Logic problem (1 Viewer)

[Evilgrin 72]

Agree with this premise, but disagree based on the OP. We know one of the two is a boy, which throws out GG (or 25% of the population.) To stick with the spirit of the OP, I would only be able to approach males on the street and pose the same question.

We know more than just that one of the two is a boy: we know which one. Namely, the one standing in front of us.That throws out not only GG, but also GB, leaving us with onlly BB and BG.

If you understand the bolded, you understand why the answer is 50%. This is really all that needs to be said. Plus, Maurile drove the point home by using the less-common 2 L spelling of only.

 

[Magic Football Shoes]

Just skimming this thread makes me want to punch you idiots in the face.

At least you aren't reproducing while you are busy with this ridiculous discussion.

 

[Cyclones]

1. Regardless of what you think the answer is, there is no good statistical reason to change your answer from 1/2 to 2/3 or vice versa based on the nature of the conversation, the presence or non presence of the son, etc, UNLESS it is revealed to you that the son is either the oldest or youngest sibling.

This may not be immediately obvious, but there's nothing unique about "birth order" that has the effect of making the answer 1/2. Any kind of ordering does this. Whether it's "older child / younger child" or "child that's present / other child that isn't present" or "child that attends Harvard / other child that doesn't attend Harvard" or, most relevantly, "child the father is talking about / other child that the father isn't talking about." The only way the answer is 2/3 is when the question doesn't give you any information that distinguishes one from the other in any way (again, even just the fact that one is being spoken about by the father and the other one isn't is enough to distinguish them in this regard). If the question was, "A family has two children. At least one is a boy. What is the probability that they are not both boys?" then the answer is 2/3. But that's a fundamentally different question than the one in the OP.

If we look at raw stats with regard to two sibling families, they are as follows (within a point or two either way, birth order discounted.)

25% are BB

50% are BG

25% are GG

It's easy to look at that and say the answer is 2/3 since there are twice the number of BG families as there are either of the other 2. When birth order is brought into question, it looks more like this

25% BB

25% GG

25% BG

25% GB

In that case, I think the cased can be made that the answer is 1/2 if we know the birth order of the boy, since it disqualifies two equal groups and leaves two equal groups.

Again, any ordering that distinguishes one from "the other" in any way has this same exact effect. You're listing them as:
older child younger child B B 25%B G 25%G B 25%G G 25%But you could also list them as:

child the father is telling you about	other child the father isn't telling you about	B                                     	B	25%B                                     	G	25%G                                     	B	25%G                                     	G	25%

and it's the same exact thing.

Ignoratio injected an interesting point - we should also count the BB and GG births in order, as follows, in equal distribution.

G1-G2

G2-G1

B2-B1

B1-B2

G1-B1

G2-B2

B2-G1

B1-G2

Giving us 8 groups to pull from. However in this case, you throw out G1-G2 and G2-G1, leaving 6 groups - 4 of which still contain girls. I still contend the answer is 2/3.

The answer isn't 2/3 in this case. You're given a boy (say, B1) and asked about the gender of his sibling. How many B1's in that table have a brother? How many B1's have a sister?

Ok, so what you're saying above is that if we know the order (B1,) the answer is 50%. If we don't know the order (boy he is referring to can be either B1 or B2,) it's 2/3. Is there an echo in here?

 

[Ignoratio Elenchi]

Tim: Hi.

Dave: Hey, how's it going.

Tim: Pretty good. I see that you're male. How many siblings to you have?

Dave: One.

Tim: Are you older or younger than your sibling?

Dave: I'm not telling.

Tim: That means there's a 2/3 chance that your sibling is female.

Dave: Okay. What if I told you that I'm the older sibling?

Tim: Then there'd be a 1/2 chance that your sibling is female.

Dave: Interesting. What if I told you that I'm the younger sibling?

Tim: In that case as well, there'd be a 1/2 chance that your sibling is female.

Dave: So if I tell you "I'm either older or younger (but I'm not saying which)," there's a 2/3 chance that my sibling is female. But if I tell you either "I'm older" or "I'm younger," there's a 1/2 chance that my sibling is female?

Tim: Exactly.

Dave: Interesting.

Good point Maurile - so perhaps we can just fall back on the fact that there are twice as many BG families as BB families, and say that the answer is 2/3, regardless of the order? I'm getting pretty fogged up here.

It honestly seems like you just assume that the correct answer is 2/3, and you keep trying to make every explanation fit that answer. Have you genuinely considered the possibility that the answer really isn't 2/3, and is in fact 1/2? I feel bad because you seem like you're on the right track over and over but then you keep coming up with 2/3 in situations when the correct answer is 1/2. Like, if you meet a boy with exactly one sibling, the probability that his sibling is a girl is 1/2. That's a fact that can be proven many ways, both empirically and mathematically. In its most basic sense, they're independent events - the gender of a child doesn't impact the gender of another child in his family, right? When a family has a boy, and then the mom gets pregnant again, she's not more likely to have a girl just because she previously had a boy, right? Yet you keep trying to find some way to explain that the probability is 2/3. It isn't. I think that's why you're getting fogged up, you're trying to make the questions fit your answer instead of finding the answer that fits the questions.

 

[Leroy Hoard]

Just skimming this thread makes me want to punch you idiots in the face.At least you aren't reproducing while you are busy with this ridiculous discussion.

You have no idea what some of us have under our desks.

 

[Ignoratio Elenchi]

1. Regardless of what you think the answer is, there is no good statistical reason to change your answer from 1/2 to 2/3 or vice versa based on the nature of the conversation, the presence or non presence of the son, etc, UNLESS it is revealed to you that the son is either the oldest or youngest sibling.

This may not be immediately obvious, but there's nothing unique about "birth order" that has the effect of making the answer 1/2. Any kind of ordering does this. Whether it's "older child / younger child" or "child that's present / other child that isn't present" or "child that attends Harvard / other child that doesn't attend Harvard" or, most relevantly, "child the father is talking about / other child that the father isn't talking about." The only way the answer is 2/3 is when the question doesn't give you any information that distinguishes one from the other in any way (again, even just the fact that one is being spoken about by the father and the other one isn't is enough to distinguish them in this regard). If the question was, "A family has two children. At least one is a boy. What is the probability that they are not both boys?" then the answer is 2/3. But that's a fundamentally different question than the one in the OP.

If we look at raw stats with regard to two sibling families, they are as follows (within a point or two either way, birth order discounted.)

25% are BB

50% are BG

25% are GG

It's easy to look at that and say the answer is 2/3 since there are twice the number of BG families as there are either of the other 2. When birth order is brought into question, it looks more like this

25% BB

25% GG

25% BG

25% GB

In that case, I think the cased can be made that the answer is 1/2 if we know the birth order of the boy, since it disqualifies two equal groups and leaves two equal groups.

Again, any ordering that distinguishes one from "the other" in any way has this same exact effect. You're listing them as:
older child younger child B B 25%B G 25%G B 25%G G 25%But you could also list them as:

child the father is telling you about	other child the father isn't telling you about	B                                     	B	25%B                                     	G	25%G                                     	B	25%G                                     	G	25%

and it's the same exact thing.

Ignoratio injected an interesting point - we should also count the BB and GG births in order, as follows, in equal distribution.

G1-G2

G2-G1

B2-B1

B1-B2

G1-B1

G2-B2

B2-G1

B1-G2

Giving us 8 groups to pull from. However in this case, you throw out G1-G2 and G2-G1, leaving 6 groups - 4 of which still contain girls. I still contend the answer is 2/3.

The answer isn't 2/3 in this case. You're given a boy (say, B1) and asked about the gender of his sibling. How many B1's in that table have a brother? How many B1's have a sister?

Ok, so what you're saying above is that if we know the order (B1,) the answer is 50%. If we don't know the order (boy he is referring to can be either B1 or B2,) it's 2/3. Is there an echo in here?

No, we don't have to know the order. It doesn't have to be B1 or B2, either one is fine. Pick one randomly if you want. Or don't pick one at all and treat them the same as if we didn't know the order. How many B's in that table have a brother? How many B's in that table have a sister?

 

[Magic Football Shoes]

Just skimming this thread makes me want to punch you idiots in the face.At least you aren't reproducing while you are busy with this ridiculous discussion.

You have no idea what some of us have under our desks.

I'm 10,000% sure it isn't a woman.Do the math on that, #####es!

 

[Cyclones]

Tim: Hi.

Dave: Hey, how's it going.

Tim: Pretty good. I see that you're male. How many siblings to you have?

Dave: One.

Tim: Are you older or younger than your sibling?

Dave: I'm not telling.

Tim: That means there's a 2/3 chance that your sibling is female.

Dave: Okay. What if I told you that I'm the older sibling?

Tim: Then there'd be a 1/2 chance that your sibling is female.

Dave: Interesting. What if I told you that I'm the younger sibling?

Tim: In that case as well, there'd be a 1/2 chance that your sibling is female.

Dave: So if I tell you "I'm either older or younger (but I'm not saying which)," there's a 2/3 chance that my sibling is female. But if I tell you either "I'm older" or "I'm younger," there's a 1/2 chance that my sibling is female?

Tim: Exactly.

Dave: Interesting.

Good point Maurile - so perhaps we can just fall back on the fact that there are twice as many BG families as BB families, and say that the answer is 2/3, regardless of the order? I'm getting pretty fogged up here.

It honestly seems like you just assume that the correct answer is 2/3, and you keep trying to make every explanation fit that answer. Have you genuinely considered the possibility that the answer really isn't 2/3, and is in fact 1/2? I feel bad because you seem like you're on the right track over and over but then you keep coming up with 2/3 in situations when the correct answer is 1/2. Like, if you meet a boy with exactly one sibling, the probability that his sibling is a girl is 1/2. That's a fact that can be proven many ways, both empirically and mathematically. In its most basic sense, they're independent events - the gender of a child doesn't impact the gender of another child in his family, right? When a family has a boy, and then the mom gets pregnant again, she's not more likely to have a girl just because she previously had a boy, right? Yet you keep trying to find some way to explain that the probability is 2/3. It isn't. I think that's why you're getting fogged up, you're trying to make the questions fit your answer instead of finding the answer that fits the questions.

Ok, I think I am coming around, and the phrasing of the problem is tantamount to coming up with the right answer. - If a two sibling family includes at least one boy, the probability that he has a sister is 2/3

- If a random boy tells you he has exactly one sibling, the probability that he has a sister is 1/2

Wait, I am falling apart again. Are these 2 statements not in conflict, regardless of which one we think is right?

 

[Ignoratio Elenchi]

Ok, I think I am coming around, and the phrasing of the problem is tantamount to coming up with the right answer. - If a two sibling family includes at least one boy, the probability that he has a sister is 2/3

Who is the "he" you're referring to? As I think Maurile said earlier, this is a syntax error. You start off talking about a family, and then go on to reference "he." There's no antecedent for that pronoun. This question, as phrased here, is meaningless.

- If a random boy tells you he has exactly one sibling, the probability that he has a sister is 1/2

Correct. It may not seem it, but this is functionally equivalent to the question in the OP, which is why the answer to the question in the OP is 1/2.ETA: Although I think you already agree with this last part. That was the whole point of the four question procession I posted earlier, which you agreed were all fundamentally the same question and all had the same answer. We started with the OP, and then made small changes to the wording (but not the meaning) until we got to this one. Now that you agree that the answer to this one is 1/2, you should also see why (or at least agree) that the answer to the question in the OP is 1/2.

 

[Cyclones]

Ok, I think I am coming around, and the phrasing of the problem is tantamount to coming up with the right answer.

- If a two sibling family includes at least one boy, the probability that he has a sister is 2/3

Who is the "he" you're referring to? As I think Maurile said earlier, this is a syntax error. You start off talking about a family, and then go on to reference "he." There's no antecedent for that pronoun. This question, as phrased here, is meaningless.

- If a random boy tells you he has exactly one sibling, the probability that he has a sister is 1/2

Correct. It may not seem it, but this is functionally equivalent to the question in the OP, which is why the answer to the question in the OP is 1/2.

Ok, if I change the first statement above based on your bolded criticism to If a two sibling family includes at least one boy, 2/3 of the time it will also include a girl

Does that make sense? And if so, why does it seem in conflict with the second statement?

ETA, I am in agreement now that the answer to the question posed is 1/2, but I can't say that I know why. I bet my previous employer would be happy to know that the $20k they spent to get me Six Sigma certified doesn't even yield an answer to such a simple problem.

 

[dgreen]

Does that make sense? And if so, why does it seem in conflict with the second statement?

How about this? What's the % chance X is G in each scenario:BXXBXX (at least one is B)I think your first statement is figuring out XX (at least one is B) and the second statement is trying to figure out BX or XB (which are pretty much the same thing).Yes? No?

 

[Ignoratio Elenchi]

If a two sibling family includes at least one boy, 2/3 of the time it will also include a girl

Does that make sense?

Yes.

And if so, why does it seem in conflict with the second statement?

It seems that way because you're not recognizing that the second statement imposes a "this one / the other one" order on the siblings, while the revised statement you provided here does not. One of the formulations gives you a property of the set of both children, and then asks you a question about another property of the set of both children. (The set of both children contains at least one boy, what is the probability that the set of both children does not contain two boys?)

The other formulation gives you a property of one of the children, and then asks you a question about a property of the other child. (The child the man is talking about is a boy, what is the probability that the other child he isn't talking about is a girl?)

As you can see, these are fundamentally two different things.

 

[Hawk Fan]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

 

[Cyclones]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

We (myself and IE) seem to have come to an agreement on two statements being accurate;2/3 of all two sibling families of which at least one is a boy will also include a girlandIf you run into a random boy on the street who tells you he has exactly one sibling, 1/2 of the time it will be a girl. So let's bear that out - you run into 100 random boys on the street with one sibling, you end up with50 BB50 BGAnd the next day, I run into 100 random girls on the street who tell me they have exactly one sibling, you end up with50 GG50 GBIf you add these up, you get 50 BB50 GG100 BGThe correct answer to the statement in the OP is unequivocally 1/2. The above demonstrates this and is in direct line with actual real data. IE/Maurile, thanks for bringing me around.

 

[chet]

Summary: any statements that include info about the family only (i.e. a 2-child family has at least one son) will have a 2/3 chance of the other kid being the opposite sex.

Any statements that include info that is specific to one of the kids will have a 1/2 chance of the other sibling being the opposite sex. i.e. a man has two children and one of his kids, who happens to be a boy, goes to Harvard.

Correct?

 

[Cyclones]

Summary: any statements that include info about the family only (i.e. a 2-child family has at least one son) will have a 2/3 chance of the other kid being the opposite sex. Any statements that include info that is specific to one of the kids will have a 1/2 chance of the other sibling being the opposite sex. i.e. a man has two children and one of his kids, who happens to be a boy, goes to Harvard. Correct?

Yes. The hard part for me was getting around what I felt was a conflict between IE's explanation and actual data. The problem I had was that I wasn't considering the entire poplulation of that data, only a subset.We could back into it another way and disprove the 2/3 group. You run into 99 boys on the street with exactly one sibling, and 2/3 of the time it is a girl;33 BB66 BGThe next day you run into 99 girls on the street with one sibling, and 2/3 of the time it is a boy;33 GG66 BGThis would bring the distribution to 33 BB33 GG132 BGWhich we know is simply not right.ETA - I don't mean to toot my own horn here, but this seems like the easiest way the 2/3 answer has been debunked. The numbers don't bear out when applied to a real life situation.

 

[Ignoratio Elenchi]

Summary: any statements that include info about the family only (i.e. a 2-child family has at least one son) will have a 2/3 chance of the other kid being the opposite sex.

Techinically a ayntax error. When you say "the other kid" who are you referring to? You haven't identified one yet, so there's no such thing as "the other one." I think what you mean to say is, "A 2-child family that has at least one boy, has a 2/3 chance of having exactly one boy." Or, perhaps closer to your wording, "A 2-child family that has at least one boy, has a 2/3 chance of having two children of opposite genders."

Any statements that include info that is specific to one of the kids will have a 1/2 chance of the other sibling being the opposite sex. i.e. a man has two children and one of his kids, who happens to be a boy, goes to Harvard.

Correct?

Correct, but even the bit about Harvard is unnecessary. The moment you said "one of his kids" you already distinguished "this kid" from "the other kid." Once you've made this distinction, you can provide the gender of "this kid" and then the probability of the gender of "the other kid" is 1/2.

 

[Maurile Tremblay]

A man with two children starts talking about his older son Frank. What is the probability that he has a girl? 1/2

I don't think this is ambiguous in any reasonable sense. When the man starts talking about his older son Frank, that's not supposed to be a secret code that actually reveals information about the sex of his younger child. "He'd tell us about Frank only if the other child was a female cellist with a fondness for tapioca." No, when he tells us about Frank, he's not telling us anything besides what the puzzle stipulates that he's telling us. Since he's telling us nothing about the younger child, that younger child — like all people about whom we have no information — is 50% likely to be female.

 

[Maurile Tremblay]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

If you assume that, the answer changes to 2/3 instead of 1/2 because by telling us that Frank is male, he's no longer telling us that his older child (or his identified child) is male; he's telling us only that at least one child is male.But that's an odd way to read the question, akin to assuming that a coin is weighted rather than fair. The question doesn't rule it out, but it shouldn't have to.

 

[Ignoratio Elenchi]

bam take it to the bank brohans

 

[12punch]

I believe the answer to the op is 2/3 - do a quick weighin with your answer so i can see if there's a consensus

 

[kutta]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

If you assume that, the answer changes to 2/3 instead of 1/2 because by telling us that Frank is male, he's no longer telling us that his older child (or his identified child) is male; he's telling us only that at least one child is male.But that's an odd way to read the question, akin to assuming that a coin is weighted rather than fair. The question doesn't rule it out, but it shouldn't have to.

Or said another way, Hawk is saying is that the father will ONLY talk about boys, even if he has girls. If he is a GG father, he won't talk to you. I think that's like the following situation with black and red cards:BB

BR

RB

RR

We usually say here, randomly pick and stack and randomly flip a card, and the chances are 1/2 that the cards will be different.

Hawk is saying to pick up your stack, and if you have a B, lay it down. What are the chances the other card is an R. (If you don't have a B, try again).

I think in his example that he is correct - the other card will be R 2/3 of the time.

Or as MT said above, it is like saying "I have at least one B card" as opposed to randomly selecting the B card.

 

[Cyclones]

I believe the answer to the op is 2/3 - do a quick weighin with your answer so i can see if there's a consensus

Best ever

 

[Cyclones]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

If you assume that, the answer changes to 2/3 instead of 1/2 because by telling us that Frank is male, he's no longer telling us that his older child (or his identified child) is male; he's telling us only that at least one child is male.But that's an odd way to read the question, akin to assuming that a coin is weighted rather than fair. The question doesn't rule it out, but it shouldn't have to.

Or said another way, Hawk is saying is that the father will ONLY talk about boys, even if he has girls. If he is a GG father, he won't talk to you. I think that's like the following situation with black and red cards:BB

BR

RB

RR

We usually say here, randomly pick and stack and randomly flip a card, and the chances are 1/2 that the cards will be different.

Or as MT said above, it is like saying "I have at least one B card" as opposed to randomly selecting the B card.

I did the above a few hours ago and posted the results;In any random set of 2 cards, if either one was a B, but both weren't R, the other was R 2/3 of the time.

If I draw the cards in order, if the first is a G, I threw it and the 2nd out, but if it was a B, I included those in the results, and it was about 50/50 BB/BR.

 

[jon_mx]

Two situations:

1. Man comes down the street and announces he has two children and at least one is a boy, the answer is still 50-50 on the gender of the other child.

2. Man comes down the street and announces he has two children. We ask him if he has at least one boy and he says yes, then the answer is 2/3s that the other child is a girl.

A random revelation about the gender of one of the children provides zero information about the gender of the other child.

 

[Drifter]

Fact: 2/3 of families comprised of 2 children will be either be 2 boys or a boy and a girl.

Fact: The chance of any single child being a boy or a girl is 50%.

The problem in arriving at the 2/3 number comes from our inability to view the problem from the perspective of an existing data set rather than our inclination to view the the possibility of the one instance.

If the man had a boy and was expecting a child, the odds become what is the chance of any one child being a boy or girl - 50%; but since we are talking about an existing data set, not a possible data set, the chance falls into the possibilities of all the combinations available.

 

[TinHat]

now let's get people right on the 2 pt conversion thread in the pool....

 

[Maurile Tremblay]

Two situations:1. Man comes down the street and announces he has two children and at least one is a boy, the answer is still 50-50 on the gender of the other child. 2. Man comes down the street and announces he has two children. We ask him if he has at least one boy and he says yes, then the answer is 2/3s that the other child is a girl. A random revelation about the gender of one of the children provides zero information about the gender of the other child.

I don't see a difference between #1 and #2. Saying at least one is a boy is the same as answering yes when asked whether at least one is a boy.Also, in both cases, which child is "the other" one?

 

[12punch]

Fact: 2/3 of families comprised of 2 children will be either be 2 boys or a boy and a girl.

Fact: The chance of any single child being a boy or a girl is 50%.

The problem in arriving at the 2/3 number comes from our inability to view the problem from the perspective of an existing data set rather than our inclination to view the the possibility of the one instance.

If the man had a boy and was expecting a child, the odds become what is the chance of any one child being a boy or girl - 50%; but since we are talking about an existing data set, not a possible data set, the chance falls into the possibilities of all the combinations available.

That's a fact?

 

[kutta]

Two situations:1. Man comes down the street and announces he has two children and at least one is a boy, the answer is still 50-50 on the gender of the other child. 2. Man comes down the street and announces he has two children. We ask him if he has at least one boy and he says yes, then the answer is 2/3s that the other child is a girl. A random revelation about the gender of one of the children provides zero information about the gender of the other child.

I don't see a difference between #1 and #2. Saying at least one is a boy is the same as answering yes to whether at least one is a boy.Also, in both cases, which child is "the other" one?

I think he is saying in #1 that the man could have randomly said "at least one child is a girl" but he chose to say, "at least one child is a boy." I know we are really hung up on the "at least one" deal, but I agree that if he could have randomly chosen to either statement above, the odds are 1/2 that the children are of different sexes.That is different from the case of being asked if he has at least one boy, because he must answer "yes" in all boy cases.

 

[CBusAlex]

This may not be immediately obvious, but there's nothing unique about "birth order" that has the effect of making the answer 1/2. Any kind of ordering does this. Whether it's "older child / younger child" or "child that's present / other child that isn't present" or "child that attends Harvard / other child that doesn't attend Harvard" or, most relevantly, "child the father is talking about / other child that the father isn't talking about."

Only if the ordering is random (i.e., either child is equally likely to be the oldest.) An ordering like "child that plays in the NFL / child that doesn't play in the NFL" would make the answer 2/3, since the NFL player is necessarily a boy. I'm not convinced there's enough information in the OP to determine whether "child the father is talking about" is random or not.

 

[Cyclones]

Fact: 2/3 of families comprised of 2 children will be either be 2 boys or a boy and a girl.

Fact: The chance of any single child being a boy or a girl is 50%.

The problem in arriving at the 2/3 number comes from our inability to view the problem from the perspective of an existing data set rather than our inclination to view the the possibility of the one instance.

If the man had a boy and was expecting a child, the odds become what is the chance of any one child being a boy or girl - 50%; but since we are talking about an existing data set, not a possible data set, the chance falls into the possibilities of all the combinations available.

That's a fact?

No, it's not, the correct statement would be 3/4 of all families will be comprised of either 2 boys or a boy and a girl. The other 1/4 will be 2 girls.

 

[jon_mx]

This may not be immediately obvious, but there's nothing unique about "birth order" that has the effect of making the answer 1/2. Any kind of ordering does this. Whether it's "older child / younger child" or "child that's present / other child that isn't present" or "child that attends Harvard / other child that doesn't attend Harvard" or, most relevantly, "child the father is talking about / other child that the father isn't talking about."

Only if the ordering is random (i.e., either child is equally likely to be the oldest.) An ordering like "child that plays in the NFL / child that doesn't play in the NFL" would make the answer 2/3, since the NFL player is necessarily a boy. I'm not convinced there's enough information in the OP to determine whether "child the father is talking about" is random or not.

Even if he says he has a kid in the NFL, it is still 50-50. The only way to get it to two-thirds if we ask a question. Any unconstrained revelation of one child's gender tell us nothing about the gender of the other child. Every two-child kid who plays in the NFL has a 50-50 chance of having a sister instead of a brother.

 

[5 digit know nothing]

A man tells you he has two children. He then starts talking about his son. He does not tell you whether the son is the oldest child or the youngest child. What is the probability that his other child is a girl?(I would like to add the term his "son" does not mean the other is not a son. Its a tough question to word but its not a play on words, just a logic question)

I am imagining I am the father, I have two children, I start talking about one of my children that is a boy, I know whether this child was the oldest or the youngest, and I know his sibling is consequently the youngest or the oldest of the two.The odds his sibling is a boy is the same as his sibling being a girl.(part of the population)oldest: byoungest: b(part of the population if the boy is the oldest)oldest: byoungest: g(part of the population if the boy is the youngest)oldest: gyoungest: bAm I missing something in the wording? Seems clear it is 50/50

 

[Juxtatarot]

I was in the 1/2 camp since 2009 then swung to the 2/3 camp a few days ago. I think I'm back with the 1/2ers now.

 

[Kool-Aid Larry]

Fact: 2/3 of families comprised of 2 children will be either be 2 boys or a boy and a girl.

Fact: The chance of any single child being a boy or a girl is 50%.

The problem in arriving at the 2/3 number comes from our inability to view the problem from the perspective of an existing data set rather than our inclination to view the the possibility of the one instance.

If the man had a boy and was expecting a child, the odds become what is the chance of any one child being a boy or girl - 50%; but since we are talking about an existing data set, not a possible data set, the chance falls into the possibilities of all the combinations available.

That's a fact?

No, it's not, the correct statement would be 3/4 of all families will be comprised of either 2 boys or a boy and a girl. The other 1/4 will be 2 girls.

and I think this is a perfect illustration of what happens in this thread, or in many others.people are determined to rephrase things, redefine things, and draw all kinds of analogies to the point where they confuse themselves and the issue.

by changing the question, you might possibly be changing the answer.

maybe tim and dave talking on the street is some kind of counterexample, but we should all be cognizant that it is not the op, and may provide different results than the op.

I think there's further confusion due to the populations being discussed.

at first glance, people may read the op and decide that each child is an independent event, and the sex of one gives you no hint about the other, but there's an implied population there that people don't acknowledge, on top of the fact that they may be subtly rewording the question in their head.

the reason the 2nd kid out of the womb is 50/50 is that the world's population is established at 50/50, and this is our data set.

if I want to slant the results, I restrict our population sample to a room which I fill with 20 girls and 10 guys.

if I then ask you to randomly pick a person it will no longer be 50/50, so it doesn't end up 50/50 simply because there are only 2 possibilities, and while a guy picked out of the room still doesn't determine the next selection, the distribution of the population still does.

my point being that part of the question spells out a particular population subset selected for our consideration which may or may not fall in line with earth's general population as a whole -- it might be slanted.

this particular pop restricts us to 2 child households, which include a boy.

to borrow from other posts:

the first child is equally B or G -- we should all agree here.

the second child is equally b or g -- again, hopefully agreement.

this creates 4 families, equally represented:

B1b1

B2g2

G3b3

G4g4

G4g4 is thrown out of the population under discussion.

the gist of the discussion comes down to the fact that our remaining pop consists of 3 households, but 4 boys.

which of these you focus on is what determines a 1/2 or 2/3 answer, or how you count the B1b1 household -- counted once or twice?

if I meet dave on th estreet, he could be one of 4 boys, and our logic questions will yield 1/2.

if I'm a genius woman who polls 3 households, 2 will claim a girl and 1 won't -- leading to 2/3.

the B1b1 father didn't respond twice because he has 2 boys.

in the op we are questioning the father (household), and if he is the father of B1b1 we need to know how to count this.

B1 and b1 are indistinguishable except in his mind -- in his mind he pictures one particular child, but as the outside observer we only know the criteria are "my son", and "sibling of my son".

both kids meet both criteria equally and they get single counted as one household.

if one kid was present, then 'the sibling' would be the one not present and they would not meet the criteria equally, so our counting and populations would be different.

 

[EYLive]

It depends. Do you fold your son when you eat him while walking down the street?

 

[Hawk Fan]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

If you assume that, the answer changes to 2/3 instead of 1/2 because by telling us that Frank is male, he's no longer telling us that his older child (or his identified child) is male; he's telling us only that at least one child is male.But that's an odd way to read the question, akin to assuming that a coin is weighted rather than fair. The question doesn't rule it out, but it shouldn't have to.

Or said another way, Hawk is saying is that the father will ONLY talk about boys, even if he has girls. If he is a GG father, he won't talk to you. I think that's like the following situation with black and red cards:BB

BR

RB

RR

We usually say here, randomly pick and stack and randomly flip a card, and the chances are 1/2 that the cards will be different.

Hawk is saying to pick up your stack, and if you have a B, lay it down. What are the chances the other card is an R. (If you don't have a B, try again).

I think in his example that he is correct - the other card will be R 2/3 of the time.

Or as MT said above, it is like saying "I have at least one B card" as opposed to randomly selecting the B card.

Yes, thank you. And I disagree with Maurile that this assumption is somehow perverting the original question. No matter what, you have to make an assumption about how the father decides to talk to you about his son.A man tells you he has two children. He then starts talking about his son. Did he randomly pick a child and that child happened to be a son? Then the answer is 1/2. But to come to this conclusion, you have to assume that his talking about his son is completely irrelevant to the problem, that the children are to be looked at as 2 independent events rather than as a group of 2. I continue to argue that this is an ambiguous problem, but I still think the better answer is 2/3.

 

[Ignoratio Elenchi]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl?

He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?

He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?

He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

 

[Hawk Fan]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl? He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

All near 1/2. There is minimal ambiguity about what child is being referred to. These are more like the "Tuesday problem" but even closer to 1/2. Though he could have two children who love the Yankees (that would be sad), graduated high school, won a golf tournament, have cancer (also sad), etc.He's talking to you about his son in very general terms because he is proud of him. What's the probability that his other child is a girl?

 

[Ignoratio Elenchi]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl? He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

All near 1/2. There is minimal ambiguity about what child is being referred to. These are more like the "Tuesday problem" but even closer to 1/2. Though he could have two children who love the Yankees (that would be sad), graduated high school, won a golf tournament, have cancer (also sad), etc.He's talking to you about his son in very general terms because he is proud of him. What's the probability that his other child is a girl?

1/2, obv.

 

[5 digit know nothing]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl?He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

/thread

 

[jon_mx]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

If you assume that, the answer changes to 2/3 instead of 1/2 because by telling us that Frank is male, he's no longer telling us that his older child (or his identified child) is male; he's telling us only that at least one child is male.But that's an odd way to read the question, akin to assuming that a coin is weighted rather than fair. The question doesn't rule it out, but it shouldn't have to.

Or said another way, Hawk is saying is that the father will ONLY talk about boys, even if he has girls. If he is a GG father, he won't talk to you. I think that's like the following situation with black and red cards:BB

BR

RB

RR

We usually say here, randomly pick and stack and randomly flip a card, and the chances are 1/2 that the cards will be different.

Hawk is saying to pick up your stack, and if you have a B, lay it down. What are the chances the other card is an R. (If you don't have a B, try again).

I think in his example that he is correct - the other card will be R 2/3 of the time.

Or as MT said above, it is like saying "I have at least one B card" as opposed to randomly selecting the B card.

Yes, thank you. And I disagree with Maurile that this assumption is somehow perverting the original question. No matter what, you have to make an assumption about how the father decides to talk to you about his son.A man tells you he has two children. He then starts talking about his son. Did he randomly pick a child and that child happened to be a son? Then the answer is 1/2. But to come to this conclusion, you have to assume that his talking about his son is completely irrelevant to the problem, that the children are to be looked at as 2 independent events rather than as a group of 2. I continue to argue that this is an ambiguous problem, but I still think the better answer is 2/3.

It is only 2/3 if there is some rule that says he must talk about his son if he has one. Unless the problem states that, why would we assume such a restriction on mentioning a son over a daughter. If the subject of the son just randomly comes up as it appears to be in the OP, then it is 50-50. In this problem there is no ambiguity. You have to assume facts which have not been mentioned to get to 2/3rds.

 

[Kool-Aid Larry]

I'm arguing for ambiguity. I'm curious what Maurile thinks the answer is if you assume: The father only cares about boys and will only talk about a son (if he has one). Haven't I provided a counterexample? You are still implicitly assuming that the father treats his children the same and selects one randomly, which is an assumption that isn't warranted from the original problem.

If you assume that, the answer changes to 2/3 instead of 1/2 because by telling us that Frank is male, he's no longer telling us that his older child (or his identified child) is male; he's telling us only that at least one child is male.But that's an odd way to read the question, akin to assuming that a coin is weighted rather than fair. The question doesn't rule it out, but it shouldn't have to.

Or said another way, Hawk is saying is that the father will ONLY talk about boys, even if he has girls. If he is a GG father, he won't talk to you. I think that's like the following situation with black and red cards:BB

BR

RB

RR

We usually say here, randomly pick and stack and randomly flip a card, and the chances are 1/2 that the cards will be different.

Hawk is saying to pick up your stack, and if you have a B, lay it down. What are the chances the other card is an R. (If you don't have a B, try again).

I think in his example that he is correct - the other card will be R 2/3 of the time.

Or as MT said above, it is like saying "I have at least one B card" as opposed to randomly selecting the B card.

Yes, thank you. And I disagree with Maurile that this assumption is somehow perverting the original question. No matter what, you have to make an assumption about how the father decides to talk to you about his son.A man tells you he has two children. He then starts talking about his son. Did he randomly pick a child and that child happened to be a son? Then the answer is 1/2. But to come to this conclusion, you have to assume that his talking about his son is completely irrelevant to the problem, that the children are to be looked at as 2 independent events rather than as a group of 2. I continue to argue that this is an ambiguous problem, but I still think the better answer is 2/3.

It is only 2/3 if there is some rule that says he must talk about his son if he has one. Unless the problem states that, why would we assume such a restriction on mentioning a son over a daughter. If the subject of the son just randomly comes up as it appears to be in the OP, then it is 50-50. In this problem there is no ambiguity. You have to assume facts which have not been mentioned to get to 2/3rds.

I disagree, as stated above

 

[kutta]

We might as well

While I understand IE's and MT's argument that saying "at least one" is different from saying, "my son...," I continue to disagree with it.

The argument that when you say "I have two children and at least one is a boy," that you cannot ask "what is the other one?" because there is no "other" makes sense. I get it. But I think it clouds the issue.

1. When you say "at least one is a boy" it gives us information. We know that we have one or more boys. That is all we know. We can glean no more information.

2. Likewise, when a man comes up and starts talking about his son, it gives us information. We know that we have one or more boys. That is all we know. We can glean no more information.

Forgetting about motive and forgetting about why a man would phrase something one way or another, I contend that both scenarios above provide us with the exact same information, and hence the answers should be the same for both scenarios.

 

[Hawk Fan]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl? He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

All near 1/2. There is minimal ambiguity about what child is being referred to. These are more like the "Tuesday problem" but even closer to 1/2. Though he could have two children who love the Yankees (that would be sad), graduated high school, won a golf tournament, have cancer (also sad), etc.He's talking to you about his son in very general terms because he is proud of him. What's the probability that his other child is a girl?

1/2, obv.

What if he has two sons that he is proud of but gives no additional information to identify him? What if you have a group of fathers who are all proud of all their sons, and one of them starts rambling on in very general terms about how proud he is of his son ... This approaches the 2/3 limit if the listener cannot distinguish the 2 children. It is the "have two children and at least one is a son" case.

 

[5 digit know nothing]

We might as well While I understand IE's and MT's argument that saying "at least one" is different from saying, "my son...," I continue to disagree with it.The argument that when you say "I have two children and at least one is a boy," that you cannot ask "what is the other one?" because there is no "other" makes sense. I get it. But I think it clouds the issue. 1. When you say "at least one is a boy" it gives us information. We know that we have one or more boys. That is all we know. We can glean no more information.2. Likewise, when a man comes up and starts talking about his son, it gives us information. We know that we have one or more boys. That is all we know. We can glean no more information.Forgetting about motive and forgetting about why a man would phrase something one way or another, I contend that both scenarios above provide us with the exact same information, and hence the answers should be the same for both scenarios.

They are not the same, you are generalizing too much.re: scenario 2Imagine you and your wife are planning to have two children. You already have names picked out for your first child, Matt or Sarah depending on gender. You also went ahead and picked names for your second child, Jacob or Mary, regardless of the sex of the first child.Now tell us a story about your son, mention his name and then tell me what are the odds that your other child is a boy vs. a girl.

 

[Ignoratio Elenchi]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl? He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

All near 1/2. There is minimal ambiguity about what child is being referred to. These are more like the "Tuesday problem" but even closer to 1/2. Though he could have two children who love the Yankees (that would be sad), graduated high school, won a golf tournament, have cancer (also sad), etc.He's talking to you about his son in very general terms because he is proud of him. What's the probability that his other child is a girl?

1/2, obv.

What if he has two sons that he is proud of but gives no additional information to identify him? What if you have a group of fathers who are all proud of all their sons, and one of them starts rambling on in very general terms about how proud he is of his son ... This approaches the 2/3 limit if the listener cannot distinguish the 2 children. It is the "have two children and at least one is a son" case.

No it isn't. The "have two children and at least one is a son" is the "have two children and at least one is a son" case. The one where the dad is talking about his son is the 1/2 case.

 

[Hawk Fan]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl? He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

All near 1/2. There is minimal ambiguity about what child is being referred to. These are more like the "Tuesday problem" but even closer to 1/2. Though he could have two children who love the Yankees (that would be sad), graduated high school, won a golf tournament, have cancer (also sad), etc.He's talking to you about his son in very general terms because he is proud of him. What's the probability that his other child is a girl?

1/2, obv.

What if he has two sons that he is proud of but gives no additional information to identify him? What if you have a group of fathers who are all proud of all their sons, and one of them starts rambling on in very general terms about how proud he is of his son ... This approaches the 2/3 limit if the listener cannot distinguish the 2 children. It is the "have two children and at least one is a son" case.

No it isn't. The "have two children and at least one is a son" is the "have two children and at least one is a son" case. The one where the dad is talking about his son is the 1/2 case.

If he reveals that he has a son without any other identifying information, it is the 2/3 case.

 

[dgreen]

He's talking to you about his son because he just graduated high school. What's the probability that his other child is a girl?He's talking to you about his son because he just won a golf tournament. What's the probability that his other child is a girl?He's talking to you about his son because you're wearing a Yankees hat and his son loves the Yankees. What's the probability that his other child is a girl?He's talking to you about his son because his son was just diagnosed with cancer. What's the probability that his other child is a girl?

/thread

This thread will never end.