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In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?

I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, by Leonard Mlodinow, Vintage Books, 2008. On p.107 Mlodinow says the chances are 1 in 3.

It seems obvious to me that the chances are 1 in 2. Am I correct? Is this not exactly analogous to having a bowl with an infinite number of marbles, half black and half red? Without looking I draw out a black marble. The probability of the second marble I draw being black is 1/2.

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    $\begingroup$ In your "obvoius" case, you don't count the case where the first child is a boy but the second child is a girl. $\endgroup$ – Rawling Dec 21 '10 at 12:22
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    $\begingroup$ @Rawling Say I visit this family. I know they have 2 kids. One of them, a girl, comes into the room. The probability that the 2nd kid is also a girl is 1/2, no? Or let's say my wife and I have our first child, a girl. The probability that the next child we have will also be a girl is 1/2, no? $\endgroup$ – NotSuper Dec 21 '10 at 12:45
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    $\begingroup$ In the second case, yes; in the first case, I think the answer is also yes (given that one of the two children walk with equal probability). Take a look at en.wikipedia.org/wiki/Boy_or_Girl_paradox $\endgroup$ – Rawling Dec 21 '10 at 13:58
  • $\begingroup$ See also: math.stackexchange.com/questions/4400/… $\endgroup$ – BlueRaja - Danny Pflughoeft Dec 21 '10 at 18:08
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    $\begingroup$ So re-order the question - "in a family with two children, at least one of whom is a girl, what are the chances that the other is also a girl?" and the 1 in 3 answer might make a little more sense. $\endgroup$ – Bill Michell Sep 27 '11 at 16:01

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I think this question confuses a lot of people because there's a lack of intuitive context -- I'll try to supply that.

Suppose there is a birthday party to which all of the girls (and none of the boys) in a small town are invited. If you run into a mother who has dropped off a kid at this birthday party and who has two children, the chance that she has two girls is $1/3$. Why? $3/4$ of the mothers with two children will have a daughter at the birthday party, the ones with two girls ($1/4$ of the total mothers with two children) and the ones with one girl and one boy ($1/2$ of the total mothers with two children). Out of these $3/4$ of the mothers, $1/3$ have two girls.

On the other hand, if the birthday party is only for fifth-grade girls, you get a different answer. Assuming there are no siblings who are both in the fifth grade, the answer in this case is $1/2$. The child in fifth grade is a girl, but the other child has probability $1/2$ of being a girl. Situations of this kind arise in real life much more commonly than situations of the other kind, so the answer of $1/3$ is quite nonintuitive.

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  • $\begingroup$ Good analysis, but the OP does not include your first situation since it has extra information that all girls are at the party. That makes girls more likely to be at the party. You can't just introduce extra information like that. $\endgroup$ – Henk Jul 16 at 13:35
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In a family with 2 children there are four possibilities:

1) the first child is a boy and the second child is a boy (bb)

2) the first child is a boy and the second child is a girl (bg)

3) the first child is a girl and the second child is a boy (gb)

4) the first child is a girl and the second child is a girl (gg)

Since we are given that at least one child is a girl there are three possibilities: bg, gb, or gg. Out of those three possibilities the only one with two girls is gg. Hence the probability is $\frac{1}{3}$.

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    $\begingroup$ Your answer is correct for the OP's question, but not how YOU pose the question. Your text says "the first child is a...". That being said, if we know the first child is a girl, then out of bb, bg, gb, and gg, only gb, and gg are left available. Hence the probability is 1/2. Had you phrased it like the OP ("if we have a family with one girl (not necessarily the first child), what is the probability that both are girls?"), then your answer would be correct. $\endgroup$ – Nemi Dec 21 '10 at 18:06
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    $\begingroup$ Yes, @Nemi is correct - I think the OP has stated the question such that gb and bg are the same thing. $\endgroup$ – Maxim Zaslavsky Dec 21 '10 at 20:12
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    $\begingroup$ @Nemi You are correct that there is an important distinction between "one of the two children is a girl" and "the first child is a girl". My text says: "Since we are given that at least one child is a girl there are three possibilities: bg, gb, or gg". I did not say that we are given that the first child is a girl. The three possibilities arise from considering whether the first/second born was a girl/boy. $\endgroup$ – Travis Service Dec 21 '10 at 20:29
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    $\begingroup$ sorry, re-reading your answer now I see that you have stated it correctly. Not sure if I read it wrong earlier or you have edited it. It is most certainly correct as stated. $\endgroup$ – Nemi Dec 22 '10 at 3:54
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    $\begingroup$ While true, this doesn't explain the apparent paradox. $\endgroup$ – BlueRaja - Danny Pflughoeft Dec 23 '10 at 21:30
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I think that the reason that these puzzles are so often confusing is that they rely on the limitations of the English language rather than on any mathematical difficulties. Of course this is not unique to English, and I think you should be able to find similar similar puzzles in pretty much any natural language.

Here is an example that brings out the difficulty more clearly. First, consider the following similar puzzle:

  • A family has two children, Robin and Lindsay. Lindsay is a girl. What is the probability that both children are girls?

In elementary probability class, they teach you to answer this by making a table of the four options

  Robin    Lindsay
  B        B
* B        G
  G        B
* G        G

The starred rows are the ones where Lindsay is a girl, and we compute from them that the chance that both children are girls is 1/2.

Now consider this puzzle

  • A family has two children, Robin and Lindsay. At least one of them is a girl. What is the probability that both children are girls?

The elementary probability method gives the following table, and a probability of 1/3. The difference is that we gain one more row, compared to the previous puzzle.

  Robin    Lindsay
  B        B
* B        G
* G        B
* G        G

After looking at these, you can see that the difficulty of the original puzzle comes because, in English, "one of the children" can mean several different things:

  • "one" can mean "a particular one". If you read the original puzzle like this, it becomes analogous to the first puzzle I wrote, and the answer will be 1/2.

  • "one" can mean "at least one". If you read the original puzzle like this, it become analogous to the second puzzle I wrote, and the answer is 1/3.

  • "one" can mean "exactly one". If you read the original puzzle like this, the answer is 0.

There is a common convention in mathematics that "one" usually means "at least one". For example, this is the sense intended in the following sentence, which is a typical example of mathematical English: "if a natural number $n$ is a multiple of a prime number $p$, and $n = ab$, then one of $a$ and $b$ is divisible by $p$." We would not read this as saying that exactly one of $a$ and $b$ is divisible by $p$.

I don't believe this convention is very common in non-mathematical English. If I say, "one of my children is a girl", in normal English this means that the other is a boy. Similar discrepancies between mathematical and non-mathematical English come up with our use of the word "or" and our use of the phrase "if/then". When we teach mathematics, we have to spend time explaining this mathematical argot to students, so they can use the same English conventions that we do.

Probability puzzles like the one you're asking about rely on these differences of English meaning, rather than on any logical or mathematical problem. In that sense, they aren't really puzzles, they're just tricks.

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  • $\begingroup$ I now, at last, agree that the correct answer is 1/3. (See my question "2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes?" at goo.gl/yyOlK and my comment beginning with "@all: It suddenly hit me that I didn't need that truthful observer.", about half way down the page, under Willie Wong's answer.) $\endgroup$ – NotSuper Dec 27 '10 at 16:25
  • $\begingroup$ (continued) I'd award you the 100 bounty points for the clarity of your explanation of the mathematics, if you hadn't added your idea that "Probability puzzles like the one you're asking about rely on these differences of English meaning, rather than on any logical or mathematical problem. In that sense, they aren't really puzzles, they're just tricks." In my case I wasn't tricked -- I immediately assumed that "one of my children is a girl" meant that "at least one of my children is a girl". $\endgroup$ – NotSuper Dec 27 '10 at 16:30
  • $\begingroup$ (continued) I assumed this because to interpret "one of my children is a girl" as meaning "exactly one of my children is a girl" would destroy the problem/puzzle in that the answer would be too obviously zero. Also, because I had first seen the question in a book about randomness and probability, I was already in math English mode. $\endgroup$ – NotSuper Dec 27 '10 at 16:30
  • $\begingroup$ @NotSuper: Thanks for your kind comments. $\endgroup$ – Carl Mummert Dec 28 '10 at 3:07
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    $\begingroup$ Should be the accepted answer. $\endgroup$ – 355durch113 Oct 17 '15 at 3:58
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Please ignore the old comments - I have radically changed my answer. It's long, but bear with me - after reading this answer, you should feel intuitively-comfortable with almost all other probabilistic fallacies, not just this one.

The first and most important thing is to define what we mean by "probability." Let's define it to mean "the expected percentage of positive outcomes when repeating an observation over a large sample" (Go ahead, read that again, more slowly). Also, let's call the method we use to chose this large sample the "model."

This may sound trivial, but it has some important implications. For example, what is the probability you will die before age 40? According to our definition, this question has no meaning: we can't observe you multiple times before age forty and record how many times you die. Instead, we observe other people below age 40, and record how many of them die.

So let's say we observe all other people on earth below age 40 (our model), and find that 1/2 of them die before hitting the big four-oh. Does this mean the probability of you dying before 40 is 50%? Well, according to this model, it does! However, this is hardly a fair model. Perhaps you live in a first-world country - now when we revise our model to include only people below 40 in first-world countries, your chances become a much-less-grim 1/10. But you are also not a smoker, and don't live in the city, and ride your bike on Sundays with your wife and crazy mother-in-law, which makes your chances 123/4567! That's much better... however, our model still doesn't take into account that you are also an avid skydiver ;)
I am pulling these numbers out of nowhere, they are not real statistics.

So the point is, asking for a "probability" only makes sense in the context of a certain model - a way of repeating our observation many times. Without that, asking for a probability is meaningless.


Now, back to the original question. Before we can assign a probability, we must choose a model; how are we choosing the families to sample from? I see two obvious choices, which will lead to different answers:

  1. Consider only families which have two children, one of whom is a girl, and choose one randomly.
  2. Consider only girls who have exactly one sibling, and choose one randomly.

Do you see the difference? In the first case, every family has the same probability of being chosen. However, in the second case, the families with two girls are more likely to be chosen than families with only one girl, because every girl has an equal chance of being chosen: the two-girl families have doubled their odds by having two girls. If children were raffle tickets, they would have bought two tickets while the one-girl families bought only one.

Thus, we should expect the probabilities in these two cases to be different. Let's calculate them more rigorously (writing BG to mean "boy was born, then girl):

  1. There are three equally probable family-types: BG, GB, and GG (BB was removed from consideration, because they have no girls). Since only one of the three has two girls, our chances of having two girls are 1/3.
  2. We have the same possibilities as above, but now GG is twice as likely as BG or GB. Thus, the probabilities are GG: 2/4, GB: 1/4, and BG: 1/4, meaning the probability of a girl-sibling is 2/4 = 1/2 (alternatively, we could have noted that there are only two equally-probable possibilities for the sibling: boy or girl).

Here lies the fallacy: the model our intuition assumes is the second one, but the way the problem is worded strongly implies the first one. When we think in terms of "randomly choosing a family (over a large number of families)," our intuition meshes perfectly with the result.


Let's take a look at another similar problem

In a family of two children, where the oldest child is a girl, what is the probability they are both girls?

Once again, I can see two different, plausible models for observing our random sample:

  1. Consider only families with two children, the oldest of whom is a girl, and choose one randomly.
  2. Consider only girls who have exactly one (younger) sibling, and choose one at random.

So once again, the question strongly implies the first model, though arguments for either could be made. However, when we actually calculate the probability...

  1. Same as before, but we've also eliminated BG, where the first child was a boy. This leaves only two equally-probable possibilities, GB and GG. Thus, the chances are 1/2.
  2. We've also eliminated BG from this case, leaving GB and GG. However, unlike the original question, GG is no longer twice as likely, since the younger child can no longer be the one who was randomly chosen. Thus, GB and GG are equally likely, and we again have a probability of 1/2 (alternatively, we could have noted that there are only two equally-probable possibilities for the sibling: boy or girl).

...we find that the choice between these two models doesn't matter, because in this case both have the same probability! Among two-child families with an older daughter, it doesn't matter if we randomly choose the family or the older-daughter, because in both cases there is only one of each per family.


Hopefully that all made sense. For bonus points, try applying this reasoning to the Monty Hall problem. What is our model - how are we making repeated observations? Why does it clash with our intuition?

For even more bonus points, try to figure out the following question; the math isn't too difficult, but it took me a long while to figure out why, intuitively, the answer should be correct:

In a family of two children, one of whom is a girl named Florida, what are the chances of two girls?

(If you have troubles, post a question and leave a link to it in the comments, and I'll try to answer it there as best I can :) )

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    $\begingroup$ I don't completely agree with this analysis. Mathematically, there is no problem at all. Once we list the possible events and the favorable events, the probability is just the corresponding fraction. So the difficulty in these puzzles is just that English hasn't evolved a clear way to describe a probability space. There are two reasonable probability spaces here, and usual English is not clear about which of them is intended. But regardless what anyone knows about the children, I can still compute the probabilities in both these spaces, to get 1/3 and 1/2, respectively. $\endgroup$ – Carl Mummert Dec 23 '10 at 15:06
  • $\begingroup$ @Carl: Then you must believe that the Monty Hall problem is ambiguous in exactly the same way, no? I'm afraid I have to disagree with that. $\endgroup$ – BlueRaja - Danny Pflughoeft Dec 23 '10 at 17:15
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    $\begingroup$ I do feel that the difficulty in the Monty Hall problem is also not "mathematical", in the sense that as soon as any probability model has been rigorously stated then the probabilities can be calculated using basic methods. The arguments about the Monty Hall problem stem from a lack of rigour in the statement, which I think is often intentional in order to make an otherwise easy probability exercise into a "puzzle". $\endgroup$ – Carl Mummert Dec 23 '10 at 17:44
  • $\begingroup$ @Carl: In both cases, it's not that it's ambiguous or not mathematically-defined, it's just counter-intuitive. $\endgroup$ – BlueRaja - Danny Pflughoeft Dec 23 '10 at 21:29
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    $\begingroup$ The two puzzles are similar, and you've made a nice comparison. They each have two simple probability models, and it's clear which gives the clever, counter-intuitive solution. But if you admit any more complexity, there's a lot of ambiguity in both problems. For example, given a room full of people convinced that switching doors is the right strategy, an adversarial Monty Hall can make sure no one wins by offering the switch only to those who first choose the car. In real life, I would assume ratings and TV gimmicks affect what choices I'm given, and I have no probability model for that! $\endgroup$ – Jonas Kibelbek Dec 23 '10 at 23:04
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Here's some idea.

Suppose you sample $4n$ families with two children, where $n$ is a very large integer. Then, with a very high probability, about $n$ of them have two boys, about $n$ have two girls, and about $2n$ have one girl and one boy. Since you ignore the families with two boys, the desired probability is given by $n/(3n) = 1/3$.

EDIT: The above solution corresponds to the following situation. You visit a large number of families. In each family, you check if there are two kids. If no, you ignore this family. If yes, you check if one of the kids is a girl. If no, you ignore this family. If yes, you check if both kids are girls.

EDIT: Among all the OP's examples, only one is relevant here. Namely, "I visit this family. I know they have 2 kids. One of them, a girl, comes into the room. The probability that the 2nd kid is also a girl is 1/2, no?". Well, in this situation the probability is indeed $1/2$, assuming the following interpretation. You visit a large number of families. In each family, you check if there are two kids. If no, you ignore this family. If yes, you wait until one of the kids comes into the room. If it is a boy, you ignore this family; otherwise you check if both kids are girls. It is readily understood that this leads to a probability of $1/2$.

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    $\begingroup$ Of course, a lot depends on how we interpret the question. $\endgroup$ – Shai Covo Dec 21 '10 at 12:52
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    $\begingroup$ I'm hoping that someone will show me why my 3 other examples are different from the case in question. a) the bowl of marbles b) my visit to this family c) my first child is a girl. The second? $\endgroup$ – NotSuper Dec 21 '10 at 12:59
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    $\begingroup$ @Shai But why? Why is (b) different from (a) and (c)? $\endgroup$ – NotSuper Dec 21 '10 at 13:09
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    $\begingroup$ @NotSuper: It all depends a lot on the precise formulation, as Shai already pointed out. Let's say you visit a friend who flips a coin twice. Given that at least one of the outcomes is Heads, the probability of two Heads is indeed 1/3. $\endgroup$ – Hendrik Vogt Dec 21 '10 at 13:35
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    $\begingroup$ @NotSuper: Shai is right indeed: You can see only one of them, say the one closer to you, and it's a head. So you know more than in the experiment I formulated. In that experiment, I assumed that you know that at least one of the two outcomes is Heads, but if it's only one head, then you don't know which coin it is! In your experiment you know that that coin lying closer to you is a head, so it's a different experiment. $\endgroup$ – Hendrik Vogt Dec 21 '10 at 15:38
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One possible solution for this problem might be though using conditional probability.

Let $B_i$ and $G_i$ represent $i$th boy and $i$th girl respectively. Here the sample space could be expressed as $S =$ {$(B_1,B_2),(B_1,G_2),(G_1,B_2),(G_1,G_2)$}

Let $A$ = An event of having both girl chil. $B$ = An event of having at-least one girl child.

$A = ${$(G_1,G_2)$} and $B =${$(B_1,G_2),(G_1,B_2),(G_1,G_2)$}

So what you are asking is $$P(A / B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} $$

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It really depends on what knowledge you have to restrict the probability space. (The original question is a little ambiguous, but I agree that 1/3 is the best answer.)

The marble example and the example of the firstborn daughter are the same situation-- the second trial is independent, and the odds are 1/2. In these cases you know the first trial is independent, and you can ignore it.

The original question intends you to imagine a randomly chosen 2 child family from among all families with at least 1 daughter. For example, suppose there is a social science study on 2 child families with at least 1 daughter-- in this situation, about 1/3 of the families will be daughter-daughter, 1/3 will be daughter-son, and 1/3 will be son-daughter. You have to consider the full probability space of two trials (d-d,d-s,s-d,s-s) and eliminate the s-s possibility. This leaves a 1/3 chance of 2 daughters.

For other variations on this question, you have to decide the appropriate probability space before answering. (One common, confusing variation is to suppose the family has two children, one of which is a son born on a Tuesday-- what are the odds of two sons?)

In your example where the first child in the room is a daughter, if we suppose that the order the children enter has nothing to do with sex, then the probability the other child is a daughter is 1/2. However, maybe the family encourages their daughters to play inside and their sons outside-- then you could assume the other child is a son, since he hasn't come in yet. Maybe the culture has a tradition that the oldest daughter in the household (if there is a daughter) offers guests drinks-- then you know the family has at least one daughter, and the odds that they have another daughter is 1/3. Once you've chosen your assumptions, you can define the probability space and compute the answer.

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    $\begingroup$ @Jonas It isn't at all clear that "The original question intends you to imagine a randomly chosen 2 child family from among all families with at least 1 daughter." Instead I imagined I hadn't seen my nephew in a decade, and we hadn't kept in touch. But I had heard that he had 2 kids and one was a girl. I was going to visit him soon and was wondering about the other child. Girl? or boy? I figured the chances of a girl was 1 in 2, and I still do. I found the Wikipedia article linked to by Rawling enlightening. $\endgroup$ – NotSuper Dec 21 '10 at 15:15
  • $\begingroup$ @NotSuper: Yes, I agree it's ambiguous. It is clear that I gave Mlodinow's intended interpretation only because he gave the answer 1/3. But your nephew scenario is ambiguous as well-- how did you hear that one child is a girl? You could learn that among their children is at least one daughter (maybe they send an old dollhouse to your grandchildren), which gives 1/3. Or, you could specify one of their children (oldest, first to answer the phone, etc., as long as it's independent of sex) and learn she is a girl, which gives 1/2. $\endgroup$ – Jonas Kibelbek Dec 21 '10 at 15:42
  • $\begingroup$ These correspond to the two probability spaces described in my answer and in the Wikipedia article. When you hear that one child is a girl, can you cross off son-son and son-daughter, or can you just cross off son-son? (With other information and messier probability models, we could get other answers than 1/2 and 1/3 as well. But these are the two most sensible models to apply.) $\endgroup$ – Jonas Kibelbek Dec 21 '10 at 15:46
  • $\begingroup$ @Jonas "How did you hear that one child is a girl?" I wrote "I had heard that he had 2 kids and one was a girl." I think that implies that someone told me that (or possibly wrote that to me in a letter or email), and that's all I know about my nephew's children. Isn't that the standard interpretation? $\endgroup$ – NotSuper Dec 24 '10 at 13:34
  • $\begingroup$ @NotSuper: To quote from Carl's answer, "one child" can be interpreted several ways in English: "one particular child," "at least one child," or "exactly one child." If the letter or e-mail gives you enough context to pick one interpretation, then it's easy to compute the probability. Your intuitive answer of 1/2 is a correct answer, provided you recognize you are using the "one particular child" interpretation. But, since this is a puzzle or riddle, the smallest assumption ("at least one child"), which gives the counter-intuitive answer, is the riddler's preferred answer. Do you agree? $\endgroup$ – Jonas Kibelbek Dec 24 '10 at 16:34
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In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?

0%. You already said that ONE is a girl. The other must be a boy.

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    $\begingroup$ If a family has two children and both are girls, it is not incorrect to say "one of the children is a girl." "One" doesn't mean "exactly one." $\endgroup$ – manthanomen May 6 '13 at 23:11
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    $\begingroup$ Does one mean at least one, exactly one, at most one, a specific one? I think Judie's point was that the parameters of the problem are not clearly stated -- you have to make an assumption about the intent. The given answers generally assume that in this case one was intended to mean one or two, but maybe we don't know that from the question by itself unless we recognize it due to prior exposure to this type of problem. $\endgroup$ – nobar May 11 '13 at 22:34
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This is really a continuation of BlueRaja's answer, where I'm changing the question slightly to give another illustration of the underlying point. Consider the following two procedures:

Procedure A: I flip two coins, a nickel and a penny. After the flip, I pick one at random, and report its value to you.

Procedure B: I flip two coins, a nickel and a penny. If exactly one lands heads, I report that one's value to you, i.e. I say either "the nickel landed heads" or "the penny landed heads" as appropriate. If both land heads, I report on one of them chosen at random. If neither lands heads, I say "both landed tails."

Now, if I follow Procedure A and say "the nickel landed heads", then the probability that the penny landed heads is 50/50. But if I say the very same thing after following Procedure B, the probability that the penny landed heads is 1 in 3.

Now take the following problem: I flip two coins and say to you "the nickel landed heads". What is the probability that the penny landed heads? Clearly there's no way to answer without knowing or assuming something about the procedure I'm following.

(All this having been said, I agree that the best answer to the original question is 1/3.)

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If you are given that the first child is a girl or that the second child is a girl, then the obvious answer of 50% is correct.

In this case we are given that one or both of the children is a girl, but not given any information about which one(s) -- so a simple evaluation of the 4 possible scenarios (for a two-child family) reveals the answer of 33%.

The confusion stems from the slightly ambiguous language used. The phrase one of the children was used to mean one or both of the children. Based on that language, our brains trick us into thinking that a specific one of the children is given and therefore we can evaluate the other child independently.

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  • $\begingroup$ A link to this Wikipedia article was given in the main comments, but I wanted to point out that the article addresses the psychology of this problem, in an attempt to expose the seeming "paradox" of common intuition. $\endgroup$ – nobar May 13 '13 at 2:20
  • $\begingroup$ youtu.be/be2wuOaglFY?t=37m0s "This problem is famous for being ill-posed." $\endgroup$ – nobar Mar 3 '17 at 1:43
  • $\begingroup$ There is some ambiguity about whether I should have said "4 possible scenarios" or "3 possible scenarios". $\endgroup$ – nobar Mar 3 '17 at 1:46
  • $\begingroup$ As noted elsewhere, this ambiguities of this problem are reminiscent of the Monty Hall problem. In both cases, the primary challenge is fully understanding the parameters. $\endgroup$ – nobar Oct 18 at 14:34
  • $\begingroup$ BTW: The evidence being presented in stages as it is, and ignoring the ambiguities, why aren't more people answering this using Bayes' theorem? $\endgroup$ – nobar Oct 18 at 14:42
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We can solve this from first principles (maybe they're technically "second" principles, who knows).

Define:

  1. $p_G$ be the probability that a child is a girl
  2. $p_B$ be the probability that a child is a boy
  3. $N_G$ be the number of girls in a family
  4. $N_B$ be the number of boys in a family
  5. $N_T = N_G + N_B$

We want to find: $\mathbb{P}(N_G=2 | N_G \geq 1 , N_T=2)$

$$\mathbb{P}(N_G=2 | N_G \geq 1 , N_T=2) = \frac{\mathbb{P}(N_G=2 , N_G \geq 1 , N_T=2)}{\mathbb{P}(N_G \geq 1 , N_T=2)}$$

$$=\frac{\mathbb{P}(N_G=2 , N_T=2)}{\mathbb{P}(N_G \geq 1 , N_T=2)}$$

$$=\frac{p_G^2}{p_G^2 + 2p_Gp_B}$$

If we take $p_G = p_B = \frac{1}{2}$ then we get the book's answer of $\frac{1\over4}{\frac{1}{4}+\frac{1}{2}}$

$$=\frac{1\over4}{3\over4} = \frac{1}{3}$$

NOTE: In the above, the second factor in the denominator is derived since the order in which the children are born does matter.

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  • $\begingroup$ The question does not say "$N_G \geq 1$", though, it just says "one of the children is a girl". Once we have interpreted this to mean $N_g \geq 1$, we have already solved the problem, as sketched here. The issue is not whether $P(N_G = 2| N_G \geq 1, N_T = 2) = 1/3$ - the issue is what probability the English is asking for in the first place. $\endgroup$ – Carl Mummert Aug 21 '18 at 20:51
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There are at least three interpretations of the statement that "one child is a girl". Two of them lead to the 1/2 and 1/3 answers which have been discussed at length, but I found only one brief answer covering the third interpretation (though I did only skim).

The meaning of the statement, and consequently the correct answer, can be altered by considering it to answer different questions.

  1. "How many of your children are girls?" "One of my children is a girl."
  2. "Is one of your children a girl?" "Yes, one of my children is a girl."
  3. "Is the child in the stroller a boy or a girl?" "This one is a girl."

Suppose you've just had the first discussion; what do you think the other child will be in that case? There is no chance the other is a girl.

The second question would prompt most people to search their set of children for a girl and answer "yes" if they can find one or "no" otherwise. Consequently, out of the 3/4 of two-child parents who can answer yes, about 1/3 of those will have another girl.

However it's conceivable that some kind of contrarian may resolve "one of your children" even before they have considered the rest of the question, choosing one at random regardless of its gender, and answering the rest of the question with respect to that random choice. This renders the situation equivalent to the third case.

In the third case, obviously, the gender of the other child is independent of all the information we have. So it's a 1/2 chance.

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  • $\begingroup$ This is entirely incorrect. You are misunderstanding what "one of" means. $\endgroup$ – The Count Feb 22 '17 at 19:00
  • $\begingroup$ @TheCount, who's the authority on the meaning of the words, here? $\endgroup$ – sh1 Feb 22 '17 at 19:07
  • $\begingroup$ Fair point, but I would argue that the term "one of them" does not imply the exclusion of "both of them" as in the famous and idiotic puzzle "I have two coins that add to 30 cents and one is not a nickel". The question was clearly not intended to be taken that way. $\endgroup$ – The Count Feb 22 '17 at 19:08
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    $\begingroup$ @TheCount, the point I'm trying to make is that the confusion between 1/2 and 1/3 comes about because we don't exactly know what we've been told. There's a third interpretation; not necessarily more correct, but also not invalid. It's a more "it couldn't possibly mean that!" case which is meant to highlight that the question is not well framed. $\endgroup$ – sh1 Feb 22 '17 at 19:13
  • $\begingroup$ I see. Well, you make a fair point, but I suppose my confusion comes about from the fact that your answer seemingly declares your interpretation to be the correct one, instead of explaining it to me as you just did. I think including that info in the answer itself would be helpful. $\endgroup$ – The Count Feb 22 '17 at 19:14
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Let $X$ be the event "one of the children is a girl" Let $BG$ refer to the event the family has one boy and one girl (I'm not distinguishing $BG$ and $GB$), and so forth. Everything is conditioned on the family having two children.

$$ P(GG | X) = \frac{P(GG \wedge X)}{P(X)} = \frac{P(X|GG) P(GG)}{P(X|GG) P(GG) + P(X|BG)P(BG) + P(X|BB)P(BB)} = \frac{1 \cdot \frac{1}{4}}{ 1 \cdot \frac{1}{4} + P(X|BG) \cdot \frac{1}{2} + 0 \cdot \frac{1}{4}} = \frac{1}{1 + 2 \cdot P(X|BG)} $$

(note that I am assuming this isn't taken as a "trick" question, so that $P(X|GG) = 1$ is meant, rather than $P(X|GG)=0$)

So everything boils down to the probability of "one of the children is a girl" given that the family has a boy and a girl.

Now, what does the phrase "one of the children is a girl" mean? There are two typical options

  • This phrase simply means that this is not an all-boy family. In this case, $P(X|BG) = 1$, and $P(GG|X) = 1/3$.
  • The phrase means that a particular child was chosen and it could just has easily have been the boy, so that $P(X|BG) = 1/2$, and $P(GG|X) = 1/2$.

In my opinion, an English speaker would almost always interpret the phrase the first way rather than the second way.

However, there is a nasty linguistic trap: even meaning it the first way, we may often pick a girl to be the referent of "one" (e.g. so that we can speak of the "other" child). And since we've picked a girl, we get stuck in the second interpretation. This is exacerbated by the fact the second interpretation has a very appealing shortcut for reasoning out the solution.

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I feel like people are always left with a feeling of "sure, I see how you can obtain that result but why isn't my interpretation correct?" Here's my latest attempt to address this issue.

https://math.stackexchange.com/a/1801261/342573.

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You have to read carefully. Here, the probability of a girl means the probability of at least one girl. You have $$P(\hbox{one child is a girl}) = 1 - P(\hbox{no child is a girl}) = 1 - P(\hbox{two boys}) = \frac{1}{4}. $$ Now you are calculating the conditional probabilty $$P(\hbox{two girls}| \hbox{at least 1 girl} = {P(\hbox{two girls})\over P(\hbox{at least one girl})} = {1\over 3}.$$

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You reach into a bag and pull two marbles. You look at the first and see that it is black. There is a 1/2 chance that the other is red.

You reach into a bag and pull two marbles. Your friend tells you whether or not you have 2 red marbles. He says you don't have 2 red marbles. The chance you have 2 black marbles isn't 1/2 or 1/4, it's 1/3.

The assertion that at least one child is a girl is equivalent to saying that you don't have 2 boys. It is analogous to the second case above, not the first.

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I am late to the party and there are already several excellent answers to this question, but I think that it is worth adding an answer in order to introduce the idea of probability trees, which are a nice way of visualizing these kinds of conditional probabilities.

enter image description here

In this particular tree, every branch is equiprobable, and a terminal node represents one of the many possible events. From the probability tree, it can be observed that if we know absolutely nothing about a family other than the fact that there are two children, then there are four equiprobable outcomes.

However, if it is known that at least one of the children is a girl, then the event represented by the bottom branch can't happen. Of the remaining three equiprobable events, only one corresponds to the event that there are two girls. Therefore if it is known that a family has two children and it is further known that one of those children is a girl, then the probability that the family has two girls is $1/3$.

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I think that the answer is going to be 1/3

As the ways in which your neighbor can have children are as follow:

(where B= Boy; G= Girl)

BB, GG, BG, GB where BB is not possible as one of them is a girl. So the probability of the other child being a G is GG out of the sample space= {GG, BG, GB}

I know some of you might argue that BG and GB are similar things but we don’t know whether the Sister is the elder one or the smaller one. So, it’s gonna work like that.

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Here is a more "straightforward" answer. The sample space is $\Omega=\{gg,gb,bg,bb\}$ for the first and 2nd child where g is a girl, b is a boy. Each element is equally likely to happen with probability of 1/4 before you seeing the children of this family.

The problem asks the probability of having both girls if you were told one of the children is a girl. The trick of this problem is that you still do not know this girl is from the first child or the 2nd child. Thus, this shows us a new sample space $\Omega'=\{gg,gb,bg\}$. Then the probability of having both girls is just 1/3.

More interesting thing is to study where 1/2 comes from. I present a scenario where the answer is 1/2. If you were told that the first child is a girl, the sample space becomes $\Omega'=\{gg,gb\}$. In this case, the probability of having both girls is 1/2.

Of course, formal probability analysis and Monte Carlo simulation will give a more rigorous answer. But I hope the above shows you some good intuition about this classic problem. Best!

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There are many ways to explain this, but let me share with you how I do it in my class. I set $q=P(\mbox{You say that he has at least one daughter} \mid \mbox{He has $\{B,G\}$})$.

If you sample at family level, then you ask the father if he has at least a daughter or you check official records. In that case $q=1$.

If you sample at child level, by for instance meeting the father in the company of one child or otherwise happening to find out about one child, then $q=1/2$. In this case you will half the time miss this family because you happen to see the son first.

Using Bayes' formula, we get $$ P(\{B,G\}) \mid \mbox{You say that he has at least one daughter}) = \frac{P(\mbox{You say that he has at least one daughter} \mid \{B,G\}) P(\{B,G\})} {P(\mbox{You say that he has at least one daughter})}. $$

Since $$P(\mbox{You say that he has at least one daughter}) = P(\mbox{You say that he has at least one daughter} \mid \{B,G\}) P(\{B,G\}) + P(\mbox{You say that he has at least one daughter} \mid \{G, G\}) P(\{G, G\}), $$

we get $$ P(\{B,G\} \mid \mbox{You say that he has at least one daughter}) = (q \times 1/2)/(q \times 1/2 + 1 \times 1/4) = 2q/(1 + 2q), $$

and it follows that $$ P(\{G, G\} \mid \mbox{You say that he has at least one daughter})=1/(1 + 2q). $$

If $q=1/2$, we get $P=1/2$, and $q=1$ gives $P=1/3$.

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  • $\begingroup$ The issue with the question is that it does not say "at least one daughter", and instead says "one of the children is a girl". Once we re-interpret the problem to say 'at least one is a girl' we have already solved it. $\endgroup$ – Carl Mummert Aug 21 '18 at 20:48

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