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Mathematics Post #7814, on Mar 12, 2026 in TG

Boy Born on a Tuesday: 66.6% vs 51.8% Probability Standoff

Why is this Mathematics meme funny?

Level 1: The Detail That Shouldn't Matter

One man proudly answers a riddle about whether a kid is a boy or a girl, and another man calmly tells him a slightly different number — all because the riddle mentioned the boy was born on a Tuesday. The first man's face melts, because how could a Tuesday change anything? It's like saying a coin flip changes odds because the coin is wearing a tiny hat. The funny part is that the tiny hat really does change the math — and watching a confident know-it-all discover that is one of life's purest joys.

Level 2: Why a Tuesday Changes Anything

The key concept is conditional probability: the chance of something given what you already know. Learning information shrinks the set of possible worlds, and you recompute odds inside the smaller set.

With "one child is a boy," the possible families are boy-boy, boy-girl, and girl-boy (girl-girl is ruled out). Two of three have a girl → 66.6%.
With "one child is a boy born on a Tuesday," you list families by gender and birth weekday. Doing the bookkeeping leaves 27 possible family configurations, and 14 of them include a girl → 14/27 ≈ 51.8%.

The unintuitive part is that the weekday feels irrelevant — and for any single child it is — but as a filter on which families qualify, it reshapes the candidate pool. It's the same trap as debugging with logs: the events you get to see are not a neutral sample of the events that happened. If you've ever been burned by a dashboard that only counted requests that completed, you've met this paradox in production clothing.

Level 3: Weaponized Trivia, Reversed

The format is the beloved Limmy's Show "steel is heavier than feathers" sketch, and the casting is the joke. In the original, Limmy (gray shirt, facing camera) is the confidently wrong one. Here the meme flips the energy: gray-shirt delivers the 66.6% answer with the smugness of someone who has read one probability blog post — the guy who corrects others at parties — and the black-shirted man lands the 51.8% like a counter-sniper, leaving Limmy with the sketch's signature thousand-yard stare of a man whose mental model just segfaulted.

That's the social dynamic this meme skewers, and it's painfully familiar in tech circles: the mid-curve correction. The 50% answer is the naive take; 66.6% is the "actually 🤓" take; 51.8% is the "actually-actually" take that punishes the first corrector. It's the probability version of replying "well, technically HTTP/2 multiplexes over one TCP connection" to someone who just finished correcting someone else about keep-alive. Interviewers have abused the two-child family of puzzles for decades precisely because every level of sophistication is wrong from the next level up — the question selects less for statistical skill than for whether the candidate has seen this specific landmine before. Data scientists recognize the deeper lesson: conditioning on information correctly is genuinely hard, the answer depends on how the data reached you (selection bias, in the wild), and a single innocuous-looking field added to the dataset — a weekday column, of all things — can silently shift every downstream estimate.

Level 4: Counting to 14/27

Both numbers in this meme are defensible, which is exactly why it's a fight. The setup is the Tuesday boy problem, a sharpened variant of Martin Gardner's classic two-child paradox, and the whole thing lives or dies on how you construct the sample space.

Naive version first. Two children, four equally likely gender orderings: BB, BG, GB, GG. "One is a boy" eliminates GG, leaving three equally likely worlds, in two of which the other child is a girl — hence the gray-shirt confidence:

That's right. It's 66.6%

Now the man in black. Take each child as a (gender, weekday) pair: $2 \times 7 = 14$ equally likely types per child, $14^2 = 196$ ordered pairs per family. Condition on "at least one child is a boy born on a Tuesday." Count those worlds: first child is a Tuesday-boy ($14$ options for the second) plus second child is a Tuesday-boy ($14$ for the first), minus the double-counted both-Tuesday-boys case:

$$ 14 + 14 - 1 = 27 $$

In how many of those 27 is the other child a girl? Tuesday-boy first with any girl (7 days) plus any girl first with Tuesday-boy second (7 days): $14$. So

$$ P(\text{other is a girl}) = \frac{14}{27} \approx 51.85% $$

The apparently useless weekday detail changes the answer because it acts as a near-identifier: the more specific the description of the boy, the closer you get to having pointed at a particular child, and "the other child" then collapses toward an independent 50/50. Push the specificity further ("a boy born at 3:07:42 AM on a Tuesday") and the probability marches asymptotically to 1/2. The remaining war — fought in every comment section this meme has ever appeared in — is about the selection procedure: 66.6% and 51.8% both assume Mary was filtered from the population of qualifying families. If she just volunteered a fact about a randomly chosen child, the answer is a boring 50%. The paradox is less about probability than about the unstated likelihood model behind a sentence — pure Bayesian ambiguity wearing a puzzle costume.

Description

A two-panel meme using stills from the Limmy's Show 'steel vs feathers' sketch. Top panel: Limmy in a grey shirt faces the camera with overlaid text: 'Mary has 2 children. She tells you that one is a boy born on a tuesday. What's the probability the other child is a girl? That's right. It's 66.6%'. Bottom panel: a second man in a black shirt turns to him and says 'It's 51.8%', leaving Limmy visibly confounded. This references the famous 'Tuesday boy' conditional probability paradox: naive two-child reasoning gives 2/3 (66.6%), but conditioning on the extra 'born on Tuesday' information yields 14/27 ≈ 51.85% - a counterintuitive result beloved by probability nerds and interview-question sadists

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BayesianInference StatisticalAnalysis DiscreteMathematics TechnicalInterviews PopCultureReference MemeFormats DataScience

Comments

120

Anonymous ★ Top Pick Like every estimate in engineering, the answer changes the moment someone adds one irrelevant-sounding detail to the ticket

Anonymous ★ Top Pick

Like every estimate in engineering, the answer changes the moment someone adds one irrelevant-sounding detail to the ticket
@Art3m_1502 4mo

Never understood combinatorics
@Art3m_1502 4mo

But biologically it's 50%
1. @Daonifur 4mo
  
  Correct. After 1 the odds increase or decrease, depending on previous results though. Less likely to repeat the same thing basically
  
  @Art3m_1502 4mo
  
  So answer should be 25% because 2nd child has also 50% chance for being a boy?
  
  @Daonifur 4mo
  
  50% chance for being a girl but also higher chance for both to not repeat the same result
  
  @Art3m_1502 4mo
  
  Ahhh, got it
  
  @Daonifur 4mo
  
  Yeah, it otherwise would be a 50% chance every time with the exception of likelihood two things will be the same number. You could go further and calculate also based on family history and a bunch of other factors too
  
  @tonko22 4mo
  
  But the event are independent, isnt it? Explaining that (independent) events do not affect the probability of future events in a sequence is challenging because the human brain is wired to find patterns and assume that streaks will "even out". This cognitive error is known as the Gambler's Fallacy.
  
  @Daonifur 4mo
  
  It's both, depending on how you look at it
  
  @Daonifur 4mo
  
  Same thing with a coin toss
  
  @purplesyringa 4mo
  
  the interesting part is that since genetics are involved, they're not actually independent. e.g. https://www.science.org/doi/10.1126/sciadv.adu7402 says We observed that a balanced offspring sex [i.e. FM (female-male) or MF] was the most frequent family composition in sibship of size of 2, but a clustering of single sex (e.g. MMM and FFFFF) was generally more frequent in subship of size 3 or larger
  
  @tonko22 4mo
  
  Nice, 0.519 it says. Probably the meme refers to that exact value too
  
  _ 4mo
  
  Is that really because of genetics, or because people are more likely to have another child if the first two have to same sex ?
  
  @purplesyringa 4mo
  
  The paper says they removed the youngest child from the consideration to account for the possibility of families stopping reproduction because their child was of a gender they didn't like, which I guess worsens this issue
  
  @purplesyringa 4mo
  
  see the "Sensitivity analysis" section of the paper
  
  @purplesyringa 4mo
  
  Therefore, in two separate analyses, we (i) excluded obvious “coupon collectors” (i.e., women who only stopped producing offspring after both offspring sexes are reached) and (ii) more conservatively excluded the last birth of every woman I guess they handled this issue at least somewhat
  
  @azizhakberdiev 4mo
  
  it's no posterior probability, second child is still either a girl or a boy
  
  @purplesyringa 4mo
  
  https://en.wikipedia.org/wiki/Gambler%27s_fallacy
  
  @VentusTheSox 4mo
  
  This has nothing to do with it.
  
  @Daonifur 4mo
  
  Yes it does, I explained why
2. @kitbot256 4mo
  
  It is forty something. Depending on woman’s age actually. 1. There are more boys than girls indeed, about 105:100. 2. Also the events are not fully independent, the kids tend to have the same sex as the previous births and the tendency grows stronger with the mother’s age.
  
  @purplesyringa 4mo
  
  Is that actually true? Wikipedia mentions the opposite figure here
  
  @kitbot256 4mo
  
  https://www.pewresearch.org/short-reads/2013/09/24/the-odds-that-you-will-give-birth-to-a-boy-or-girl-depend-on-where-in-the-world-you-live/ You are right, this info is dated. And also incorrect, thanks AI While historically, there have been about 105 boys born for every 100 girls worldwide — which creates a “sex ratio at birth” of 1.05 — the share of boy babies has increased in recent decades. 2011 data from the World Bank show the global sex ratio at birth is now 1.07, or 107 boys born for every 100 girls.
@VentusTheSox 4mo

When thinking on probability you can arrange the event results in a table G : girl b : Boy Plotting this out: B B B G G B G G
1. @VentusTheSox 4mo
  
  We know that one is a boy so GG is ruled out The remaining ones are BB BG GB BG GB Are both valid answers that result in one boy and one girl so 66.6% chance (2 of the 3 remaining options result in a boy girl set) This is the part 1
  
  @Art3m_1502 4mo
  
  It just breaks my mind how answer can be different depending how we put the question
  
  @Art3m_1502 4mo
  
  In general it's 50% chance for a child to be girl but if we want to guess a childs gender when we already know that 1st is a boy it's 66%
  
  @VentusTheSox 4mo
  
  The joke is that something that biologically is a 50% chance. Mathematics take that and make it no longer make sense (66.6% And then you apply mathematics even harder and it returns to normalcy 51.8%
  
  @Art3m_1502 4mo
  
  But 66% makes sense for the problem
  
  @Daonifur 4mo
  
  We should test their DNA to determine the answer more accurately
  
  @Art3m_1502 4mo
  
  But 51.8% yeah, it's funny
  
  @purplesyringa 4mo
  
  I'm sorry, does no one here know math?
  
  @VentusTheSox 4mo
  
  Part 2 Because we specify that the boy is born on a Tuesday you now have to count all the combinations of the two births in of a boy girl pair, while also accounting for the 7 days of the week. (I CBA do plot that out in text) By doing that you get 51.8%
  
  @VentusTheSox 4mo
  
  https://youtu.be/JSE4oy0KQ2Q?si=RGRhsVeOcOGIgPCX
  
  @VentusTheSox 4mo
  
  Here is the full video which explains this joke
  
  @VentusTheSox 4mo
  
  Everyone take 7 minutes of your day to watch this video by this amazing channel about the topic
  
  @M4lenov 4mo
  
  Thank you
  
  @purplesyringa 4mo
  
  Well, that can't be right. With the weekday added, the elements of the probability space are two tuples of (gender, weekday). There's 14^2 elements in the probability space in total, and by saying there's a boy born on Tuesday, we're limiting it to 14 * 2 - 1 = 27 elements (((Boy, Tuesday), (..., ...)) and ((..., ...), (Boy, Tuesday)), and ((Boy, Tuesday), (Boy, Tuesday)) is counted twice). Out of this set, exactly 7 * 2 = 14 elements contain girls (((Boy, Tuesday), (Girl, ...)) and vice versa). This gives 14 / 27 = 2 / 3 as well
  
  @VentusTheSox 4mo
  
  14/27 =/= 2/3
  
  @purplesyringa 4mo
  
  yeah, thanks. now I'm confused
  
  @VentusTheSox 4mo
  
  =/= means does not equal
  
  @purplesyringa 4mo
  
  I understand what you're saying, I was mistaken
  
  @purplesyringa 4mo
  
  I guess this makes sense because you're making states more distinguishable
  
  @purplesyringa 4mo
  
  I'll proudly state I believed 3 * 7 = 27 for like five minutes. I should probably get more sleep
  
  @death_by_oom 4mo
  
  What I imagine the accounting department at Enron was circa 2001
  
  @SwedishSock 4mo
  
  Does that mean that we can make the likelihood whatever we want just by tossing in whatever measurements we want into it?
@purplesyringa 4mo

ah, apparently I can't count. thanks
@Daonifur 4mo

The first set on the left shows 7 sets in one row, then 6
1. @Daonifur 4mo
  
  Which can be represented as 7, 6. The inverse being 6, 7
@VentusTheSox 4mo

B⁰ meaning Boy born on Sunday B¹ meaning boy born on Monday This table is showing all results in which have at least B² meaning at least one boy born on Tuesday You can't show the same result twice that's why there is one result 'missing' Meaning 14/27 Instead of 14/28 (50%)
1. @death_by_oom 4mo
  
  I think I lost all 3 of my remaining brain cells reading this thread
  
  @feralape 4mo
  
  Lost you mean you finally used them up?
  
  @death_by_oom 4mo
  
  No, I mean, this is maths brainrot
  
  @VentusTheSox 4mo
  
  Watch the video
  
  @death_by_oom 4mo
  
  I understand the concept of the paradox, but the way the you are building the sample space is not correct. If you don't have info on the ordering of the children, then you cannot include it in the sample space and (Bx, Gy) and (Gy, Bx) are the same and are double counted in the video, now if we have the information on if the boy was born first or second, then either all BG are invalid or all GB are invalid. In both those cases the probability is back to 50%
  
  @purplesyringa 4mo
  
  That's not quite the right interpretation. The idea behind the choice of the sample space is that the birth of two children are treated as independent events, and so the sample space for event A followed by event B must be the cartesian product, that is, ordered tuples of events (A, B). By merging (girl, boy) and (boy, girl), you'd be implicitly saying that birthing a boy and a girl is just as likely as birthing two boys, even though it's equivalent to winning a coin flip once (first birth is "free", second birth to get a different gender) rather than twice (getting boys both times).
  
  @death_by_oom 4mo
  
  No, I am not. The point is that by not considering (Bx, Gy) and (Gy, Bx) the same you are implying that there is some kind of difference between them. From the text of the problem there is no difference between them, so they are double counting
  
  @Daonifur 4mo
  
  There's also already a boy in the existing subset first, so no need for the statistics with girl first
  
  @purplesyringa 4mo
  
  Look at it this way. If I'm flipping a fair coin and, for whatever reason, instead of the sample space {heads, tails}, I use the sample space {heads, (tails and it's daytime), (tails and it's nighttime)}, then it's redundant, but as long as P(tails and it's daytime) + P(tails and it's nighttime) = 0.5, I'll get the exact same answers. Drawing a difference between boy+girl and girl+boy might be redundant in some way, but it's just a trick to keep the sample space uniform. You can, of course, replace the sample space with {{boy, boy}, {girl, girl}, {boy, girl}}, and make the 3rd option twice as likely as others. That'll give you the same results: P({boy, boy}|{boy, boy} or {boy, girl}) = 0.25 / 0.75 = 1 / 3
  
  @death_by_oom 4mo
  
  You are assuming that the children are distinguishable, this was not given in a problem. Don't make assumptions about the sample space
  
  @purplesyringa 4mo
  
  Let's make it clearer. What exactly are you disagreeing with? Are you disagreeing with the claim that the probability of having a girl and a boy is twice as large as the probability of having two girls?
  
  @death_by_oom 4mo
  
  The 51.8% paradox and the 66.6% one
  
  @death_by_oom 4mo
  
  They stem from the same assumption of distinguishablity of objects
  
  @purplesyringa 4mo
  
  In my previous comment, I've mentioned that you can drop distinguishability as long as you can still assume that P(has girl and boy) = 2 * P(has two girls). Do you agree with this equality or not?
  
  @purplesyringa 4mo
  
  The issue with this is that if we know nothing about the probability space, we don't even know if it's uniform. You could argue that we're not told that children are distinguishable in the problem statement, but then we don't know if we're on Mars where there are three sexes, or where there are 10 times as many boys as girls, or where everyone can only birth people of a single sex. There necessarily has to be some inherent knowledge here, and the knowledge we're choosing is that births are independent. That implies distinguishability.
  
  @Daonifur 4mo
  
  The only one making sense being the latter btw
  
  @Daonifur 4mo
  
  But only if you keep in mind bg bb gg set would discredit the last one and still produce 50/50
  
  @death_by_oom 4mo
  
  No it fucking doesn't. GB and BG are equivalent unless told otherwise. If you say that the boy was born first for example, then you have a way to distinguishe the children and suddenly GB and BG are not the same, but at the same time GG and GB are not valid, and we are back to 50%
  
  @Daonifur 4mo
  
  I already explained this in the reply I added to expand upon it, as well as you already stating this earlier. It only makes sense in bg bb gg format, which excludes gg
  
  @purplesyringa 4mo
  
  You cannot assume this. You're not told they are equivalent, just as you're not told they are not equivalent, therefore you can't assume this, therefore the problem is not solvable without assumptions. You have to make some choice here, and that for that choice to consistently represent the real world, you have to make them distinguishable
  
  @death_by_oom 4mo
  
  That's the issue, I am not assuming anything, you are assuming that two children are distinguishable from each other
  
  @purplesyringa 4mo
  
  "GB and BG are equivalent" sounds like you're assuming equivalence
  
  @purplesyringa 4mo
  
  or, screw equivalence, you're assuming the sample space is uniform, which is worse and what we should actually be focusing on
  
  @purplesyringa 4mo
  
  so I repeat my question: do you think there are as many families with a boy and a girl as there are with two girls? does that match your intuitive understanding of the world or your experiences?
  
  @death_by_oom 4mo
  
  In the real world children are distinguishable, and we know that. In a math problem if you give me no way to distinguishe the children then I won't be able to, and will be forced to count them as indistinguishable
  
  @purplesyringa 4mo
  
  You're a free man, no one forces you to do anything, you can say the problem is not solvable as given and go for a walk
  
  @purplesyringa 4mo
  
  you don't have to go out of your way to invent defaults that don't match the real world and only come to your mind because it's the simplest alternative
  
  @death_by_oom 4mo
  
  There are no invented defaults, math does not exist in our physical world, it exists outside of it, so if you are using it in the real world, then you need to define all the assumptions you are making
  
  @purplesyringa 4mo
  
  Assuming the probability space is uniform is a (wrong, in this scenario) default
  
  @death_by_oom 4mo
  
  You are making the same assumption
  
  @death_by_oom 4mo
  
  If we don't define the probability of a boy and girl as .5 and the probablity of a child being born on a certain day of the week as 1/7 the paradox falls apart anyways
  
  @Daonifur 4mo
  
  The only one making any sense would be 66% and only if the boy didn't already exist
  
  @death_by_oom 4mo
  
  If you are such a stickler for the rules, just add assume uniform model in the end of the problem, my point still stands
  
  @purplesyringa 4mo
  
  If you add the uniform assumption, then of course I'll agree with you. It's just that it doesn't agree with reality, so it no longer works as a gotcha, but rather becomes an abstract mathematical trick, at which point what are we even doing.
  
  @death_by_oom 4mo
  
  You are assuming uniformity as well, so what you are says, you agree with me
  
  @purplesyringa 4mo
  
  I'm assuming a) uniformity of genders of a single child, b) independency and distinguishability of child births. I am not assuming uniformity or non-uniformity of the overall probability space, not directly
  
  @death_by_oom 4mo
  
  I hope this finally resolves it. i took your assumtions
  
  @purplesyringa 4mo
  
  > [...] the “14/27 paradoxical result” arises only when the problem ambiguously mixes cases without specifying which child is meant. yes, this ambiguity is the whole point...
  
  @death_by_oom 4mo
  
  Honestly, truce? I am kinda tired of this argument. I already written a pdf on this shit
  
  @purplesyringa 4mo
  
  bro wrote a pdf yea, same
  
  dev_meme 4mo
  
  Wait till they're offered to choose door and allowed to change their decision later 🌚
  
  @Art3m_1502 4mo
  
  Girl has 0 chances when there's boy next door
  
  @death_by_oom 4mo
  
  If you have no input to distinguishe them, then that are equivalent, not assuming anything
  
  @purplesyringa 4mo
  
  bro thinks children and fermions
  
  @Daonifur 4mo
  
  GB can't be used in the existing set because B already exists
  
  @purplesyringa 4mo
  
  can you elaborate?
  
  @Daonifur 4mo
  
  Think of it as an equation. B is already given and you're trying to determine all possible sets with G and B. You'll get BG, BB
  
  @Daonifur 4mo
  
  My example doesn't include all combinations of both B and G because B already exists
  
  @purplesyringa 4mo
  
  I understand what you're saying but you couldn't have said it more confusingly
  
  @Daonifur 4mo
  
  Boy already exists, ooga Chance of girl, booga Boy boy or boy girl, unga 50/50 booga 50 percent ooga
  
  @purplesyringa 4mo
  
  When you're saying this, you're implicitly saying that "boy boy" and "boy girl" are equivalently likely. Phrased otherwise, you're saying that having two children of same gender is twice as likely as having two children of different gender. Does that sound right to you based on your experience?
  
  @Daonifur 4mo
  
  You're starting to get it now. It's almost as if two variables have a 50/50 or 50% chance of getting picked, just like a coin toss
  
  @purplesyringa 4mo
  
  Do you have a coin nearby?
  
  @Daonifur 4mo
  
  Do you? Preferably of solid gold
  
  @purplesyringa 4mo
  
  Do me a favor, run an experiment like 20 times. Throw two coins and count the occurrences of "two heads", "two tails", and "head and tail". Tell me if you'll get a 33/33/33 distribution or a 25/25/50
  
  @purplesyringa 4mo
  
  do me a favor as well, do this too
  
  @Daonifur 4mo
  
  If only the boy didn't exist already or you were looking for the chances of a specific pairing if two kids were made
  
  @death_by_oom 4mo
  
  If you take all the families that have at least 1 boy out of 2 children, like in The problem, then yes, BB and BG are .5 each
  
  @Daonifur 4mo
  
  Otherwise 66% makes sense if you're determining the likelihood you'll have a girl or boy when having two kids
  
  @Daonifur 4mo
  
  Not with one though, because boy already exists
  
  @death_by_oom 4mo
  
  Change the problem to red and blue pencils instead of children and recalculate. Your answer must not change, since mathematically the problems are the same
  
  @death_by_oom 4mo
  
  But with indistinguishable objects the (x, y) and (y, x) are equivalent
  
  @Daonifur 4mo
  
  The only major issue with this is it's not a coin toss, so there's no real controllable factor to determine this and so it still just goes back to 50%, despite all the gambling and statisical likelihood percentages
  
  @death_by_oom 4mo
  
  Exactly my point, they are independent
@zzerga 4mo

The correct answer should be even less than 50%, because on average, there are more newborn boys than girls (in human race)
@Art3m_1502 4mo

Canon event
@b7sum 4mo

#whalegang 🐳
@IkenieWo 4mo

Fascinating thread. Brightest example of how bureaucracy is solving a made up problem with math without applying common sense
@srt57t 4mo

@victorbratov @purplesyringa conditional probability is meaningless until the selection rule is specified. You are only allowed to study probability if you have two kids and at least one son born on Tuesday. You are selected at random from all eligible contestants. What is the probability that your other child is a girl? and you get 14/27 if it's a real life talk about your kids, then Tuesday condition doesnt bring anything meaningful to the context - it's just noise. Therefore you can guess the gender of the other child with 50% possibility - no info whatsoever, 2 variants

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Boy Born on a Tuesday: 66.6% vs 51.8% Probability Standoff

Why is this Mathematics meme funny?

Level 1: The Detail That Shouldn't Matter

Level 2: Why a Tuesday Changes Anything

Level 3: Weaponized Trivia, Reversed

Level 4: Counting to 14/27

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