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Two kids, with a nod to Teanchi and Beanchi


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20 replies to this topic

#11 bushindo

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Posted 17 July 2012 - 05:01 AM

The OP intended to deny bias among reporters, since each can comment on any two-child family,
Even tho some statements are denied to some reporters for some families.

However ...

The greater intent of this puzzle was to cool the debate somewhat, housed in the Teanchi-Beanchi thread.

We don't know the algorithm used by the reporter, and now we understand that it matters.
Most of us "1/3" zealots assume the reporter was like Red.
But there is no basis for that [or any other] assumption.

Spoiler for my answer


Some comments
Spoiler for Conditional probabilities

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#12 bonanova

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Posted 17 July 2012 - 03:29 PM

Did I forget to mention that uncle Jed had put their names in a hat and drew one of them to speak? :)

More interestingly, does the "One Boy, One Girl" thread any longer have a definitive answer?


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
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#13 Rockmond

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Posted 17 July 2012 - 04:54 PM

If Ned said it, probability is 1 that it is GG;
If Red said it, it is either BG, GB, or GG, probability is 1/3 it is GG
If Ted said it, you know the older child is G, probability is 1/2 it is GG
If Zed said it, you know the coin selected child is G. He mentions sex of other child (G) so probability of GG is 1

1/4*1+1/4*1/3+1/4*1/2+1/4*1=17/24 = probability that it is GG.
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#14 bushindo

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Posted 17 July 2012 - 07:05 PM

Did I forget to mention that uncle Jed had put their names in a hat and drew one of them to speak? :)

More interestingly, does the "One Boy, One Girl" thread any longer have a definitive answer?



If Ned said it, probability is 1 that it is GG;
If Red said it, it is either BG, GB, or GG, probability is 1/3 it is GG
If Ted said it, you know the older child is G, probability is 1/2 it is GG
If Zed said it, you know the coin selected child is G. He mentions sex of other child (G) so probability of GG is 1

1/4*1+1/4*1/3+1/4*1/2+1/4*1=17/24 = probability that it is GG.


Regarding these answers
Spoiler for derivation of solution

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#15 Yoruichi-san

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Posted 17 July 2012 - 08:11 PM

Lol...settled one debate and started another... ^_^

We were assuming the original choice was random, just like in my previous extreme example, the bag I drew the ball out of was randomly chosen. But after the result was observed, the probability of it coming from either bag was not equal.

In this case, the 4 brothers do not contribute equally to the probability that the "girl" is reported, hence when "girl" is reported, they are not equally likely to have spoken.

Spoiler for Visual explanation


I do really like how the coin toss in number 4 is a "red herring"...it doesn't matter if you always pick tallest, use an unfair coin, or use some voodoo ritual...in the end it amounts to the same thing :thumbsup: .

Edited by Yoruichi-san, 17 July 2012 - 08:12 PM.

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Women are definitely stronger. We are [Fe]males, after all...

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#16 bonanova

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Posted 18 July 2012 - 05:02 AM

I totally get that the likelihood, or ease, of saying girl depends on the reporter's algorithm.
But if Jed drew their name from a hat, the four are equally likely to have been the reporter.
And then, likely or not, the OP tells us that the reporter said girl.

Repeating, we are not given that the reporter said girl and then asked which reporter is the most likely to have spoken.
The OP saying we don't know which one spoke was meant to be even stronger: we don't even have a clue.

So I need to understand better Bushindo's analysis.

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#17 Yoruichi-san

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Posted 18 July 2012 - 06:20 AM

When the reporter drew the name out of the hat, it was equally likely to have been any of the 4 brothers, and it was equally likely to have been any of the 4 child pairings, hence it was equally likely to have been any of the 16 possibilities I outlined above.

But after it was announced girl, you can eliminate the cases where "boy" would have been reported. Therefore the remaining 7 cases are equally likely, but not equally distributed among the 4 brothers.

Even if we are not asked which reporter is likely to have spoken, it is still part of the problem of figuring out the likelihood of the other child being a girl.

The key is that the cases were "girl" is reported are equally likely, not the reporter.



Take my example with the balls and the bags above. Let me make it more similar to this problem by asking "If a draw a blue ball, what is the likelihood that the next ball will be blue?" Bag A has all red balls, bag B has all blue balls. We are told that we are drawn a blue ball, likely or not, but after you know you drew a blue ball, it is definitely NOT equally likely to have come from either bag. Saying it would would give a prob of (1/2)*0+(1/2)*1=1/2 for the next ball being blue, which is clearly erroneous. Bag B contributes 100% of the prob of getting blue balls, so if the ball was blue, it is 100% likely to have come from bag B, and hence the next ball drawn from the bag will 100% likely be blue.

Edited by Yoruichi-san, 18 July 2012 - 06:30 AM.

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Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

#18 Yoruichi-san

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Posted 18 July 2012 - 06:43 AM

Doh...had a revelation. Let's bypass the whole "which brother spoke" thing and just say...

Spoiler for Visual explanation, furthered


*Takes a deep breath* There, much simpler :thumbsup: .
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Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

#19 bonanova

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Posted 18 July 2012 - 05:28 PM

I'm trying to see where my thinking has gone astray:

p(other is girl) =
p(other is girl if Ned spoke) x p(Ned spoke) +
p(other is girl if Red spoke) x p(Red spoke) +
p(other is girl if Ted spoke) x p(Ted spoke) +
p(other is girl if Zed spoke) x p(Zed spoke).

If the reporter had worn a name tag, we're done: we know p(other is girl if x spoke) for each x.
We're only uncertain about the name tag, it seems. Jed makes each tag probability 25.

Bushindo, I perceive an extra level of conditionals.
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- Bertrand Russell

#20 bushindo

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Posted 18 July 2012 - 06:42 PM

I'm trying to see where my thinking has gone astray:

p(other is girl) =
p(other is girl if Ned spoke) x p(Ned spoke) +
p(other is girl if Red spoke) x p(Red spoke) +
p(other is girl if Ted spoke) x p(Ted spoke) +
p(other is girl if Zed spoke) x p(Zed spoke).

If the reporter had worn a name tag, we're done: we know p(other is girl if x spoke) for each x.
We're only uncertain about the name tag, it seems. Jed makes each tag probability 25.

Bushindo, I perceive an extra level of conditionals.


You're right, there is an extra layer of conditionals. We need to incorporate the information about what was said ("One of the kids is a girl") into the calculations as well.

Spoiler for elaboration

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