If you know only that there are two children, then the sex of the older and the sex of the younger are independent events. = 1/2
If you are told that "at least one child is a boy," the gender of the other child is no longer independent. = 2/3
If you learn that "the older child is a boy," then you have definite knowledge about the older child, but you know nothing about the younger. The younger child's gender is again independent. = 1/2
If you are told that "at least one child is a boy," the gender of the other child is no longer independent. = 2/3
If you learn that "the older child is a boy," then you have definite knowledge about the older child, but you know nothing about the younger. The younger child's gender is again independent. = 1/2
I think the question is ambiguous, but I’d say 50% is marginally the better answer.
<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com
fficeffice" /><o
></o>
It seems to me that the folks who answer 67% are adding an assumption that is not present in the original wording, namely something like:
<o></o>
If a person has two children and one is a boy and one is a girl, and he commences talking about one of them to you, it will be the boy.<o></o>
<o></o>
Because look what happens when you do not make that assumption:
<o></o>
It is indeed true, as has been pointed out ad nauseum in this thread by the 67% proponents, that of the three possible combinations of two children once girl-girl is eliminated, two out of three have exactly one boy and one girl.
<o></o>
But the very fact that this person is telling you about a son means the boy-boy combination is more likely than either of the other two. Because if he’s the father of two boys, the child he happens to tell you about will always be a boy, whereas if he’s the father of one of each, the child he happens to tell you about will only be a boy 50% of the time (unless we—unjustifiably—make the above assumption that a father of a boy and a girl will always talk to you about his son).
<o></o>
<o></o>
Or to put the point another way: The 67% proponents claim that the original scenario is equivalent to the following question-and-answer:
<o></o>
Do you have exactly two children? (If yes, go on.)
<o></o>
Is at least one of those children a boy? (If yes, go on.)
<o></o>
Do you have two boys, or a boy and a girl?
<o></o>
And they contend, rightly, that you will find that approximately two-thirds of such folks will have a boy and a girl.
<o></o>
But I say that is not equivalent to the original scenario. Though it’s ambiguous, I think the following is closer to the original scenario:
<o></o>
Do you have exactly two children? (If yes, go on.)
<o></o>
Tell me about one of them. (If he talks about a boy, go on.)
<o></o>
Do you have two boys, or a boy and a girl?
<o></o>
I contend that only approximately 50% of these folks will turn out to have a boy and a girl.
<o></o>
<o></o>
In conclusion, the 67% proponents are correct in their math that from the premises that a person has exactly two children, and at least one of them is a boy, one can derive the conclusion that there is a 2 out of 3 chance he has one boy and one girl. However, they are incorrect that those premises are all the relevant information that can be gleaned from the original scenario.
<o></o>
The additional premise they’re ignoring is the way you found out at least one of his children is a boy is that the first child he happened to talk about to you was a boy, which conveys crucial additional information that impacts the probabilities.
<o></o>
Hence, 50% is the better answer.
<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com
fficeffice" /><o></o>It seems to me that the folks who answer 67% are adding an assumption that is not present in the original wording, namely something like:
<o
></o>If a person has two children and one is a boy and one is a girl, and he commences talking about one of them to you, it will be the boy.<o
></o><o
></o>Because look what happens when you do not make that assumption:
<o
></o>It is indeed true, as has been pointed out ad nauseum in this thread by the 67% proponents, that of the three possible combinations of two children once girl-girl is eliminated, two out of three have exactly one boy and one girl.
<o
></o>But the very fact that this person is telling you about a son means the boy-boy combination is more likely than either of the other two. Because if he’s the father of two boys, the child he happens to tell you about will always be a boy, whereas if he’s the father of one of each, the child he happens to tell you about will only be a boy 50% of the time (unless we—unjustifiably—make the above assumption that a father of a boy and a girl will always talk to you about his son).
<o
></o><o
></o>Or to put the point another way: The 67% proponents claim that the original scenario is equivalent to the following question-and-answer:
<o
></o>Do you have exactly two children? (If yes, go on.)
<o
></o>Is at least one of those children a boy? (If yes, go on.)
<o
></o>Do you have two boys, or a boy and a girl?
<o
></o>And they contend, rightly, that you will find that approximately two-thirds of such folks will have a boy and a girl.
<o
></o>But I say that is not equivalent to the original scenario. Though it’s ambiguous, I think the following is closer to the original scenario:
<o
></o>Do you have exactly two children? (If yes, go on.)
<o
></o>Tell me about one of them. (If he talks about a boy, go on.)
<o
></o>Do you have two boys, or a boy and a girl?
<o
></o>I contend that only approximately 50% of these folks will turn out to have a boy and a girl.
<o
></o><o
></o>In conclusion, the 67% proponents are correct in their math that from the premises that a person has exactly two children, and at least one of them is a boy, one can derive the conclusion that there is a 2 out of 3 chance he has one boy and one girl. However, they are incorrect that those premises are all the relevant information that can be gleaned from the original scenario.
<o
></o>The additional premise they’re ignoring is the way you found out at least one of his children is a boy is that the first child he happened to talk about to you was a boy, which conveys crucial additional information that impacts the probabilities.
<o
></o>Hence, 50% is the better answer.
I think the question is ambiguous, but I’d say 50% is marginally the better answer.
<o
It seems to me that the folks who answer 67% are adding an assumption that is not present in the original wording, namely something like:
<o
If a person has two children and one is a boy and one is a girl, and he commences talking about one of them to you, it will be the boy.<o
<o
Because look what happens when you do not make that assumption:
<o
It is indeed true, as has been pointed out ad nauseum in this thread by the 67% proponents, that of the three possible combinations of two children once girl-girl is eliminated, two out of three have exactly one boy and one girl.
<o
But the very fact that this person is telling you about a son means the boy-boy combination is more likely than either of the other two. Because if he’s the father of two boys, the child he happens to tell you about will always be a boy, whereas if he’s the father of one of each, the child he happens to tell you about will only be a boy 50% of the time (unless we—unjustifiably—make the above assumption that a father of a boy and a girl will always talk to you about his son).
<o
<o
Or to put the point another way: The 67% proponents claim that the original scenario is equivalent to the following question-and-answer:
<o
Do you have exactly two children? (If yes, go on.)
<o
Is at least one of those children a boy? (If yes, go on.)
<o
Do you have two boys, or a boy and a girl?
<o
And they contend, rightly, that you will find that approximately two-thirds of such folks will have a boy and a girl.
<o
But I say that is not equivalent to the original scenario. Though it’s ambiguous, I think the following is closer to the original scenario:
<o
Do you have exactly two children? (If yes, go on.)
<o
Tell me about one of them. (If he talks about a boy, go on.)
<o
Do you have two boys, or a boy and a girl?
<o
I contend that only approximately 50% of these folks will turn out to have a boy and a girl.
<o
<o
In conclusion, the 67% proponents are correct in their math that from the premises that a person has exactly two children, and at least one of them is a boy, one can derive the conclusion that there is a 2 out of 3 chance he has one boy and one girl. However, they are incorrect that those premises are all the relevant information that can be gleaned from the original scenario.
<o
The additional premise they’re ignoring is the way you found out at least one of his children is a boy is that the first child he happened to talk about to you was a boy, which conveys crucial additional information that impacts the probabilities.
<o
Hence, 50% is the better answer.
Thank God someone around here can READ. I don't think anyone was ever questioning the math of the 67%ers...
The additional premise they’re ignoring is the way you found out at least one of his children is a boy is that the first child he happened to talk about to you was a boy, which conveys crucial additional information that impacts the probabilities.
Tim...There's maybe 2 or 3 people here who are questioning the math assumption of the 67%ers. I'm not one of them.
Thanks for posting such a thought provoking little riddle. I'm sure you meant to post the classic version of this where the answer is in fact 67% but the way you worded it can only result in one answer.
Not sure if it was just a bad copy/paste or what but it certainly added some oomph to offshore the last few days.
KUTGW
Can't believe I just read this thread. Here are my thoughts.
One is definitley a boy. The other can be a boy or girl.
He can have
BG, GB, BB. However, although yes thouse are 3 possibilities which is put first as the oldest has nothing to do with the question of the percentage of boy vs girl.
This means the other child can either be a boy or a girl 50%. Doesn't matter who is older. It's either a boy or a girl 50%.
If it was what are the odds that the oldest is a boy. Then it would be 66%.
One is definitley a boy. The other can be a boy or girl.
He can have
BG, GB, BB. However, although yes thouse are 3 possibilities which is put first as the oldest has nothing to do with the question of the percentage of boy vs girl.
This means the other child can either be a boy or a girl 50%. Doesn't matter who is older. It's either a boy or a girl 50%.
If it was what are the odds that the oldest is a boy. Then it would be 66%.
Depends how he starts his sentence when talking about the son. "The boy" "my boy" would be obvious answers that the other would be a girl. "My oldest" "my favorite" would be there is 2 boys. Otherwise, the possibility is truly 50% unless you count the possibility that it could be a BOYGIRL... true?
The outcome of child 2 is surely NOT dependent of child 1 unless you take the statistical probability of how many boys:girls are born. Then it could be broken down into the country, the region, the city, the block, etc.
Overall, the true probability is 50%. You would not say "my boy was born with 10 fingers but my other child was born with 9." That probability is a lot less and most people would be 10 and 10 because having 10 fingers is what most of us come out with... not because your sibling did.
