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One Girl - One Boy


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

Spoiler for solution
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#51 unreality

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Posted 29 July 2007 - 07:54 PM

thanks

all these posts by dsu and martini are getting way complicated, i was mostly doing that so i could understand it myself
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#52 propsguy

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Posted 30 July 2007 - 07:25 AM

The probability that the second child is a girl is 50%. Just as the probability that the first kid was a girl was 50%. Just as the probability that the 6th kid they have (should they be so inspired) would also be a girl would be 50% too.

The probability of having two children that are both girls, now that's a different story. There, the probability would be 25%

The puzzle gives the gender of the first kid as a given. The probability of any one child being a girl is 50%. First, Second, Sixth, it doesn't matter.

50%. Just like a coin toss. On any given flip, the probability is (theoretically) 50% that it will come up heads. The previous flips have no bearing on the probability of the current given flip.
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#53 propsguy

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Posted 30 July 2007 - 07:38 AM

The answer is 25%. Before they had children the probability for a girl first is 50%. The probability for a girl second is 50%. BUT, to have 2 girls the probability moves to half of half, or 25%.

Comparing a coin toss is a common mislead. A single coin toss is of course 50/50. But 2 consecutive tosses is 50/50 for the first, and 25/75 for the second. Taken as a set (as you must in a family), you cannot discount the total probability by looking at the single toss.

Much like this:
I have 3 boxes (a - b - c).
2 have nothing in them.
1 has a million dollars.
If you guess the correct one, you get the money.

You pick one.
Out of the remaining 2, I show you that one is of course empty.
I give you the option of changing your answer.
Do you have better, worse, or the same odds if you DO change?

Better. The first guess you have a 1 in 3 chance of being right. Once I remove one, your odds move up to 2 in 3 by selecting the OTHER box, because it would have been like selecting the 2 boxes you didn't pick.

If I had offered for you to trade your ONE pick for the TWO you didn't pick, before showing one was empty would you?


Ahhh:

You are right in your "Monty Hall" puzzle, but wrong with the One boy one girl puzzle:
That the couple have 2 children is a given, One is a girl is a given, the variable is only the gender of one child, the second, which is set at 50%. The question isn't what is the chances of any couple having 2 female children, which would in fact be 25%.
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#54 Martini

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Posted 30 July 2007 - 11:04 AM

The puzzle gives the gender of the first kid as a given.


No, it doesn't. Please read the riddle again and this thread in its entirety so the solution and the reasons for it don't have to be explained all over again.
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#55 unreality

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Posted 30 July 2007 - 04:44 PM

just read my post near the end of page 5. it explains everything. i think this topic is over. really. it cant go much anywhere else...
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#56 propsguy

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Posted 30 July 2007 - 10:31 PM

there is a lot of confusion between the two sides of this problem, and i've read all ur arguements, and it isnt about conditionals or conjunctionals or anything


its about wording.


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.



this is the problem.

let me requote the important part (middle line)

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


"they have two kids." They already have them. They're not expecting kids... they already have both of them born. The kids could be 20 and 18 years old for all we care.


"One of them is a girl"
okay... so if we have two kids, already born, and we can rule out B/B:
G/B
B/G
G/G


"what is the probability that the other kid is also a girl."

not: "what is the probability that the other kid will be a girl."

In that case it would be 1/2, since the other child does not affect the probability. But it doesnt say that. It said "is also a girl". The child is already born.

it all comes down to two sides:
Side A (answer is 1/2): We're looking at the second child's birth as an independent event.
Side B (answer is 1/3): We're looking at both together.

and I am sorry to say Side B is right (as I was on Side A for quite some time)

Why?

Because the second child has already been born. We dont care what was the probability that it was boy/girl while it was in its mother's womb. We know thats 1/2. The fact is, the gender has already been determined. We're just trying to figure out the probability they are both girls.

Since the options are:

G/B
B/G
G/G

we can safely say, G/G has a 1/3 chance.

There is no fallacy. There is no bla-bla-bla. It all comes down to wording:

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



the "other kid" is already born. The solution doesnt call for the probability that the other kid will be a girl. It calls for the probability that they are (as in right now) BOTH girls.

We're not looking at one girl's chances. In that case there would be two options:
B
G

but we're not. BECAUSE OF THE WORDING, we are looking at both children at once:
G/B
B/G
G/G

It is 1/3.

It pains me to say this (I defended 1/2 for a while until i realized i was wrong) but I'm not being bitter and I'm trying to prove what I now see is right.

its a fine line, based on skale's wording. but the answer is 1/3. I rest my case. I think this topic should be done. The discussion cant go much elsewhere.



Thanks Unreality. And Martini.

I've read your explanation and the rest of the thread. And I know how the answer is supposed to be 1/3 But...

It does come down to the wording.

"One of them is a girl"

versus

"One of which is a girl"

The statement in the riddle is not "One of which" though. Nor is it "At least one of them is a girl"

It's a matter of assigning. As soon as you say "one of them" you've assigned, say, the first born, or the second born to be a girl. If one of them, say the first born, is in fact a girl, the chances of the other being a girl are 50%.

in:
G/B
B/G
G/G

it does look to be 1/3. But, by assigning, by saying "one of them is a girl" I submit that you can, in a way, add a second G/G set.

Or, of those 6 kids, 4 are girls. The chances of "one of them" girls having a sister is 50%.

Semantics maybe, but that's how I see the puzzle.

But then again, I may just be "one of them" guys.
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#57 Martini

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Posted 30 July 2007 - 10:52 PM

I've read your explanation and the rest of the thread. And I know how the answer is supposed to be 1/3 But...

It does come down to the wording.

"One of them is a girl"

versus

"One of which is a girl"

The statement in the riddle is not "One of which" though. Nor is it "At least one of them is a girl"

It's a matter of assigning. As soon as you say "one of them" you've assigned, say, the first born, or the second born to be a girl. If one of them, say the first born, is in fact a girl, the chances of the other being a girl are 50%.


You have not assigned the first or second as a girl by saying "One of them is a girl". The fact that one is first or second hasn't even been mentioned. Saying "one of them" means simply that, "one of the two". "One of them is a girl" is no different than "one of which is a girl".
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#58 propsguy

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Posted 31 July 2007 - 03:47 PM

I think it might be a not-so-sneaky way of saying "I read too much into the puzzle"

I thought the puzzle was working with an additional, or deeper twist:

like:

"you have two coins in your pocket totalling 30 cents in value. One of them is not a quarter"

That classic works as a puzzle, specifically because it does assign. So when I looked over the words "one of them" I took that to be a crucial part of the puzzle.

I've been wrong before and I'll be wrong again Dammit.
Sneaky?... not so much.
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#59 unreality

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Posted 31 July 2007 - 05:18 PM

It didn't say "one of them is NOT a girl"

it said "one of them is a girl"

which could also mean "one of them is NOT a guy" but that just means that one that isnt a guy, is a girl, which is the same conclusion


no, it was just a probability problem, which, because of wording, has the answer of 1/3
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#60 willy

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Posted 04 August 2007 - 01:31 AM

this is why i have a problem with this aspect of statistical analysis. let's see what we have here.

1] if you know the oldest child is a girl, the probability the other is also a girl is 1/2.
2] if you know the youngest child is a girl, the the probability the other is also a girl is 1/2.
3] if you do not know which child is a girl, the probability changes to 1/3?

sorry, but this defies any logic i understand.

i encountered a problem similar to this called the 3 siblings which gave an even more ridiculous reason for one answer over another. no sense confusing this issue more though.

but ask yourself logically. why does assigning change the probability? and don't just regurgitate all the past answers. i know the statistical processes pretty well and as i said before, this is not just a debate you guys are having. there are many websites dedicated to both sides of this question.

another similar problem is the 2 bears.

2 bears are standing in the woods. one is brown and one is black. you are told that one of them is a male. what is the probability the other is also a male.

virtually the same as this problem. as i said earlier, most mathematicians will agree with the 1/3 answer. but there are many that challenge it and contend it is 1/2. they're having the same debate over it that you're having. in fact marilyn van zant has found out the many people disagree with her 1/3 answer. and it's not just a matter of semantics either.
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