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One Girl - One Boy


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

Spoiler for solution
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#301 dawh

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Posted 08 August 2008 - 08:18 PM

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. :rolleyes: )

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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#302 toxicoxyde

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Posted 08 August 2008 - 10:27 PM

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?


"One of them is a girl" is a statement. And as thus we don't know why this statement is supplied so a condition can not be made. So in all instances of a statistical analysis we have to consider the statement as a random event and are just as likely to be informed "One of them is a boy".

"Given that one of them is a girl" is a condition. Saying that in all instances of a statistical analysis we will always be supplied with the information that one of them is a girl if their is at least one girl.

The difference is with the wording of this problem. All other instances of this equation that I have read on the Internet give a condition (witch is something I overlooked before in my haste.... so Dr. Math is right!). That is not the case with wording of problem in the OP.
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#303 GazzaTheBook

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Posted 08 August 2008 - 11:01 PM

dahw

Fairly robust response. However:

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?


Please read one of my previous posts. I believe this sets out the equivocal nature of the question.


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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#304 Tearz

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Posted 09 August 2008 - 07:06 AM

Ok I can't resist having another go :)

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.



1. If a family has two kids and the older one is a girl, what is the probability that the younger child is a girl?
2. If a family has two kids, one of them is a girl, what is the probability that the other kid is also a girl?

To be fair, 1/2 is in fact the correct answer to the first question. Let us represent a boy with 'B' and a girl with 'G', with the older child coming first, and assume that the boy:girl ratio is precisely 50:50. This produces four possible combinations: BB, BG, GB and GG. However, in the case of the first question, we are told that the older child is a girl, thus rendering the combinations BG and BB impossible. We are thus left with GB and GG, in which one out of the two equal possibilities contains a girl as the younger child. The probability of the younger child being a girl is thus 1/2.

Now let us look at the second question, which states that at least one of the children is a girl. This means that out of the four possibilities, only BB is impossible owing to the fact that it does not contain a boy. As the second question does not state whether the girl is the older or the younger child, it is possible to have any one of GB, BG or GG. In other words, the girl we know of could have an older brother, a younger brother or a sister.

Note that the last possibility, GG, should only be counted once. This point can be confusing and thus merits a further explaination. First, let us look at GB and BG:

GB = there is an older girl who has a younger brother.
BG = there is a younger girl who has an older brother.

Clearly, these two situations are different, and thus represent two distinct possibilities. However, let us treat the ways in which GG might occur in the same manner:

GG = there is a younger girl who has an older sister.
GG = there is an older girl who has a younger sister.

Unlike the first pair of sentences, the ones for GG both describe the same situation - the words we use to describe GG simply depends on which of the girls we think the question has already referred to. GG is therefore only one possibility out of three, and thus has a 1/3 probability of occurring. On the other hand, having an older girl and a younger brother is different to having an younger girl and a older brother.

Two children- combinations are BB, BG, GB, GG.
At least one child is a girl - combinations are GG, BG, GB.
Probability that the other chid is a girl 1/3.

Now I am going to ask another question. This question differs slightly from question number 2 (as above), but again asks us the probability that the other child is a girl.

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

Now let us look at the third question, which states that at least one of the children is a boy. This means that out of the four possibilities, only GG is impossible owing to the fact that it does not contain a boy. As the second question does not state whether the boy is the older or the younger child, it is possible to have any one of GB, BG or BB. In other words, the boy we know of could have an older sister, a younger sister or a brother.

Note that the last possibility, BB, should only be counted once. This point can be confusing and thus merits a further explaination. First, let us look at GB and BG:

GB = there is a younger boy who has an older sister.
BG = there is an older boy who has a younger sister.

Clearly, these two situations are different, and thus represent two distinct possibilities. However, let us treat the ways in which BB might occur in the same manner:

BB = there is a younger boy who has an older brother.
BB = there is an older boy who has a younger brother.

Unlike the first pair of sentences, the ones for BB both describe the same situation - the words we use to describe BB simply depends on which of the boys we think the question has already referred to. BB is therefore only one possibility out of three, and thus has a 1/3 probability of occurring. On the other hand, having an older boy and a younger girl is different to having an older girl and a younger boy, and the probability of the family including a girl is therefore 2/3.

Two children - combinations are BB, BG, GB, GG.
At least one child is a boy - combinations are BB, BG, GB.
Probability of one child being a girl is 2/3.

This question is slightly different from the orginal question, and although it gives us a different answer (2/3) I have posted it to support the argument that there are three possibilities.

It tells us that one of the children is a boy. It asks the probability that the other child is a girl. In this case the boy can have a brother an older sister or a younger sister thus 2/3.
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#305 bonanova

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Posted 09 August 2008 - 07:44 AM

You say:

-----------------

"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.

-----------------

No it's not a different answer. Consider:

"They have two kids ... "
This means that with equal probability the kids are same or mixed gender.

"... one of them is a girl ..."
This means that the kids are either both girls or mixed gender, the second option having twice the probability of the first.

"... what is the probability that the other kid is also a girl?"
This means they are both [one + the other = both] girls, and that probability [against the twice as likely mixed case] is 1/3.
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#306 Tearz

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Posted 09 August 2008 - 07:52 AM

Has any one heard of The Monty Hall Problem.

I am posting this is support of the 1/3 camp.

I have taken this from wikipedia. To view the whole Wikipedia page please go to:

http://en.wikipedia....ty_Hall_problem

Monty Hall problem
From Wikipedia, the free encyclopedia

Posted Image

The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal. The name comes from the show's host, Monty Hall. The problem is also called the Monty Hall paradox, as it is a veridical paradox in that the solution is counterintuitive.

A well-known statement of the problem was published in Parade magazine:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? (Whitaker 1990)

Because there is no way for the player to know which of the two unopened doors is the winning door, most people assume that each door has an equal probability and conclude that switching does not matter. In fact, in the usual interpretation of the problem the player should switch—doing so doubles the probability of winning the car, from 1/3 to 2/3.

When the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with Ph.D.s, wrote to the magazine claiming the published solution was wrong. Some of the controversy was because the Parade version of the problem is technically ambiguous since it leaves certain aspects of the host's behavior unstated, for example whether the host must open a door and must make the offer to switch. Variants of the problem involving these and other assumptions have been published in the mathematical literature.

The standard Monty Hall problem is mathematically equivalent to the earlier Three Prisoners problem and both are related to the much older Bertrand's box paradox. These and other problems involving unequal distributions of probability are notoriously difficult for people to solve correctly, and have led to numerous psychological studies. Even when given a completely unambiguous statement of the Monty Hall problem, explanations, simulations, and formal mathematical proofs, many people still meet the correct answer with disbelief.

Now you may say that this is nothing like the original posted question, but I have posted it because the solution for both "paradox's" is quite counter-intuitive.

The Monty Hall Problem demonstrates that you cannot trust your instincts in even fairly simple situations involving chance. You must sit down and work out the details to check that your intuition is correct.

Now apparently most people said there was now a 1/2 (50-50%) chance of choosing the door with the car, but by showing them that door 3 had a goat, it increased the contestants odds from 1/3 to 2/3.

Do you get what I'm trying to say?

If you go to the above link it goes into much more detail with lots of math stuff (equations) that I don't understand so would never post, incase I would have to later explain them (and I wouldn't be able to).

I thought by referring to another probability paradox might help the 1/2 camp understand the 1/3 camps answer.
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#307 bonanova

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Posted 09 August 2008 - 08:16 AM

Tearz,

It was posted here a year ago.
It has a long discussion thread and similar groups of intuitants who absolutely know it's a wash.
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#308 Tearz

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Posted 09 August 2008 - 09:03 AM

I wasn't trying to start a new post, I was using an example of a paradox to support my argument, just like others have used the coin toss, penny experiment, coloured marbles in a bag etc..... Whether it is now believed to be a wash or not, my use of this paradox was purely in support of my argument how I see it. I will refrain from using it in the future.
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#309 Scraff

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Posted 09 August 2008 - 03:24 PM

"One of them is a girl" is a statement. And as thus we don't know why this statement is supplied so a condition can not be made.

<snip>

"Given that one of them is a girl" is a condition.

Interesting. "One of them is a girl" is a statement and not a condition, but if that sentence were to start with "Given that" then that statement for some reason becomes a condition?



"One of them is a girl" is a statement. And as thus we don't know why this statement is supplied so a condition can not be made.

So, unless we know the reason the statement was supplied, we're not dealing with conditional probability? This is plain silly. We're dealing with conditional probability.



So in all instances of a statistical analysis we have to consider the statement as a random event and are just as likely to be informed "One of them is a boy".

Why does that matter? It doesn't. What matters is what we were told. We could have been told one of them is a boy, but we weren't. This information isn't useless just because the information could have been different.

The difference is with the wording of this problem. All other instances of this equation that I have read on the Internet give a condition (witch is something I overlooked before in my haste.... so Dr. Math is right!). That is not the case with wording of problem in the OP.

"One of them is a girl" is a condition.

Dr. Math is right? Here's how it's worded on his website:

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

Why is 2/3 the correct answer in that case and 1/3 not in the case of the OP's riddle?
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#310 toxicoxyde

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Posted 10 August 2008 - 08:38 PM

Interesting. "One of them is a girl" is a statement and not a condition, but if that sentence were to start with "Given that" then that statement for some reason becomes a condition?


Yes. By changing the statement to "Given that one of them is a girl" adds structure to the information. The information is structured in such a way that we will always be given that their is a girl. However if the information being supplied is as random as the couple then half the time when the have a GB/BG mix we will be told that one of them is a boy. Why because their is no structure to it.

So, unless we know the reason the statement was supplied, we're not dealing with conditional probability?


Yes and no. If you walk up to a guy on the street and he says.
"Hi my name is Teanchi and this is my wife Beanchi. We have two kids, one of them is a girl" (aka the riddle)

Then your odds of the other one being a girl is best guess 50%. Why? Because in a GB/BG mix it is just as likely we will be informed that one of them is a boy. Their is no structure to the information.

But if you walk up to a guy and he says.
"Hi my name is Teanchi and this is my wife Beanchi. We have two kids."
Then you ask
"Is at least one of them a girl"
and he reply
"Yes"

Then the odds are 1/3 because their is structure to the information. The fact that he could have answered "no" changes your odds 1/3.


"In a two-child family, one child is a boy. What is the probability that the other child is a girl?"
Why is 2/3 the correct answer in that case and 1/3 not in the case of the OP's riddle?

Because "In a two-child family, one child is a boy" is a statement (however false) that sets the guidelines for the question. It says that in any given two-child family, one child is a boy. Kind of like saying "In a bathroom, their is toilet paper". Their a plenty of bathrooms out their without toilet paper (I hate it when that happens) but the initial statement sets the condition that all bathrooms have toilet paper. Now I know what you are going to say. "how is the statement in the riddle any different". Well because the statement riddle pertains only to Teanchi and Beanchi ("they have two kids"). So it more like saying "In your bathroom, their is toilet paper. What is the probability that I have toilet paper in mine".

http://www.mathpages...me/kmath036.htm
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