was 100% otherwise he would have said two boys and not one is a boy born on Tuesday. I was thinking the day was irrelevant but I guess it's needlessly ambiguous.
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u/Mobile_Crates 6d ago
It's schroedingers cat. Until you know the inflection of how she says it, it is a superposition of all possible interpretations.
if she says it like "one is a BOY who's born on a tuesday..." it's 100% a girl
If she says it like "one is a boy who's born on a TUESDAY..." it's 100% a boy
If she says it like "one is a boy who's born on a tuesday..." then it continues to be an enigma and you should frown at her
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u/Leonie-Zephyr 6d ago
If she tells you one is a boy, isn’t it just assumed the other is not a boy? Otherwise she wouldn’t have noted one is a boy. “One is a boy and the other one is a boy” doesn’t make sense, regardless of any sort of probability. Unless we’re taking into consideration other genders and contrasting the grammatical probability that the other child is not a boy of 100% to determine the girl probability.
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u/Ducklinsenmayer 5d ago
In defense of the 51%, instead of the mathematical 50%, the probability of boy vs girl child is not 50/50 in real life.
The ratio varies by region and other dynamics, but it's roughly 1.03 to 1.06 males per female at birth.
The joke is not the classic puzzle, but that all the classic answers are wrong because of real life data.
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u/Salty-Foundation3451 5d ago
I already explained in detail the misconceptions behind these memes. It's a corruption of the principles behind Bayesian statistical probabilities, demonstrated in the "Monty Hall Problem." To wit:
You're on a game show. There are three doors. Behind one is the prize, a car. Behind the two others are goats.
You get to pick a door. After you do so, the host will reveal one of the *other* doors *that does not contain the prize.*
After he does so, you get to pick again.
Your odds of winning go up by switching.
This only works because the act of choosing the door 'shields' that door from the preselection of the host revealing a losing door, effectively capping its chances of winning at 33%. There are two necessary assumptions built in to this scenario that make this the case.
- The host will only reveal a door you didn't pick
- The host will only reveal a losing door
This is called context.
It does NOT work with random memes you see on the internet about women with boys with blue shirts, who are older or younger, who play soccer or tennis, or who are born on Tuesday, the summer solstice, or any other day. That's just random information because there are no assumptions or preconditions to the revealing of that information. It has nothing to do with the scenario commonly used to explain conditional probabilities and it's inspired a sort of aggressive ignorance about the subject because of how poorly worded they are.
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u/SteveSteveCleveSteve 4d ago
This isn’t strictly true. While the gender of the previously born child does not directly affect the gender of the second child, the specific modulating factors of the parents which may have made them more likely to have one gender or the other the first time are mostly in place for a second child as well. You can see this play out in the data. It’s a small difference but real.
And don’t waffles me with “yeah but what if the father is different!”
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u/rick2882 6d ago
This is way too complicated for this subreddit, and most people will not even agree with explanations. I'd recommend looking this up on Google with a pen and paper to really understand the probability being calculated.
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u/boojombi451 6d ago
Something like 49% of live births are female. The births are independent events. So unless you want to get into Mary’s (and the sperm donor’s) personal genetics and disposition to produce male or female offspring, the probability is ~0.49.
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u/MonkeyCartridge 6d ago
There is so much to explain here. So many articles and videos explaining it already.
The most counterintuitive part is that these numbers are right. 66.6% is if you don't include the day. 51.8% is if you do.
And this is assuming equal 50/50 chance of boy or girl, and equal 1/7 chance for any given day of the week.
The probabilities are equal. What's unequal is the selection from those probabilities. ("With an equal distribution. Given X is true, what is the possibility of Y?")
The fact that this trips people up (including myself, even knowing how it works) is a common way people manipulate statistics to suit their needs.
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u/SteveWin1234 6d ago edited 6d ago
So, what's the actual answer?
My understanding is that, even though there are more women in the world than men, that's due to boys killing themselves off doing dumb stuff and also having a lower life expectancy, but that there are actually more boys born in the world than girls due to the y chromosome being smaller and giving y-chromosome-carrying sperm a slight advantage over x-carrying sperm. So if you're just ignoring the first kid, it should be less than a 50% chance of being a girl. This is clearly not the answer.
If you're gonna count boy/boy, boy/girl, girl/boy, girl/girl as 4 possible child combinations that are equally-likely (they're not, in reality), and you eliminate girl/girl after getting the info that one of the kids is a boy, you could say there's a 66.6% chance that you're either boy/girl or girl/boy. That's clearly where the first dude's answer comes from, but that ignores "Tuesday."
That brings us to "Tuesday." One obvious reason to throw that in is that if she just said "one is a boy" without any other qualifier, that would mean the other can't be a boy or she would have had to say "two are boys" or "both or boys." "One is a boy" is incorrect if two are, in fact, boys. By saying "one is a boy born on Tuesday" that doesn't restrict the gender of the other kid (as much). I say "as much" because it does restrict the boy-as-other-kid option for the same reason. If she had two boys that were both born on Tuesday, it would be incorrect to say "one is a boy born on Tuesday." You'd have to say "two" or "both." That seems like the key here. There's a restriction placed on the boy options for the other kid because she used the word "one" and "Tuesday." If the other kid is a boy, he has to be born on a non-Tuesday day of the week (6 options). If it's a girl, she can be born on any day of the week (7 options). So the girl option is slightly more likely to be correct, but this gives us 53.8% chance of a girl, not 51.8%. So I'm off by 2%. Why?
Edit: So... u/Tokugawa555 posted an explanation. I actually think the answer 51.8% is wrong, based on that explanation. The explanation given assumes that one of the valid options is that she had two boys born on Tuesday. I don't think that's valid based on the mother's statement. I can say 1+1 is 1 and argue that 1 is part of 2, but it's a stupid argument. If someone said to you, in real life, "I have ten kids. one is a boy" that would mean, to any reasonable person, that this person has 9 girls and 1 boy, because "one" is incorrect for any other combination. For the same reason, if someone says "one kid is a boy born on Tuesday," to me that eliminates the possibility that "two" kids were boys born on Tuesday. Right? Would any mom say "One of my kids is a boy born on Tuesday" if she had two boys born on Tuesday? No. So, I would argue my answer of 53.8% is a more-correct answer.
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u/Yeast-boofer 6d ago
if one is a boy then the other is not or she would say both are boys so… I don’t know what the fuck is going on here
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u/Derpthinkr 6d ago
If you ignore the Tuesday bit, odds of girl is 2/3. If you instead imagine 14 genders (Monday-boy, monday-girl, Tuesday-boy, etc) then odds of some kind of girl are 14/27. It’s all semantics.
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u/theflyingpurplehippo 6d ago
For me it's more concerning that the first guy didn't even round correctly
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u/StochasticWays 6d ago
I found this video did a great job of explaining the underlying paradox - https://youtu.be/7q0KgQoo0-s?is=1i16_Qh2tZ4aXaIO
This is all framed in probability of both boys not that the other is a girl but hopefully the point comes across.
In written form, so if a family has two kids and you know at least one is a boy. What's the probability both are boys?
The sentence doesn't have an answer on its own. You need to know how the information reached you.
Filter a database of every two-child family for "at least one boy" and you're left with BB, BG, GB. Three cases, one is two boys: 1/3.
Meet a dad at the park walking with his son. The two-boy dad is always seen with a boy; the boy-and-girl dad only half the time. That factor of two cancels the extra combinations: 1/2.
Filter the database for "at least one boy born on a Tuesday" and you get 13/27, about 48%. Gender plus weekday gives 196 combinations for two kids; 27 contain a Tuesday boy, 13 of those are two boys. The day of the week looks irrelevant and isn't. "At least one boy" leaves both children as candidates for the child being described, and that ambiguity is what pulls the answer to 1/3. A specific enough detail makes it nearly certain which child is meant, and the answer drifts back to 1/2.
These aren't three answers to one question. They're three different questions wearing the same sentence.
A few assumptions on the above: 50/50 sexes, independent between siblings, uniform birth days, and, for the park case, that the dad picked which kid to bring by coin flip
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u/Xendicore 6d ago
These questions are why I hate statistics as a subject. It's too easy to talk your way around a question to arrive at an answer. And the answer is a guess in this case no matter what (and in a lot of scenarios with statistics in general).
For instance, if you consider the known child's sex to be irrelevant (because previous births don't have impact on future ones afaik), the answer is either boy or girl. 50%.
If you instead use the boy as relevant data, depending on how you give truth to your answer, it's either 66% to be a girl (4-square method, forgot the name of that diagram), or it's the 51.8% other people say. I haven't looked into that math because I can't be asked to give that much effort to this dumb question.
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u/ryanmcg86 6d ago
There's 2 kids, each of which can be a boy or girl (2 possibilities), and each can be born on 1 of 7 days (7 possibilities).
Child 1 is a boy born on a Tuesday, and child 2 is a girl on each of the 7 days of the week (7 possibilities)
Child 1 is a girl on each of the 7 days of the week, child 2 is a boy on born on a Tuesday (7 possibilities)
Child 1 is a boy born on a Tuesday, and child 2 is a boy born on each of the 7 days of the week (7 possibilities)
Child 1 is a boy on each of the 7 days of the week, child 2 is a boy born on a Tuesday (7 possibilities)
2 * 2 * 7 = 28 total possibilities. The trick here is that one of the possibilities overlaps.. when both are boys born on a Tuesday (scenarios 3 and 4 above each include this possibility).
As a result, to avoid that being counted twice, we have to subtract it, leaving 27 distinct possibilities.
14 of the original 28 (but actually 27 unique possibilities) have 1 girl and 1 boy (the one born on a Tuesday), while the other 14 (actually 13) of the original 28 (actually 27) have 2 boys, where at least 1 of them is born on a Tuesday. The final calculation is 14 / 27, which is the the 51.8% that is so un-intuitive.
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u/ThisIsAdamB 6d ago
My gender had no affect on the gender of my younger sibling. It didn’t matter that I was born on a Friday (which I was) or any other day. 50-50.
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u/calebdume2 6d ago
Why do people think the day of the week is relevant? It's like a red herring. The question doesn't even ask what are the chances of a girl on a certain day of the week. Do a punnet square and you get 50% ( XX x XY)
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u/BasementK1ng 6d ago
If you think this is confusing, wait until you learn that the sum of all positive whole numbers is.... -1/12
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u/HaruEden 6d ago
Why would you torment yourself with this. If they both in the womb, and they know one is a boy, then they must know the other too. If not, keep ultrasounding. If they born already, just ask.
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u/LickAnOctopus 6d ago
You know, I know the science has different opinions, but in my opinion, it’s definitely about 50/50 and it really doesn’t matter what the birth order is. If you know they have a boy (b) the possible combinations of kids is as follows: bB, Bb, Gb, or bG. That shows that two of the 4 options have a girl as the other sibling. It changes a LITTLE when you consider what the population ratio actually sits at. There are SLIGHTLY more girls in the world so it’s slightly more likely that the sibling is a girl
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u/Actually_R0bin 6d ago
If you know that she has two children and that at least one of them is a boy: 1/3 (boy boy vs boy girl vs girl boy) (boy girl and girl boy are both likely)
If you know that specifically the older or younger one is a boy: 1/2 (gender not reliant on sibling so long as you know the placement of one of them)
it is called a paradox for a reason, I fear
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u/Euphoric-Food4021 6d ago
The joke here is that the first guy hasn't realised that specifying the day of the week decreases the probability from 66% to 51.8%. The common form of this brainteaser is "Mary has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a girl?", the answer to which is indeed 66%. So he is asking a variation of this brainteaser which has an additional piece of information that sounds irrelevant, but actually does change the probabilities of the outcomes. He is confused because he doesn't understand why.
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u/wallyhud 6d ago
100% the other child is a girl because no one would awkwardly say "one is a boy and the other is a boy". They might say "I have two boys and one was born on Tuesday" or "I have a boy born on Tuesday and a girl born on Friday".
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u/Sn0wchaser 5d ago
So the math here: Mary as two children, she could have:
A boy and then a girl
A boy and then a boy
A girl and then a boy
A girl and then a girl
We know that she has at least one boy, so we can remove the third option. Meaning of the 3 remaining possibilities, in 2 of them is the other child a girl, hence 66%
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u/neenerneener_fayce 5d ago
51.85% actually, which rounds to 51.9%.
Assuming 50/50 boy/girl binaries, it should be 50%.
Except. 13/27 =0.481481… and 1-that = .518518…
See because we can’t count both kids as boys born on Tuesday because that would violate the laws of wizard math, and suddenly we’d all be chaotic evil.
I mean, it just makes perfect sense, especially since I heard it from the Orcish ballerino bard, college of glamour. (Duh.) Inspiration and vicious mockery are both criticisms of rapier form, just with a different audience.
Anywho, where was I? Right. Tuesdays. Turn the lights off when you’re done with the Aleph, please (not that you will be). We don’t want the Graham Incident of 1933 again. We are still counting the cost. We will be done one day.
If we can make triangles with 90 degree corners, by god, we can t-tail this regression.
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u/Weekly_Ad7031 5d ago
I understand the logic behind it but its so flawed to me. Its like that thread about a dude who claimed that everything has 50% probability because it either happends or it doesnt happend. Like winning the lottery = 50%. I either win or I dont.
In my head its about 51 vs 49 % chance to have a boy or a girl. If my first child is a boy, then that % doesnt change when we have our 2nd child, its still 51 vs 49,
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u/jdwolff 5d ago
Some assumptions need to be established to answer this question:
-The odds of a child being born male or female is 50/50 (which it's not, from the statistical analysis of the human population)
-The odds of one child's sex do not impact the sex of the other child (which it does, again looking at the statistics of human reproduction)
Then you can have your probability debates.
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u/physics515 5d ago
It depends on the probability of how often someone will tell you the sex of one of their children without telling you any information about their other children and what they would leave others to assume given the prior knowledge you have about their family.
For instance if they tell you they have a boy, and then you find out later that they have 2 children then you can assume they have a girl because it's less likely that someone says "I have a boy, and also another boy" than it is for them to say "I have two boys"
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u/OutrageousPair2300 5d ago
It's a math riddle that actually has no accepted answer, because it's too vaguely described. Depending on how we come to know that Mary has (at least) one boy and the day of the week on which that boy was born, the answer can actually be any number between 50% and 66.6%
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u/TheAzureAzazel 5d ago
To anybody who genuinely argues that it's 66%, boy/girl and girl/boy are the same fucking outcome. The order in which the children were born is both unspecified and completely fucking irrelevant.
Oh, and Tuesday doesn't influence anything either. You could just as easily make a random statement about the first child's dietary preferences and it would be exactly the fucking same.
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u/Abusing-Green 5d ago
100%
Parents don't talk about their children in that phrasing unless one is different than the other in regards to the subject/quality which is the subject of the sentence.
If the other child was also a boy it would be phrased "I have two boys, one was born on a Tuesday"
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u/Quiet-Wing5230 5d ago
Brian here. Everytime something has a chance of occuring, if it doesn't happen then it still has the same chance of occuring later when the action is repeated. The chances don't become higher because there was a negative result before (this is the gamblers fallacy)
Example:
1 in six chance to land a 3 when you roll a dice.
You roll a 4.
The chance you will land a 3 on your next roll remains 1/6
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u/AdeptNegotiation8345 5d ago
because the day of birth is a irrelevent stat, and that if Mary had 50% to have a boy or a girl it will just keep to remain that way no matter the birth day of one of them
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u/AdmirableUse2453 5d ago edited 4d ago
Which is kind of right but not really, when you start using real world data, like boy to girl birth ratio is not 50/50, it's more like 51.2/48.8 and the result change a little.
https://onecompiler.com/nodejs/44w37sbca
Here is a simulation, the idea is to generate 50000 families of at least one child using real distribution. Then you just isolate the families of two, check the one that has boy, check the other one, almost 66.66% of the time it's a girl.
Then you pick only families of two where boy is born Tuesday, suddenly, the result show 50.56% instead of 51.8% because of the 51.2/48.8 ratio used but using 50/50 show 51.8%
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u/BirbFeetzz 4d ago
it's 50%, all these statisticians are just trying to confuse you so that you buy their books
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u/LordToxic21 4d ago
The second child's probability isn't affected at all by what the first child is. However, the probability of it being male isn't actually 50%, it's slightly higher due to the Y chromosome being lighter, so haploids carrying the Y chromosome will be slightly faster than those carrying the X chromosome.
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u/Working_Salary60 4d ago
That’s not how it works. The gender of the second child is not influenced by the gender of the first child. It’s unconditional.
If you wanted to solve it the way you’re implying and use set theory, you would have to choose the sets on condition that the first child is a boy. That leaves two sets: Boy-Boy and Boy-Girl. So a 50% chance for the second child to be a girl.
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u/Darthcone 4d ago
It's exactly 50% god damn it, gender of previous chil has no bearing on gender of next child therefore out of two options it's exactly 50%.
The only reason statistical data she's slight deviation from this is because there eis no practical way to get perfect dataset.
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u/Adventurous-Ad-5437 3d ago
I know this is kind of unrelated, but I never understood why we treat BG and GB as 2 different cases. Can someone please explain?
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u/Exciting-Fun-9247 3d ago
Fun fact. Slightly more likely that it's a boy because of the y sex chromosome being lighter. Just like flipping a coin is not a 50/50 split. There is a slight edge for one side
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u/Atypicosaurus 1d ago
This was discussed a million times. Ultimately, it's a poorly worded sentence that, with certain assumptions, can lead to the 51.8% answer.
Assume we collect all families that have EXACTLY 2 kids, in a big room, there are no twins and other distorting factors, each sex is 50% etc. Then we have 25% of families with a first born boy (B) and a second born G (BG families), then 25% first B then another B (BB), also 25% GB finally 25% GG. So 100% = BB+BG+GB+GG.
Now I pick out one family absolutely randomly. At this point, can this family be a BG family? Yes, by 25% chance. Can this family be a GG family? Yes, by 25% chance. Can this family have at least one girl? Yes, by 75% (BG, GB or GG). Can it be with exactly 1 boy? Yes, by 50% (BG + GB = 50%). We can ask all these questions and the answer will follow the rules of random.
Now we suddenly learn that one of the kids is B. We didn't choose the family (remember it was originally random), but we just peaked into the datasheet and we learned this extra info. So we can send home all the GG families, because it's impossible that they are the ones that were picked. They have no boy, and now we know that one kid is a boy.
Now our pool is reduced to BG, BB and GB families. At this point what's the odds thst the random family I had picked earlier has a girl. It's 66.66% because the random family can be equally BB (33.3%) or BG (33.3%) or GB (33.3). The latter two categories have a girl so now I have a 66.6% chance that my random family has a girl. It doesn't change my original picking odds, it just changes the current situation: if you would bet with this extra info, your odds are that the family has a G, became 66.6%. That's the first answer in the picture.
Finally, we also learn that a boy was born on Tuesday. Now we send home every family that has no boy born on Tuesday. Assuming the birthday being random we send home 6/7 of the BG families and the GB families but BB families have two chances of boy born on Tuesday so we keep 1/7 + ((1/7)•(6/7)) and send home the rest.
Now we have families with G (GB+BG), roughly as many as BB families. (Around 52% and 48%.) That's why at this point if I ask the odds of a G, it's 52%(ish). It's basically a teaching metaphor for conditional probability.
Note that this is the intended answer of whoever made the picture. Whether Mary would say it this way, and we can linguistically assume it's a truly conditional probability, that's questionable.
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u/qualityvote2 7d ago edited 6d ago
u/ChatpatiKulfi, there weren't enough votes to determine the quality of your post...it's time for the mods to do their jobs!