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# One Girl - One Boy

Best Answer Riddari , 13 July 2007 - 07:38 PM

Spoiler for solution
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348 replies to this topic

### #281 GazzaTheBook

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Posted 07 August 2008 - 01:54 PM

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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### #282 Drydung

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Posted 07 August 2008 - 02:04 PM

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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### #283 GazzaTheBook

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Posted 07 August 2008 - 02:22 PM

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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### #284 Scraff

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Posted 07 August 2008 - 02:49 PM

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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### #285 GazzaTheBook

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Posted 07 August 2008 - 03:51 PM

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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### #286 GazzaTheBook

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Posted 07 August 2008 - 09:47 PM

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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### #287 Scraff

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Posted 07 August 2008 - 10:39 PM

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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### #288 bonanova

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Posted 08 August 2008 - 12:40 AM

"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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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #289 noobsauce

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Posted 08 August 2008 - 03:47 AM

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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### #290 Tearz

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Posted 08 August 2008 - 06:36 AM

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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