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# One Girl - One Boy

## Question

Ok, so Teanchi and Beanchi are a married couple (dont ask me whose he and whose she)!

They have two kids, one of them is a girl, what is the probability that the other kid is also a girl.

Assume safely that the porbability of each gender is 1/2.

Ofcourse its not 1/2 else would make it a lousy puzzle...

Ans: 1/3

This is a famous question in understanding conditional probability, which simply means that given some information you might be able to get a better estimate.

The following are possible combinations of two children that form a sample space in any earthly family:

Girl - Girl

Girl - Boy

Boy - Girl

Boy - Boy

Since we know one of the children is a girl, we will drop the Boy-Boy possibility from the sample space.

This leaves only three possibilities, one of which is two girls. Hence the probability is 1/3

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## Recommended Posts

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To Bonanova, Scraff, Tearz et al.

Let us consider the ambiguity of the question.

If we are asked to consider both children at the same time then clearly the answer is 1/3

If we are asked to consider the probability of two girls prior to any birth then clearly the odds are 1/3

The problem states "They have two kids, one of them is a girl, what is the probability that the other kid is also a girl?".

Now, do we take each child as an indpendent event or do we look at this as a family of 2 girls? If we take the family view then the answer is 1/3 because we are combining 2 events (births). This is where the head/tail coin toss analogy works rather well.

However, if we know there is already a girl we can infer the second child's sex is independent from the first because it says "the other kid". This is to say it would be incorrect to combine the two occurences. If we combine the events we would conclude the answer to be 1/3, but due to the ambiguity of the question it would not be incorrect to give the answer as 1/2. In this example we can only give our "best" answer based on the information we have.

If the question read: " A family, the Smiths, is chosen at random from all 2 child families that contain at least one girl. What is the probability that they have 2 girls?", then the answer is 1/3 because we are making our judgement on the family as a whole and no child is nominated.

In our problem: "They have two kids, one of them is a girl, what is the probability that the other kid is also a girl" we are only being asked to suggest the sex of the other child, not in the context of the family as a whole. This is a very important distinction and boils down to semantic pedantry I know, but to dismiss the '1/2' answer is incorrect. The wording "the other kid" is extremely important since we are not being asked to evaluate them together. We are merely being asked to evaluate the probability of the sex of "the other kid". It has been nominated, therefore we are not required to look at the problem as a combination of 2 births but one independently. In this instance the "best" answer would be 1/2.

Try this: "A child from the Jones family is selected at random, and that child is a girl.” Here, the probability that the Joneses are a two-girl family is 1/2. Again, a child has been specifically nominated.

I believe the wording in the OP is somewhere between the Smiths and Jones examples that I give above and therefore a conclusive answer is not possible. It is how we, as individuals, interpret the ambiguity and then make assumptions based on our conclusions.

You've all been beating up the 1/2 camp for some time now and rightly so given the spurious nature of some of the comments. However, I would ask you to leave your 1/3 bias aside for minute and consider that the answer may not be as clear-cut as you all suggest.

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You are ignoring 1 fact that is there has to be a girl and the girl isn't specified where it is. By your calculation it would be:

Girl-Girl : 1/2 * 1/2 = 1/4 note that 1/2 is for the girl equal likely to being the 1st or second place.

boy-girl : 1/2 * 1/2 = 1/4

girl-boy : 1/2 * 1/2 = 1/4

3* 1/4 = 3/4 does 3/4 = 1?

so that adds up to 3/4. Where is the missing 1/4?

Simple the missin g 1/4 is at girl girl. You are ignoring the question and you are supposing that the girl could be the older sibling and the younger sibling at the sametime. No, only one baby has to be a girl not the other, so 1 is 100% and the other is 50% what you are doing now is supposeing that both baby are 50% for the girl-girl situations. Not to metion Your answer doesn't add up mathematically and is therefore incorrect.

Like i posted before, if you were actually reading. This question is not the same as a normal coin flipping question.

By eye, it is correct that there are 3 possible outcome, but the fact that girl-girl can appear 2 times was not metioned. Yet you continue to ignore my comment and posting the same thing for 10 posts. You should get yoiur facts straight.

noobsauce

This is utter rubbish. I believe that 1/2 has some merit as I explain in my previous post, but your reasoning is seriously flawed. Sorry

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noobsauce

This is utter rubbish. I believe that 1/2 has some merit as I explain in my previous post, but your reasoning is seriously flawed. Sorry

Hey GTB, we don't quarrel here. If you don't like somebody's answer just tell him/her what is wrong OR you may not post any reply. Saying that an answer is just "utter rubbish" does not have any merit in itself.

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GazzaTheBook and noobsauce

We know there are 2 kids and one of them is a girl. So all possible cases 3; G-G, G-B, B-G. These 3 cases are equally likely, right?

Number of events favourable to the outcome is 1, G-G, right? So the probability is 1/3.

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GazzaTheBook and noobsauce

We know there are 2 kids and one of them is a girl. So all possible cases 3; G-G, G-B, B-G. These 3 cases are equally likely, right?

Number of events favourable to the outcome is 1, G-G, right? So the probability is 1/3.

Dd/noobsauce

Apologies for the "utter rubbish" comment.

I don't disagree wtih your rationale above. It is perfectly plausible and technically correct given your question. But that is not the question in the problem.

The question is: They have two kids, one of them is a girl, what is the probability that the other kid is also a girl?".

We are asked to evaluate whether "the other kid" is a girl. In this instance I believe it is incorrect to apply your reasoning because you are analysing the probability outcomes of 2 births. The question doesn't ask us to do that, it merely asks us to work out the probability of a nominated child (the other kid) being a girl.

The "best" answer we can give is 1/2 as I have demonstrated in my previous post.

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Dear GTB,

It seems to be a problem of interpretation. Although the interpretation was pretty simple to me. Anyway what is your answer had the question been 'what is the probability that the family have 2 girls?'

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Dear GTB,

It seems to be a problem of interpretation. Although the interpretation was pretty simple to me. Anyway what is your answer had the question been 'what is the probability that the family have 2 girls?'

Dd

Then the answer is 1/3 as I have previously demonstrated in my example from a previous post:

If the question read: " A family, the Smiths, is chosen at random from all 2 child families that contain at least one girl. What is the probability that they have 2 girls?", then the answer is 1/3 because we are making our judgement on the family as a whole and no child is nominated.

As I have previously stated, the question asks us to name the sex of a particular child, namely "the other kid".

In this example the "best" answer we can give is 1/2.

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noobsauce, I asked you to respond to posts and you're ignoring me. Not the way to behave on a message board.

You are ignoring 1 fact that is there has to be a girl and the girl isn't specified where it is.

No, we haven't. neither have any of the links you were given. Because the girl has not been specified, the following are the possibilities:

GG

GB

BG

By your calculation it would be:

Girl-Girl : 1/2 * 1/2 = 1/4 note that 1/2 is for the girl equal likely to being the 1st or second place.

boy-girl : 1/2 * 1/2 = 1/4

girl-boy : 1/2 * 1/2 = 1/4

3* 1/4 = 3/4 does 3/4 = 1?

That makes no sense. There's no reason to multiply. In all families with two children, GG occurs 1/4 of the time. If at least one is a girl, it occurs 1/3 of the time.

You are ignoring the question and you are supposing that the girl could be the older sibling and the younger sibling at the sametime.

No, we are supposing the girl could be either sibling.

Like i posted before, if you were actually reading. This question is not the same as a normal coin flipping question.

By eye, it is correct that there are 3 possible outcome, but the fact that girl-girl can appear 2 times was not metioned.

Again, that makes no sense. How can it be correct by eye, but "the fact that girl-girl can appear 2 times was not mentioned"? I showed you that it appears as many times as BG and GB and told you it could be proved with an experiment and you have ignored me.

Yet you continue to ignore my comment and posting the same thing for 10 posts. You should get yoiur facts straight.

No, I and others have responded to your point directly. You are the one doing the ignoring. You have even been given links to websites that specialize in math that have explained the facts.

In our problem: "They have two kids, one of them is a girl, what is the probability that the other kid is also a girl" we are only being asked to suggest the sex of the other child, not in the context of the family as a whole. This is a very important distinction and boils down to semantic pedantry I know, but to dismiss the '1/2' answer is incorrect. The wording "the other kid" is extremely important since we are not being asked to evaluate them together. We are merely being asked to evaluate the probability of the sex of "the other kid". It has been nominated, therefore we are not required to look at the problem as a combination of 2 births but one independently. In this instance the "best" answer would be 1/2.

It doesn't matter if the question asked "what is the probability they are both girls" or "what is the probability the other child is a girl". The answer is 1/3 in either situation. We do have to look at the problem as a combination of two births because we don't know which child the girl is in the question. It could be any of the following.

GG

GB

BG

The other child is a girl 1/3 of the time.

Try this: "A child from the Jones family is selected at random, and that child is a girl.” Here, the probability that the Joneses are a two-girl family is 1/2. Again, a child has been specifically nominated.

Terrific. But this information has not been given in the riddle. Further information helps narrow down probability, but it hasn't been given by the OP. "Again, a child has been specifically nominated" is not information we have been given in the OP and can not be assumed. The fact that the OP gives an answer of 1/3 is evidence that he did not want us to make this leap and we were to take only the information given.

I believe the wording in the OP is somewhere between the Smiths and Jones examples that I give above and therefore a conclusive answer is not possible. It is how we, as individuals, interpret the ambiguity and then make assumptions based on our conclusions.

Right, we can't make assumptions. All we know that one is a girl; without making assumptions the answer is 1/3.

As I have previously stated, the question asks us to name the sex of a particular child, namely "the other kid".

There is nothing wrong with the wording in the OP.

Asking "what is the prob they are both girls?" is effectively the same as asking "what is the probability that the other kid is also a girl?"

Either way, the probability is 1/3.

And the links that were provided agree.

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Scraff

"What is the probability that they are both girls?", is not the same as saying, "What is the probability the other kid is a girl?".

There is a small, but crucial, difference. The first example requires us to look at the problem as a combination of two births. In this scenario the answer is 1/3.

The wording of the OP specifically refers to "the other kid". This is definitive. It requires us to determine the probability of "the other kid" being a girl independently as an individual and not as a combination of 2 births.

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I see that the 1/3 camp is unsually quiet. The understanding of semantics is crucial to the answer.

Which is, on this occasion, a 1/2.

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I see that the 1/3 camp is unsually quiet. The understanding of semantics is crucial to the answer.

Which is, on this occasion, a 1/2.

Yes, because if a few hours go by on a thread without a response, the logical conclusion is that everyone was stumped by the last post. There's nothing you said in your last post that hasn't already been responded to. But since you're asking, here ya go:

"What is the probability that they are both girls?", is not the same as saying, "What is the probability the other kid is a girl?".

The wording from the mathforum.org and Dr. Math are the same:

"In a two-child family, one child is a boy. What is the probability that the other child is a girl?"

They still say this:

"When the only information given is that there are two children and one is a boy...: "

That's said because that is the only information given... because both questions are essentially the same. Since the riddle does not specify order, we still have three possibilities.

There is a small, but crucial, difference. The first example requires us to look at the problem as a combination of two births.

No. With the information given, there are still three combinations of births.:

GG

GB

BG

The wording of the OP specifically refers to "the other kid". This is definitive. It requires us to determine the probability of "the other kid" being a girl independently as an individual and not as a combination of 2 births.

Of course "the other kid" is a separate individual. No one disputes that. When we speak of one child being a girl, all that means is that one is a girl. Which one is not specified. The "other one" is a girl 1/3 of the time because all that means is that both are girls. Since we still don't no order, the three possibilities above are the correct ones.

There is one girl in each of the three possibilities. The "other child" is a girl in only one case.

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"What is the probability that they are both girls?", is not the same as saying, "What is the probability the other kid is a girl?".

Gazza,

If you perceive silence from the "1/3 crowd", you might consider the possibility

that it's not because you have persuaded them - they may have just concluded

that you're impervious to logic.

It's more convincing to post proofs than it is to hint, for example, that it's not about

logic, anyway, but about semantics. You might even try to say what that means.

For starters ... your statement in red above.

Can you give a counter-example? Show a case where both "one" and "the other"

of two children are female, but it's not the case that they both are girls.

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GazzaTheBook and noobsauce

We know there are 2 kids and one of them is a girl. So all possible cases 3; G-G, G-B, B-G. These 3 cases are equally likely, right?

Number of events favourable to the outcome is 1, G-G, right? So the probability is 1/3.

Don't get that obivous thing at all. Show me the mathimatical steps that proves all 3 cases are equal.

From what I see.

when it come to B-G it is not 1/4, like the other cases, but instead 1/2. Why? Because since the first baby is a boy, the second baby has to be a girl. So 1/2 * 1 = 1/2

So it would be 1/4, 1/4, 2/4. We note that the all the probabilities add up to one.

So the answer would end up as 1/4?

No one seemed understood my obove posts, and I have come to understand the problem. But nevertheless 1/3 is not a correct outcome.

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Why have I been quiet?

Could it be because I have a life outside the den, hmmm maybe I have children, a job, the fact that I live in NZ and my time zone is different from other den members or because trying to convince the 1/2 camp that they should be in the 1/3 camp is a lost cause.

If I have offended anyone in the 1/2 camp with prior posts, then I apologise. If I did, it was purely in humour and nothing else.

I will not claim to be a maths expert but believe "maths" experts on the internet have solved this puzzle in the same way I have. (referring to links Scraff has previously posted and others).

I have thought about posting another argument in support of the 1/3 camp again, but all I would be doing is repeating myself over and over.

I have considered the 1/2 answer and have read through the 1/2 camp posts. I do believe the correct answer is 1/3. I have asked the 1/2 camp to go to the links Scraff posted and read what the experts have said, if they have websites that support their argument (1/2) then I would be happy to go to these sites.

I could just stop my argument, but its so sad walking away knowing I haven't done my best in helping the 1/2 camp realise the true answer .

Maybe we should have a poll to vote, there have been almost 300 posts and 26,000 visits.

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Show me the mathimatical steps

since the first baby is a boy,

No one seemed understood my obove posts

nevertheless 1/3 is not a correct outcome.

Four statements ... four responses:

1. The math does not matter, if you start with a false premise.
2. The first baby's sex is not given.
3. Possibly because you are not answering the question in the OP.
4. You say.
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In the Blue corner we have the 1/3 guys. And in the Red corner we have the 1/2 guys.

This is a 6-round fight, and the 3-knockdown rule has been suspended because no one

could agree on the probability that a third knockdown could even occur, given that two

knockdowns had preciously occurred. Now let's hear the rules:

In each round, the Blue camp will lead with a punch and the Red camp will be given the

opportunity to counter-punch. In the event the Red team fails to throw even one legal

counter-punch, the Blue team shall be declared the winner.

If the Red team shall lose, but insist on a rematch, the fight will be held at a new venue.

Now, go to your corners, and let's have a clean fight.

The bell sounds ...!

1. What does the OP say?

1. A family has two children.
2. Boys and girls have equal birth probability.
3. One of them is a girl.
Then it asks: What is the probability that the other is a girl?

Pertaining to the question, the OP says this, and only this.

If you disagree, say why, and call it Counter-punch 1.

2. What does "boys and girls have equal birth probability" mean?

It means that, before any other conditions were stated, the gender of each child

is boy or girl, with equal likelihood. As a consequence, [do the math, it's easy]

1. same-gender families and mixed-gender families have equal likelihood.
2. half of the same-gender families will be boys; the other half, girls.
If you disagree, say why, and call it Counter-punch 2.

3. What does "one of them is a girl" mean?

It means that it is not the case that the children are both boys. And nothing else.

Therefore,

If it's a same-gender family, the children are girls.

If you disagree, say why, and call it Counter-punch 3.

4. What gender distributions now exist, other than 2 girls or mixed-gender?

No other gender distribution exists.

If you disagree, say why, and call it Counter-punch 4.

5. What is the likelihood of a 2-girl family relative to that of a mixed gender family?

[see points 2 and 3 above]

The likelihood of a 2-girl family is 1/2 the likelihood of a mixed-gender family.

If you disagree, say why, and call it Counter-punch 5.

6. If there are two and only two possible outcomes, say A and B,

and B has twice the likelihood of A, what is the probability of A?

It is 1/3.

If you disagree, say why, and call it Counter-punch 6.

If you cannot counter any of the six punches but still think the 2-girl case has probability of 1/2, then create your rematch in the Others venue, and name it Alternative approaches to logic.

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Scraff, Bonanova et al,

Apologies. I did not mean to confuse you with my earlier post.

Semantics: noun

1. (linguistics) The science of the meaning of words. Semantics is part of linguistics.

2. The study of the relationship between words and their meanings.

3. The individual meanings of words, as opposed to the overall meaning of a passage.

Take a close look at 3.

3. The individual meanings of words, as opposed to the overall meaning of a passage.

We all know that the premise of the OP is to apply logic and come up with the answer 1/3.

However, when we deconstruct the sentence it is possible to to answer 1/2. Here's why:

"They have two kids, one of them is a girl, what is the probability that the other kid is also a girl?"

Lets take two specific parts of this passage: "one of them" and "the other".

The children are referred to separately. No reference is made to "BOTH" children. If the OP stated "What is the probability they are both girls" then the best answer is 1/3. But it doesn't. The first and second elements of the statement are unconnected, independent. We are asked to assess the probability that "the other kid" is a girl in isolation because in the second part of the statement no reference is made to the original girl. This is crucial. We are not asked to make a correlation or combination. We are only asked to evaluate the probability of "the other kid" being a girl.

In this instance, "the other kid" has an equal chance of being either a boy or a girl.

You may consider this to be semantic pedantry, but the wording of such problems must avoid ambiguity. If the OP read:

"There is a 2 child family who have at least one girl. What is the probability that they are both girls?"

Then this is an open-and-shut case. 1/3. Now look at our problem again:

"They have two kids, one of them is a girl, what is the probability that the other kid is also a girl?"

Not the same is it? Same intention, different wording, different answer.

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Punch 1 - no counter-punch thrown.

Punch 2 - no counter-punch thrown.

Punch 3 - no counter-punch thrown.

3. What does "one of them is a girl" mean?

It means that it is not the case that the children are both boys. And nothing else.

Therefore,

If it's a same-gender family, the children are girls.

If you disagree, say why, and call it Counter-punch 3.

One is a girl changes nothing except to eliminate two boys.

Nothing in the counter=punch says otherwise, so the punch stands.

Punch 4 - no counter-punch thrown.

Punch 5 - no counter-punch thrown.

Punch 6 - no counter-punch thrown.

Score: Blue 6 Red 0.

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One is a girl is equivalent to at least one is a girl.

If the more restrictive One and only one is a girl meaning is assumed,

then the probability of another girl is 0, not 1/2.

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One child and the other child do not refer to specific children, who then can be discussed separately,

as would be the case if it said the older child is a girl, what is the probability that the younger child is a girl.

One does not give the gender of a specific child.

One gives the count of the minimum number of children who are not boys.

One child is a girl has no logical consequence other than to eliminate the case of two boys.

What is the probability the other is a girl is logically identical to what is the probability of two girls.

I'll grant that my statements are a semantical interpretation of the puzzle.

But they are more than just a description of what personally makes sense.

They are supportable by truth tables. They have logical stance as well.

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In this instance, "the other kid" has an equal chance of being either a boy or a girl.

This is the crux of your reasoning, which I believe that I follow.

Taken in isolation, it's true. It's a premise of the OP.

And it's true, before the conditions of the OP have been stated.

But consider: the same is true of whichever child is referred to as the "one child".

But the OP changes that probability from 1/2 into certainty when it says "one child" is a girl.

At the same time, and with equal force, it changes the probability of the gender of whichever child is referred to as "the other kid".

You say the two are not related, that they can be taken separately.

But logic does not support it - the conditions in the OP affect them both.

You're inspecting the trees, so to speak, as if there were no forest.

But if one is a girl, then if the other is a girl, they are both girls.

You somehow want to not see that the OP is asking a question that is logically equivalent to both being girls.

But there simply is not room to have that difference: one of them + the other of them = both of them.

So, yes, it's not the same.

And if it were consistent with the OP it would lead to a different answer.

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One is a girl is the same as at least one is a girl.

If the more restrictive case One and only one is a girl is assumed, then the probability of another girl is 0, not 1/2.

Not so fast, Batman!

One is a girl = gender assigned to a child confirmed.

At least one is a girl = gender of either child unconfirmed.

One and the other = independent events.

You see the forest, but can't make out the individual trees, so to speak.

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I'll grant that my statements are a semantical interpretation of the puzzle.

At last!

This was the only concession I was seeking from the 1/3 fraternity.

That due to semantical interpretation it is possible to arrive at an alternative answer because of the ambigiuty in the wording of the puzzle

and that if you deconstruct the sentence it is possible to arrive with 1/2.

I have given numerous examples where the answer is 1/3, but the wording in our puzzle is sufficiently ambiguous to be open to semantical interpretation and therefore it is only possible to give a "best" answer. In this case I believe the "best" answer is 1/2.

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Counter punch 6:

We know there are 3 probabilities, but we do not know whether the chances of the 3 probablities are the same.

Can you provide proof that all 3 cases have the same probabilities?

I suppose that it is obivous to you guys that all 3 cases equal yet I don't understand it.

If you can I'd happily change to the 1/3 crowd.

BTW: 1/2 is not the right answer. I was wrong and I'll admit it. 1/2 is definitely not right.

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At last!

This was the only concession I was seeking from the 1/3 fraternity.

That due to semantical interpretation it is possible to arrive at an alternative answer because of the ambigiuty in the wording of the puzzle

and that if you deconstruct the sentence it is possible to arrive with 1/2.

I have given numerous examples where the answer is 1/3, but the wording in our puzzle is sufficiently ambiguous to be open to semantical interpretation and therefore it is only possible to give a "best" answer. In this case I believe the "best" answer is 1/2.

Sorry, but I really have to take issue with this. I would say that the English language is imprecise in general because of how it developed. So just because you think there is an ambiguity in the phrasing, I don't think that should mean that the OP has multiple correct interpretations (or rather, that you interpretation is more correct than the mathematical community's consensus. )

Here is the exact wording:

"They have two kids, one of them is a girl, what is the probability that the other kid is also a girl.

Assume safely that the porbability of each gender is 1/2."

Now forgive me if I haven't been following this thread from beginning to end (I hope you can understand why:P) so this might have been used before, but as I read that sentence, I see an equivalent sentence that reads:

"They have two kids. Given that one of them is a girl, what is the probability that the other is also a girl?

Assume safely that the probability of each gender is 1/2."

This is conditional probability. Let's give the two kids gender-unspecific names, Kelly and Jean (abbreviated k and j) and say that "k is a girl" (G). Say that the event that "j is a girl" is A. The OP is asking for Pr(A|G). For the two events to be statistically independent, then Pr(A|G) = Pr(A). We know that this is not the case because if you know that one of them is a girl, that affects the choices for what the other child can be. This is why the answer should be 1/3, not 1/2.

1. Pr(A) = 1/2 (Given from OP)

2. Pr(A|G) = 1/3 (30 pages of explanation)

3. Pr(A|G) != Pr(A) (see 1 & 2)

If you can show that these two sentences have different semantic meanings, then I'll accept your answer, but as it is, I don't see that being the case.

Original: They have two kids, one of them is a girl, what is the probability that the other kid is also a girl?

Mine: They have two kids. Given that one of them is a girl, what is the probability that the other is also a girl?

Are they different?

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