Boy or Girl?

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Eduk
Posts: 2466
Joined: December 8th, 2016, 7:08 am
Favorite Philosopher: Socrates

Re: Boy or Girl?

Post by Eduk »

ok so time for the answer I guess.

As LuckyR says you have 4 possible combinations to start with, all with the same probability (1/4 each). BB, BG, GB and GG. This is hopefully straightforward?
We can rule out GG because we know one of the children is a boy, but we can't rule out BG or GB because we don't know which of the children is a boy. This leaves the chance that the remaining child is a boy as 1/3.

The reason this is confusing is due to the fact that we test BOTH children to see if one is a boy. And we only progress to asking the question if the test comes back true. It feels like we are asking a new unconnected question when we ask if the other child is a boy but we aren't.

A different way of phrasing this question would be to take 100 people with two children. In our perfect world of exact 50/50 boy girl ratios we would have 4 groups.
a. 25 of BB
b. 25 of BG
c. 25 of GB
d. 25 of GG
This I assume everyone would agree with?
So if I simply remove the 25 instances of GG from the population. Then it's clear that a family selected at random would have a 1/3 chance of being in group a, b or c. To satisfy the original question we need to select group a only, hence 1/3 chance.

Would everyone agree with the answer?
Unknown means unknown.
Fooloso4
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Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
As LuckyR says you have 4 possible combinations to start with, all with the same probability (1/4 each). BB, BG, GB and GG.
There is only one variable, with only two possibilities: B or G. The birth order does not matter. BG is the same as GB since we are not asked to determine birth order.
The reason this is confusing is due to the fact that we test BOTH children to see if one is a boy. And we only progress to asking the question if the test comes back true.


Why? We are told that one is a boy from the beginning. We can assign a probability of 100% that at least ONE of the children is a boy. B + x is the same as x +B.
Eduk
Posts: 2466
Joined: December 8th, 2016, 7:08 am
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Re: Boy or Girl?

Post by Eduk »

Fooloso4, ignore the original phrasing of the question for now. As this is the confusing part.

Please indicate which of the following you agree with.

Given a perfect split of exactly 50/50 boys to girls, if we take 100 families with two children (only) we can make the following statements.

1. In total there are 100 girls and 100 boys.
2. Group A has 50 boys (25 boys born 1st and 25 boys born 2nd)
3. Group B has 25 boys (first born) and 25 girls (2nd born)
4. Group C has 25 girls (first born) and 25 boys (2nd born)
5. Group D has 50 girls (25 girls born first and 25 girls born 2nd)
6 If you add groups A to D together you get 100 girls and 100 boys.
7. I haven't missed a boy or a girl and I haven't created a boy or a girl.
8. You could in real life actually sort these groups by hand (assuming you had willing participants).
9. If you felt like it you could actually do this on pieces of paper. As in tear up 100 pieces of paper and write BB, BG, GB and GG on 25 each. By the way you don't need 100 pieces of paper, the maths is the same with 4, it's just 100 is more illustrative.
Unknown means unknown.
Fooloso4
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Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

EduK:
Fooloso4, ignore the original phrasing of the question for now.
But how the question is posed makes the difference between ½ and ⅓. See, for example, wiki "boy or girl paradox".
Eduk
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Re: Boy or Girl?

Post by Eduk »

Yes, but they are talking about ambiguity in the question leading to different results. Once the ambiguity is resolved the answer is clear.

Do you feel my post had ambiguity? I did say that if it did this was a mistake on my part and I would gladly resolve any ambiguities?
Unknown means unknown.
Fooloso4
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Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
Do you feel my post had ambiguity? I did say that if it did this was a mistake on my part and I would gladly resolve any ambiguities?
To be clear, the ambiguity is not the result of a mistake on your part.

From the wiki article:
Its answer could be 1/2, depending on how you found out that one child was a boy. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[4] and Nickerson.
...
The intuitive answer is 1/2.[2] This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child (i.e., boy and girl),[2][9] and that the probability of these outcomes is absolute, not conditional.[10]

Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:
From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.
Eduk
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Re: Boy or Girl?

Post by Eduk »

It's not fundamentally ambiguous. It relates to exactly how you found out that one child was a boy. My two initial questions are hopefully not ambiguous.

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

This is the ambiguity in a nut shell. Do you agree with the above two statements in quotes?
Unknown means unknown.
Fooloso4
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Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
It's not fundamentally ambiguous. It relates to exactly how you found out that one child was a boy.
Are we returning to your initial formulation now or am I still to ignore it?

I found out that one child was a boy before you posed the problem.
Do you agree with the above two statements in quotes?
The two statements pose different problems. Is your example closer to the first or the second? Why?
Gertie
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Re: Boy or Girl?

Post by Gertie »

Fooloso4 wrote:Gertie:
Then again it could be like the Monty Hall game show door paradox thingy, which I could never get my head round, so I'm not sure.
I’ve just been playing around with the Monty Hall game. I think I understand the math and the simulations support the conclusion but I am still reluctant to accept the claim that it is better to switch.
Horrible isn't it! :lol:

I dunno about the maths, but I sort of get the thinking behind it, maybe...

Say it's 3 cups with a pea under one (I find this easier to grasp for some reason), but you don't know which. You pick a random cup, then one of the others is removed. You've narrowed the odds of the pea-cup remaining out of the other two, but not narrowed any odds on your original pick.

I can think that in my head, but it still doesn't ring true, cos there's 2 cups
and the pea's under one, so fifty/fifty surely!

But as the link says, if you instead have say 20 cups, and you pick one, and
then 18 pea-free cups are removed, the one which is left then looks much more
likely to be the pea-cup than your original random 1/20 pick.

Perhaps being just 3 to start, then leaving 2, we don't intuitively notice the narrowing of the odds with such a small number? And simultaneously our 50/50 either/or intuition is triggered?


Having said all that, if the simulations showed fifty/fifty, I'd accept that too and find an explanation to match!



from the wiki -
The intuitive answer is 1/2.[2] This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child (i.e., boy and girl),[2][9] and that the probability of these outcomes is absolute, not conditional.[10]
I think that's what I was sort of saying, once you know one is a boy in this specific case (which was my reading of the question), then it's 50/50.
Eduk
Posts: 2466
Joined: December 8th, 2016, 7:08 am
Favorite Philosopher: Socrates

Re: Boy or Girl?

Post by Eduk »

Do you agree with the above two statements in quotes?


The two statements pose different problems. Is your example closer to the first or the second? Why?
My example is supposed to be exactly the first statement.

I ask the mother in question if at least one of her two children are boys. She would answer yes if her first born was a boy and yes if her 2nd born was a boy (and obviously yes if both are boys). She would only answer no if both children were girls.

-- Updated January 30th, 2017, 7:30 pm to add the following --

Monty Hall is a very similar problem. I'd rather discuss it after any confusion is removed from the boy/girl paradox though :)
Unknown means unknown.
Fooloso4
Posts: 3601
Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
My example is supposed to be exactly the first statement.
But its not.
I ask the mother …
It is not a matter of what you ask but of what you tell us. What you tell us is that one of her children is a boy. We do not have to factor that in. It is closer to the second statement, although not exact since you were not selecting from a particular group but rather determined that she had two children and one was a boy.

As quoted above:
The intuitive answer is 1/2. This answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex of the second child (i.e., boy and girl).
Eduk
Posts: 2466
Joined: December 8th, 2016, 7:08 am
Favorite Philosopher: Socrates

Re: Boy or Girl?

Post by Eduk »

So which child is a boy? The first or second born?
Unknown means unknown.
Fooloso4
Posts: 3601
Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
So which child is a boy? The first or second born?
It doesn’t matter. From the Wiki article:
Thus, if it is assumed that both children were considered while looking for a boy, the answer to question 2 is 1/3. However, if the family was first selected and then a random, true statement was made about the gender of one child in that family, whether or not both were considered, the correct way to calculate the conditional probability is not to count all of the cases that include a child with that gender. Instead, one must consider only the probabilities where the statement will be made in each case.
See the table and calculation that follows it (I do not know how to properly format it here). The probability is ½.

In your case, the family was first selected and then and then a random, true statement was made about the gender of one child in that family.

I admit that I may be missing something, and in that case I don’t want to push it.
Eduk
Posts: 2466
Joined: December 8th, 2016, 7:08 am
Favorite Philosopher: Socrates

Re: Boy or Girl?

Post by Eduk »

no a random true statement was not made in my example. She was asked if at least one of her children was a boy, we do not know in advance what her answer will be.

-- Updated January 31st, 2017, 2:53 pm to add the following --

For example if I stopped 1000 women in the street and asked if they had two (and only two) children and 100 answered yes. I then asked those 100 if at least one of their children was a boy. In the case of perfect distribution of boy/girl which is assumed in the questions. Then 25 women would say no and 75 women would say yes. In my example I have asked you a question about the 75 which remain.
Unknown means unknown.
Fooloso4
Posts: 3601
Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
no a random true statement was not made in my example. She was asked if at least one of her children was a boy, we do not know in advance what her answer will be.
The random true statement was made by the person you asked, which you reported to us.
… we do not know in advance what her answer will be.
You did not know when you asked the question but the way you presented it to us the reader did know. We are told that one is a boy and asked about the probability of the other. The article emphasizes that the answer is dependent on how the question is asked and what information is provided.
For example if I stopped 1000 women …
This is a different question. That was the point of quoting the following from the article:
Specifically, Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:
[A] From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of ½.


Your original formulation was B. Your restatement is A.

All I am doing now is quoting from the Wiki article. It serves no purpose for me to continue to do so.
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