# Probability problem, please explain it to me! **URL:** \ **Category:** Factual Questions\ **Created:** [January 14, 2026, 1:44pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683 "2026-01-14T13:44:21Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Hermitian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hermitian/32/470_2.png) [@Hermitian](https://boards.straightdope.com/u/Hermitian)\ **Post date:** [January 14, 2026, 1:44pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/1 "2026-01-14T13:44:21Z")
There is a probability problem on the internet that I really don’t understand. Please explain it to me in a way that makes rational sense to me. I am stealing this text from the webpage linked below. “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? Most people think the answer is 50%. People familiar with the boy or girl paradox may guess 66.6%. But in this particular variation, the answer is about 51.8%. So people are wondering: where are the numbers coming from?” On [this](https://mindyourdecisions.com/blog/2025/12/26/boy-or-girl-paradox-born-on-tuesday/) webpage (which also has a video), he goes through the math to show why it is 14/27 or ~51.8%. And just by the way, it assumes the boy/girl birth ratio to be 50%. I understand the classic answer being 66.6%. I really do not understand how stating that the boy was born on a Tuesday changes the answer. The boy had to be born on a day. There is nothing special about Tuesday. So if we replace Tuesday with any other day, the math will end up the same (51.8%). So if the answer is the same for any day, it will always be the same, which conflicts with the general answer of 66.6%. Please someone help me make rational sense of the answer! ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [January 14, 2026, 1:57pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/2 "2026-01-14T13:57:32Z")
Firstly in order to reframe the puzzle properly, you should ask her whether she has at least one boy born on a Tuesday, the fact she’s giving the information herself skews the reasoning for why she might say it. There are 27 equally likely different ways that there can be at least one Boy (Tue) 1/27 the other is a boy (tue) 14/27 it’s a girl 7 times older, 7 times younger 12/27 its a boy (not tue) 6 times younger, 6 times older. 14/27=51.85% It’s the same logic as the other problem, where removing the chance of both children being girls or (boys not born on Tuesday) increases the odds of the subsection “girls” ---
**Author:** ![glee](https://avatars.discourse-cdn.com/v4/letter/g/b5a626/32.png) [@glee](https://boards.straightdope.com/u/glee)\ **Post date:** [January 14, 2026, 2:00pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/3 "2026-01-14T14:00:54Z")
There is an text explanation on the website (scroll down to see it), so you don’t have to watch the video. It’s seems that the information about Tuesday doesn’t matter. But consider this. First, if somebody has two children there are various possibilities. In birth order, there could be: boy, then boy boy, then girl girl, then girl girl, then boy. So when we learn that one child is a boy, we drop the ‘girl, then girl’ possibility. Next we can add what days of the week each child was born on. (I’m not going to list them here as there’s a lot!) However the point is that a few possibilities will drop out because they must include a boy born on Tuesday. And that’s why the probability changes… ---
**Author:** ![Hermitian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hermitian/32/470_2.png) [@Hermitian](https://boards.straightdope.com/u/Hermitian)\ **Post date:** [January 14, 2026, 2:08pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/4 "2026-01-14T14:08:03Z")
I feel like the responses are just restated the probability math, which I sort of follow. It still doesn’t make sense to me. Can someone respond to my “any day” proposal. Instead of her saying “he was born on a Tuesday” she could have said Sunday or Monday. If we use those days instead we get: Boy (born on Sunday): Chance of other child being a girl =51.8% Boy (born on Monday): Chance of other child being a girl =51.8% Boy (born on Tuesday): Chance of other child being a girl =51.8% Boy (born on Wednesday): Chance of other child being a girl =51.8% …and so on. So for every day of the week, the chance of the other child being a girl is 51.8%. But that conflicts with the general answer of 66.6%. Can someone help me reconcile these answers? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 14, 2026, 2:09pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/5 "2026-01-14T14:09:04Z")
> [@Hermitian](#): > > Please someone help me make rational sense of the answer! The standard resolution to this “paradox” is that the probability depends on the exact assumptions of the underlying probability space and selection procedure. So, for example, having a special phonebook which lists all the 2-child families where one of the kids is a boy born on a Tuesday might, not too surprisingly, not lead to the same calculation as randomly knocking on some door and asking the kid who comes out on what day he was born. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 14, 2026, 2:15pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/6 "2026-01-14T14:15:34Z")
> [@Hermitian](#): > > So for every day of the week, the chance of the other child being a girl is 51.8%. But that conflicts with the general answer of 66.6%. Can someone help me reconcile these answers? There is, in fact, a section on Wikipedia about the “boy or girl paradox” illustrating why, under those assumptions, the more (at first glance seemingly irrelevant) information you give the lower the probability that the other child is a girl: > **[Boy or girl paradox | Information about the child](https://en.wikipedia.org/wiki/Boy_or_girl_paradox#Information_about_the_child)** > > Suppose we were told not only that Mr. Smith has two children, and one of them is a boy, but also that the boy was born on a Tuesday: does this change the previous analyses? Again, the answer depends on how this information was presented – what kind of selection process produced this knowledge. > Following the tradition of the problem, suppose that in the population of two-child families, the sex of the two children is independent of one another, equally likely boy or girl, and that the birth date... ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [January 14, 2026, 2:16pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/7 "2026-01-14T14:16:19Z")
ETA: largely ninja’ed by @DPRK’s second post that I didn’t see until I finished this one. > [@Hermitian](#): > > Instead of her saying “he was born on a Tuesday” she could have said Sunday or Monday. Nope. It _looks_ like she could have said that. But she could not have done that and also tell the truth. Because to do so, she’d have needed to have 7 boys, 1 per day, and _also_ simultaneously have just one boy. Once you drop down to just [specific day of week regardless of which specific day of week], now you’re in 51.8 space, not 66.6 space. The answer is different again if she had said “I have one boy born on Tue or Thu but I’m not telling you which.” @DPRK really nailed it conceptually. Unsurprising given his expertise. Both the puzzle setter and the puzzle solver need to be very precise about what is known, what is unknown, and what the possibility space really is. Almost any reductive thinking by the solver amounts to removing essential information and thereby coming to a wrong conclusion. Of course for puzzle-making, that amounts to inviting people to bite off on the easy but wrong answer. So of course lots of puzzles exploit that aspect. That’s also why so many internet puzzles are screwed up messes. The setter doesn’t know what they’re doing, or it gets copied with details unwittingly changed. Or details deliberately changed to hide the plagiarism, but the changer doesn’t recognize their changes to the problem trigger changes to the answer too. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 14, 2026, 2:47pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/8 "2026-01-14T14:47:30Z")
Like Socrates, I have to deny any expertise, but thanks 🙂 The real point of paradoxes is that they _force_ you to example the underlying assumptions formally: Russell’s Paradox, the Coastline Paradox, the Banach–Tarski Paradox, you name it. If you do not do it, there may not be a well-defined answer. Another famous probability one is, "Randomly break a stick into three pieces. What is the probability that the three pieces form a triangle? In this problem, as the Wikipedia link explains, under the “traditional assumptions” of this problem, if the extra information about the boy has probability \varepsilon, where, for example, \varepsilon=1 if you just say “Mary has two children, one of whom is a boy” but \varepsilon=1/7 if you say the boy was born on a Tuesday, then an application of Bayes’s Theorem results in a conditional probability of \displaystyle\frac{2-\varepsilon}{4-\varepsilon} that both children are boys. Again, to backtrack, there is a difference between “Mary has two children, one of whom is a boy” and (under reasonable assumptions, again) seeing Mary with a boy and her telling you that is one of her two children. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [January 14, 2026, 2:54pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/9 "2026-01-14T14:54:30Z")
> [@DPRK](#): > > Again, to backtrack, there is a difference between “Mary has two children, one of whom is a boy” and (under reasonable assumptions, again) seeing Mary with a boy and her telling you that is one of her two children. That is a fun one. There’s also the (possibly inadvertent, possibly deliberate) ambiguity in the first sentence: “Mary has two children, one of whom is a boy”. Which can reasonably be interpreted as either - Mary has two children, _exactly_ one of whom is a boy - Mary has two children, _at least_ one of whom is a boy The former is the more formal “mathy” reading, while the latter is the more common conversational English reading. Which did the puzzle setter mean? Does that person even know what they did? ---
**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\ **Post date:** [January 14, 2026, 3:05pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/10 "2026-01-14T15:05:45Z")
Is there an assumption that the other child cannot be a boy born on a Tuesday, or would that affect anything? ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [January 14, 2026, 3:11pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/11 "2026-01-14T15:11:37Z")
The 1/14 chance that the other child is also a boy born on Tuesday is the key to it. It excludes the possibility that both children are girls born on Tuesday and everything else is symmetrical between boys and girls. The same logic as the non-tuesday problem applies, just with a narrower range of exclusions it will be a lot closer to 50-50. ---
**Author:** ![Elmer\_J.Fudd](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/elmer_j.fudd/32/1211_2.png) [@Elmer\_J.Fudd](https://boards.straightdope.com/u/Elmer_J.Fudd)\ **Post date:** [January 14, 2026, 3:17pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/12 "2026-01-14T15:17:50Z")
I hate this probability problem…mostly because Tuesday is a girl’s name. ---
**Author:** ![Hermitian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hermitian/32/470_2.png) [@Hermitian](https://boards.straightdope.com/u/Hermitian)\ **Post date:** [January 14, 2026, 4:05pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/13 "2026-01-14T16:05:09Z")
I follow the math (mostly), but I have come to the conclusion that I am not smart/intuitive enough to be able to look at the answer and say “Oh. Now I see why that makes sense.” It isn’t the first time, and won’t be the last. Thank you all. ---
**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\ **Post date:** [January 14, 2026, 4:27pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/14 "2026-01-14T16:27:51Z")
> [@Hermitian](#): > > I follow the math (mostly), but I have come to the conclusion that I am not smart/intuitive enough to be able to look at the answer and say “Oh. Now I see why that makes sense.” These probability problems always key on the wording of the original question. If that is off by even a little, the problem will make no sense. It’s all about the question the person is trying to answer, and removing any ambiguity in the language. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [January 14, 2026, 6:36pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/15 "2026-01-14T18:36:51Z")
Probability math be damned. If a couple has one child, a boy born on a Tuesday, and the couple is expecting a second child, then there is, in real world populations, about a 51 to 52% that the child will be a boy (slight real world boy predominance). A year later the probability that the second child was a boy did not change. ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [January 14, 2026, 7:39pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/16 "2026-01-14T19:39:03Z")
> [@DSeid](#): > > Probability math be damned. > > If a couple has one child, a boy born on a Tuesday, and the couple is expecting a second child, then there is, in real world populations, about a 51 to 52% that the child will be a boy (slight real world boy predominance). > > A year later the probability that the second child was a boy did not change This is wrongly structured and shows complete ignorance of the standard 2 child problem, let alone the Tuesday wrinkle. You don’t look at one child then another or specify “second child”. If you do it will be 50%. ---
**Author:** ![DSeid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dseid/32/20194_2.png) [@DSeid](https://boards.straightdope.com/u/DSeid)\ **Post date:** [January 14, 2026, 7:51pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/17 "2026-01-14T19:51:47Z")
See I don’t think it is. She has two children. She chooses one of the two kids to identify, say flipping a coin, and tells us the day of the week and gender of that child. In this case boy on a Tuesday. Whatever the gender, whatever the day of the week, the gender of the other child remains what it is, which was slightly more than 50% chance a boy in real world populations. ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [January 14, 2026, 7:55pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/18 "2026-01-14T19:55:42Z")
That’s sloppy phrasing by the OP. The better way to think of it is if for whatever reason you need a boy born on a Tuesday for a project. You ask a lady “how many children do you have?” 2 “are any of them boys” yes “were any of them born on Tuesday” yes In this case she is not volunteering any information you specifically didn’t ask for and that’s what makes the maths work. ---
**Author:** ![Gorsnak](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gorsnak/32/7587_2.png) [@Gorsnak](https://boards.straightdope.com/u/Gorsnak)\ **Post date:** [January 14, 2026, 7:57pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/19 "2026-01-14T19:57:01Z")
The standard 2 child problem messes with your intuition about probability because it messes with assumptions about how you come to know something. Let’s change the subject to a different scenario with equivalent probabilities. I’ve got a bag, into which I place two marbles. Each marble has a 50% chance of being blue, and a 50% chance of being yellow. You draw one marble from the bag and see that it is blue. What is the probability of the second marble also being blue? This is the situation you described, and the answer is that the probability is indeed 50%, because you’ve ruled out YY and YB from the list of probabilities, leaving just BB and BY. Now imagine instead that you ask me if there’s at least one blue marble in the bag. I look into the bag and answer, “Yes.” Now what is the probability of there being two blue marbles in the bag? The answer is 33%, because my answer ruled YY out from the list of possibilities, leaving BB, BY, and YB still in play. Adding in the days of the week changes the math, but not the general structure of the problem. The framing of the 2 child question frequently leaves the opening statement structured grammatically such that it’s ambiguous whether you’re in the first or second situation, and our natural inclination is to prefer the first interpretation. ---
**Author:** ![FlikTheBlue](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/fliktheblue/32/10972_2.png) [@FlikTheBlue](https://boards.straightdope.com/u/FlikTheBlue)\ **Post date:** [January 14, 2026, 8:17pm UTC](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683/20 "2026-01-14T20:17:39Z")
> [@Gorsnak](#): > > Adding in the days of the week changes the math, but not the general structure of the problem. It’s not clear to me why the extra information changes the math (that is I don’t get why the above examples use a denominator of 27 rather than 28). None the less, I assume that adding more detailed information would make the odds even closer to 50%. Say that instead of a boy born on Tuesday, we specify a boy born on Tuesday between 10 and 10:59:59 AM. Would that make the probability of a girl closer to 50% rather than 51.8%? [Next page](https://boards.straightdope.com/t/probability-problem-please-explain-it-to-me/1026683.md?page=2)