How can the “Boy or Girl” paradox be worded unambiguously?

The first time I encountered the Boy or Girl paradox I was struck by how ambiguous all of the formulations in English are. As Wikipedia notes:

Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:

• From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
• From all families with two children, one child is selected at random, and the sex of that child is specified. This would yield an answer of 1/2.

Many formulations have a less technical problem: when a parent says that they have "two children and one of them is a boy", it's nearly impossible to read that in English without thinking "therefore the other one is a girl".

Can you rephrase the paradox so that the proper randomizing procedure is specified and the Bayesian logic is the crux of the problem?

I was at a dinner party the other night and I spoke to Mr Smith. He mentioned having two kids, both with unisex names: Sam and Alex. I remember that he told a story about "my son", but I don't recall which child he was talking about, and I don't know if he has a daughter or not. What are the odds that he has two sons?

This avoids selection entirely, and avoids bringing twins into the mix. It does, however, hinge on your ability to accept Sam and Alex as being short for Samantha and Alexander, Samantha and Alexandra, Samuel and Alexander, or Samuel and Alexandra. It makes it easier to enumerate the possibilities though, since you now have names -- I got lost in the weeds for a while discussing this problem with why having a boy first and then a girl was counted separately from having a girl first and then a boy and didn't quite realize I was talking about a different version of the question at all.

I was at a dinner party the other night and I spoke to Mr Smith. He mentioned having two kids, a boy named Jim and a child named Sam whose gender I don't recall. What are the odds that he has two sons?

Which is obviously 50%.

• Excellent formulations. The only problem I can see with the 50% version is A Girl Named Jim scenario. But this is my new favourite solution. – Jon Ericson Jun 10 '14 at 19:50
• "A boy named Jim" :) – Yamikuronue Jun 10 '14 at 19:50
• A father with two boys is twice as likely to tell you a story about a son, so although you've cleared up the ambiguity as asked for, you've changed the probabilities. – greg m Jun 10 '14 at 23:19
• I know it's off topic but I just have to. – 355durch113 Jul 24 '14 at 19:06
• @gregm except that if he has 2 sons, he would probably specify which one, or at least allude to the fact that he has 2 by saying "One of my sons", therefore decreasing the odds that he has 2 by not doing this. – dberm22 Jun 8 '15 at 14:33

Bob walked into the clothing store. He looked a bit lost. "Hi, I'm Alice, can I help you?" said Alice. "Remember, you can get anything you want, at Alice's Restaurant." She looked at Bob again and swallowed her usual "excepting Alice."

Bob laughed. "I was wondering about that name, but now I get the reference. Can you help me pick out a dress for a girl? My sister Carol is staying over with her two kids, and one of them spilled chocolate milk, ruining her favorite dress. I want to surprise her with a new one. You see, I don't have any kids myself, I'm single." Bob wondered why he had emphasized that.

They talked, and looked at several dresses, but Bob just couldn't pick one. "You know, maybe she should pick one out herself, I'm sorry," he said. Now Alice didn't want to lose a customer, and she didn't want to lose Bob. "I tell you what," she said, "bring both girls over to pick out a dress and I'll give you a discount."

To make a long story short, Bob did, the girls got dresses, Bob and Alice have been married for 7 years now, and they all lived happily ever after, proving that you really can get anything you want, at Alice's Restaurant.

But what are the chances that Alice would've made a wrong guess and the whole thing would've fallen apart?

Below is my first try, but I took Jon Ericson's advice and created the above.

Alice goes into a toy store and asks the clerk, Bob, to help her pick out a toy. Alice tells Bob she needs a toy for one of the two kids of Carol, a friend of hers. Since Alice has only girls herself, she needs his help picking out a toy for a boy. Bob helps her find something and they chat a while.
As a friendly gesture, Bob offers Alice a discount if she buys the same item twice, so she can give the other kid a present as well.
What is the chance that Alice has any use for Bob's offer?

• That's pretty good. It might be a little better if Alice were to be buying cloths, since toys are more likely to be gender neutral. There's also a minor problem with the final question since the reason for the purchase could be a birthday or other occasion only applicable to one of the children. No matter what the gender, Alice would have little use for a duplicate in that situation. – Jon Ericson May 22 '14 at 15:53
• @JonEricson I fully agree, clothing might be better, especially since I don't like gendered toys. I tried to think of an occasion where both the information and the question would come naturally, much like your twins example. – SQB May 22 '14 at 18:06

One solution is to reverse the subject of puzzle. Instead of focusing on the father who is withholding an important detail, make it about the strategy of teasing out information. Here's my attempt:

Last weekend our Parents of Twins Club had a father-daughter night at the ballet. Since he wasn't invited, my twin boy stayed home with my wife. When we arrived, my daughter and I discovered our seats were isolated from the rest of the club. I didn't have a chance to see any of the other fathers or their daughters.

During intermission, I got into a conversation with another father in the washroom. I recognized him from other club gatherings, but for the life of me I couldn't recall if he had a twin son or not. Taking a chance, I asked if he'd be at an upcoming father-son event the club was sponsoring next month.

Assuming the other father has only one set of twins and that non-twin siblings are not allowed at Twins Club events, what are the odds I've just made a fool of myself?

The important bit is that it puts the puzzle into a relatable, real-life situation rather than "puzzleland". The selection method is perhaps old-fashioned (my sons would enjoy ballet too!), but clear. The other father isn't being coy about the information he holds. Instead, his mere presence at the event gives us the needed information. We'd get a very different answer if the chance encounter occurred at some location that didn't ensure he had a twin daughter.

While we are at it, ambiguity in the selection criteria is also the problem with variations of the Monty Hall problem that leave out important assumptions. The puzzle only works if the host:

2. can not reveal that information explicitly until the guess is made, and
3. always presents a puzzle.

Much the same criteria are needed to make the "boy or girl" paradox unambiguous.

• Perhaps you could make the protagonist tell that the other father has fraternal twins, possibly by the colour of his badge, because otherwise you're getting the odds of identical vs fraternal twins into the mix. – SQB May 23 '14 at 6:05

"At dinner party for adults only, you find out that Mrs. Smith has exactly two children of different ages. You also find out that she was a Cub Scout Den Mother when they were younger. What are the chances she has two boys?"

The key to this version, is that the property "at least one boy" is identified as a property of the two children as a pair, not of one specific child in the pair. If it was one specific child, no matter how or who selected him or her, the answer can only be 1/2.

I know that this can be hard to accept. So I'll demonstrate with a variation of what is currently the top answer, Yamikuronue's. Note that I'm not singling that one out for any reason other than it is at the top.

"At a dinner party the other night I spoke with Mr Smith. He mentioned having two kids. He also told a story about one of them attending a special event for children of only one gender, although he didn't mention whether it was for boys or girls. What are the chances that he has two children of the same gender?"

If Mr. Smith had mentioned that it was a boys' event, the question is the same as Yamikuronue's with some details filled in about the story. If he had mentioned that it was a girls' event, it is an equivalent question that can only have the same answer. Since every possibility produces the same answer, I don't need to know what gender the event was for in order to give that answer.

But I also don't have any information about Mr. Smith's children. With two children and no information about either child's gender, the chances that he has two of the same gender are 1/2. Yet, if the answer to Yamikuronue's question could be 1/3, we'd have a paradox since a story which provides no useful information changes the chances from 1/2 to 1/3.

This paradox has a name: Bertrand's Box Paradox. While some modern treatments use that name for the problem, Bertrand used it to show that you have to know how this type of information is obtained to change the answer. And in fact, the Two Child Problem is essentially Bertrand's Box Problem, with an added box containing one of each kind of coin/gender.

The classical boy/girl question is asked like this:

I have two kids, and one of them is a boy. What is the probability that I have two boys?

The problem, as you said, is that the statement itself is incredibly ambiguous. The interpretation that most people think of (and correctly too, I'd say, taking linguistic context into account) when they hear the boy/girl problem is as such:

I have two kids, and one of them is a boy. What is the probability that the other one is also a boy?

In this case, the answer is 1/2, because "one of them is a boy" refers to a specific boy, rather than just the fact that at least one of the two children is a boy. This is equivalent to the second randomizing procedure set out in Gardner's procedure.

To word it in a way that evokes the Bayesian inference, you need to perform gymnastics with the language of the problem that give the trick away.

I have two kids, and at least one of them is a boy, with all scenarios where this is applicable being equally likely. What is the probability that I have two boys?

It's much too dry, and the ambiguity of the language is what makes the problem seem so controversial and counter-intuitive in the first place.

• Suppose that immediately before someone asked the question, there was a PA announcement "Would somebody who has two children, exactly one of whom is a boy, please find Joe Z. and ask him [the question above]"? What would be the probability that the person approaching you had two sons? How about if the PA announcement had called for someone with two children, both boys? In the absence of information about why one was approached, the probability need not be 33% or 50%, but could just as well be 0%, 100%, or anything in-between. – supercat Jul 8 '15 at 19:29

I have beein playing around with the question and came up with the following:

At a party, you see a boy misbehaving and you try to locate the parents. You see Mrs Smith.
- Hello, Mrs Smith, do you have children?
- Yes. I have two.
- Do you have a boy?
- Yes, I do.
You point at the boy.
- (a) Yes, it is! (b) No, it is not!

With the answer (a) there is a 50% chance that Mrs Smith has a daughter. However, if it is (b), there is a 2/3 chance that she has a daughter.

• I don't see that. If there were many children at the party it is overwhelmingly likely that Mrs Smith will answer (b) regardless of the number of sons/daughters she has. – Penguino Sep 14 '14 at 21:43
• The point is that meeting Mrs Smith's son is more likely if there are 2 of them. So, if we see a boy, it is more likely she has 2. – Florian F Sep 18 '14 at 17:24

Your family's chef serves two types of pancakes; blueberry pancakes and raspberry pancakes, which he serves at random to breakfast guests.

Today, the disreputable Duke Froggington II's breakfast is a stack of two pancakes, the topmost of which is blueberry. You can't see his second pancake, as it's entirely hidden beneath the first pancake. What is the probability that both pancakes are blueberry?

(Here I'm using blueberry and raspberry pancakes instead of boys and girls, and the nefarious Duke Froggington II instead of Mr. Smith, but structurally it's exactly the same puzzle, just with the theme adjusted to remove the ambiguity of the "and one of them is a boy" statement)

It seems to me that the reason why there are no versions of the puzzle which are worded unambiguously is that the puzzle isn't really a 'puzzle' when it's worded precisely; instead, it's a simple statement of basic probability. We could change the theme again to make the same puzzle be even more simply stated as:

I flipped a fair coin two times, such that there was an equal 50% chance of either face showing after each flip. The first time, it came up heads. What is the probability that it came up heads both times?

The "boy or girl" puzzle is intentionally built around a misunderstanding that the puzzle-solver is intended to make; it stops being recognisable as a puzzle when that ambiguity is removed. And that's why you don't find online examples of it without the ambiguity.

Whether it truly works as a good puzzle even when the ambiguity is present, is left as an exercise for the reader.

Mr. Smith has two children. Mr. Smith only tells you that at least one of them is a boy, but does not tell you anything else about them. Ceteris paribus, what is the probability that both children are boys?

I have added the part in bold, and this rids the problem of ambiguity. Mr. Smith has told you that, for his family, at least one of the two children is male.

This gets you the intended answer of 1/3 for the problem by removing the possible interpretation that Mr. Smith specified the gender of one child.

• I don't know. This still sounds like "one child is selected at random" to me. The selection criteria really is the hinge this puzzle turns on. If there's any ambiguity in this already counterintuitive paradox, it's going to seem unfair. – Jon Ericson May 19 '14 at 19:12
• @Jon I've updated it to include a bit more specificity - does it read well? – Aza May 19 '14 at 19:46
• Other than the Latin phrase I had to look up? ;-) This is improved, but there's still the problem of people constructing a scenario for how Mr. Smith revealed the information. The natural assumption is that he had one of his child in mind when he mentioned the gender. What we really need is to find out that he's in the class of all parents with two children and at least one son. The problem seems sufficiently tricky on it's own without introducing the possibility that System 1 fixates on the incorrect selection criteria. – Jon Ericson May 19 '14 at 20:32

Today was "bring your kids to work"-day at the office. Alice has no kids of her own, but she saw her co-worker Bob with a lovely girl.
"That's weird," she tought, "I thought he had two kids." So she approached them, asked the girl's name (Carol), and asked Bob if she was his only child. "Nah, " said Bob, "my other kid is sick at home." Alice thought for a bit and said to Carol, "wish your brother well when you get home."

What chance did Alice have in guessing correctly?

• This is the 50% case, isn't it? – Yamikuronue Jun 10 '14 at 19:51

This question is asking for examples of how to phrase the puzzle unambiguously. So, this answer is meant to be a question. :)

All the parents in my town with exactly 2 school aged children are invited to a dinner. At this dinner, it is explained that a free scholarship to a prestigious, all-boy prep school will be given away.

People with only daughters are unfortunately not eligible for the contest, but can still get a nice meal.

It is explained that if the boy who is chosen has a brother, then BOTH children will get a scholarship!

Given that each family gets 1 entry, what are the odds that the school will have to offer a second scholarship?

• @TheSimpliFire I also flagged this as "not an answer" at first, but if you check the context of the question, it actually is an answer. The question is specifically seeking formulations of a puzzle, not answers to a puzzle. – Rand al'Thor May 22 at 11:56
• @Randal'Thor Thanks for your comment. I did indeed review too fast on that one! – TheSimpliFire May 22 at 12:03
• So did I :-( I'm leaving my comment even though yours is gone, in the hope that it'll help others to realise quicker than we did. – Rand al'Thor May 22 at 12:05
• Thanks both of you. I am going to edit it to mention that before the answer. – w3c_ee May 23 at 20:03