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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

### #291 bonanova

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Posted 08 August 2008 - 06:49 AM

Show me the mathimatical steps

since the first baby is a boy,

No one seemed understood my obove posts

nevertheless 1/3 is not a correct outcome.

Four statements ... four responses:
• The math does not matter, if you start with a false premise.
• The first baby's sex is not given.
• Possibly because you are not answering the question in the OP.
• You say.

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

### #292 bonanova

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Posted 08 August 2008 - 09:21 AM

In the Blue corner we have the 1/3 guys. And in the Red corner we have the 1/2 guys.
This is a 6-round fight, and the 3-knockdown rule has been suspended because no one
could agree on the probability that a third knockdown could even occur, given that two
knockdowns had preciously occurred. Now let's hear the rules:

In each round, the Blue camp will lead with a punch and the Red camp will be given the
opportunity to counter-punch. In the event the Red team fails to throw even one legal
counter-punch, the Blue team shall be declared the winner.

If the Red team shall lose, but insist on a rematch, the fight will be held at a new venue.
Now, go to your corners, and let's have a clean fight.

The bell sounds ...!

1. What does the OP say?
• A family has two children.
• Boys and girls have equal birth probability.
• One of them is a girl.
Then it asks: What is the probability that the other is a girl?

Pertaining to the question, the OP says this, and only this.

If you disagree, say why, and call it Counter-punch 1.

2. What does "boys and girls have equal birth probability" mean?

It means that, before any other conditions were stated, the gender of each child
is boy or girl, with equal likelihood. As a consequence, [do the math, it's easy]
• same-gender families and mixed-gender families have equal likelihood.
• half of the same-gender families will be boys; the other half, girls.
If you disagree, say why, and call it Counter-punch 2.

3. What does "one of them is a girl" mean?

It means that it is not the case that the children are both boys. And nothing else.
Therefore,
If it's a same-gender family, the children are girls.
If you disagree, say why, and call it Counter-punch 3.

4. What gender distributions now exist, other than 2 girls or mixed-gender?

No other gender distribution exists.
If you disagree, say why, and call it Counter-punch 4.

5. What is the likelihood of a 2-girl family relative to that of a mixed gender family?
[See points 2 and 3 above]

The likelihood of a 2-girl family is 1/2 the likelihood of a mixed-gender family.
If you disagree, say why, and call it Counter-punch 5.

6. If there are two and only two possible outcomes, say A and B,
and B has twice the likelihood of A, what is the probability of A?

It is 1/3.
If you disagree, say why, and call it Counter-punch 6.

If you cannot counter any of the six punches but still think the 2-girl case has probability of 1/2, then create your rematch in the Others venue, and name it Alternative approaches to logic.
• 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #293 GazzaTheBook

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Posted 08 August 2008 - 10:17 AM

Scraff, Bonanova et al,

Apologies. I did not mean to confuse you with my earlier post.

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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### #294 bonanova

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Posted 08 August 2008 - 11:07 AM

Punch 1 - no counter-punch thrown.
Punch 2 - no counter-punch thrown.
Punch 3 - no counter-punch thrown.

3. What does "one of them is a girl" mean?

It means that it is not the case that the children are both boys. And nothing else.
Therefore,
If it's a same-gender family, the children are girls.
If you disagree, say why, and call it Counter-punch 3.

One is a girl changes nothing except to eliminate two boys.
Nothing in the counter=punch says otherwise, so the punch stands.

Punch 4 - no counter-punch thrown.
Punch 5 - no counter-punch thrown.
Punch 6 - no counter-punch thrown.

Score: Blue 6 Red 0.
• 0
The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #295 bonanova

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Posted 08 August 2008 - 11:19 AM

One is a girl is equivalent to at least one is a girl.

If the more restrictive One and only one is a girl meaning is assumed,
then the probability of another girl is 0, not 1/2.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #296 bonanova

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Posted 08 August 2008 - 11:23 AM

One child and the other child do not refer to specific children, who then can be discussed separately,
as would be the case if it said the older child is a girl, what is the probability that the younger child is a girl.

One does not give the gender of a specific child.
One gives the count of the minimum number of children who are not boys.

One child is a girl has no logical consequence other than to eliminate the case of two boys.
What is the probability the other is a girl is logically identical to what is the probability of two girls.

I'll grant that my statements are a semantical interpretation of the puzzle.
But they are more than just a description of what personally makes sense.
They are supportable by truth tables. They have logical stance as well.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #297 bonanova

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Posted 08 August 2008 - 11:44 AM

In this instance, "the other kid" has an equal chance of being either a boy or a girl.

This is the crux of your reasoning, which I believe that I follow.

Taken in isolation, it's true. It's a premise of the OP.
And it's true, before the conditions of the OP have been stated.

But consider: the same is true of whichever child is referred to as the "one child".
But the OP changes that probability from 1/2 into certainty when it says "one child" is a girl.
At the same time, and with equal force, it changes the probability of the gender of whichever child is referred to as "the other kid".

You say the two are not related, that they can be taken separately.
But logic does not support it - the conditions in the OP affect them both.

You're inspecting the trees, so to speak, as if there were no forest.
But if one is a girl, then if the other is a girl, they are both girls.
You somehow want to not see that the OP is asking a question that is logically equivalent to both being girls.

But there simply is not room to have that difference: one of them + the other of them = both of them.

So, yes, it's not the same.
And if it were consistent with the OP it would lead to a different answer.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #298 GazzaTheBook

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Posted 08 August 2008 - 12:18 PM

One is a girl is the same as at least one is a girl.
If the more restrictive case One and only one is a girl is assumed, then the probability of another girl is 0, not 1/2.

Not so fast, Batman!

One is a girl = gender assigned to a child confirmed.
At least one is a girl = gender of either child unconfirmed.
One and the other = independent events.

You see the forest, but can't make out the individual trees, so to speak.
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### #299 GazzaTheBook

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Posted 08 August 2008 - 12:43 PM

I'll grant that my statements are a semantical interpretation of the puzzle.

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.
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### #300 noobsauce

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Posted 08 August 2008 - 08:15 PM

Counter punch 6:
We know there are 3 probabilities, but we do not know whether the chances of the 3 probablities are the same.
Can you provide proof that all 3 cases have the same probabilities?
I suppose that it is obivous to you guys that all 3 cases equal yet I don't understand it.
If you can I'd happily change to the 1/3 crowd.
BTW: 1/2 is not the right answer. I was wrong and I'll admit it. 1/2 is definitely not right.
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