Logic problem (1 Viewer)

[Sweet J]

Age is important only because if you DO know that the child we are talking about is "the older one" then the answer can never be 2/3, even if worded correctly. So, if he said: "a man has two children. At least one is a boy. It is the older child." Then the answer can't be 2/3.

Again, you have to be careful with the phrasing."a man has two children. At least one is a boy. It is the older child." is the same as"a man has two children. One or both are boys. It is the older child."

I agree. The minute you say that the older child is a boy/girl, that means we are JUST talking about the younger child. If we are JUST talking about the gender of one individual kid, it is 50/50.

 

[sartre]

If I was a teacher who gave this question on a test, and got a simple response of either 50/50 or 2/3, I would be forced to give credit for both. But I would give extra credit to someone who identified the problem that the reader. Is forced to make an assumption.I can undersrand why you would answer 50/50. But it still requires an assumption, just like 2/3 requires an assumption. I do NOT think the answer is "somewhere between 50/50 and 2/3" because it has to be one or the other. I'd say "the answer is either one or other, depending on a necessary assumption."

See my responses to bostonfred on the last page.Concluding the answer is 2/3 requires a major assumption -- that all fathers will talk about their sons before their daughters in all scenarios.Concluding the answer is 1/2 requires no assumption. To say that it does is disabling.

 

[Sweet J]

We should all go back and read the sub-title of this thread one more time.

You know I was thinking, that perhaps the OP came to the FFA with this because someone posed the question to him, and told him his 2/3 answer was incorrect. That's why he included the "not a trick question" (it is) and "you don't know which is older or younger" (red herring) bits.Pure speculation on my part, of course.

The older/younger parts are not red herrings in the classic problem. It is vitally important information. I think in this case, it was an unintentional red herring because of the ambiguity of the OP.

Wait... are we back on "age is a magical property"?You don't need the older/younger information, it just helps folks understand that individuals are being addressed, as opposed to the set. The bare bones classic problem can be simply stated:A man has two children. At least one is a boy. What are the odds that he has a daughter?

I agree. What I'm saying is that the classical problem that you wrote would have a different answer if you knew the boy or girl was older. That's all.The way you wrote the problem is better.

I would disagree that the way satre wrote the problem results in the classic 2/3 answer.If I look at two of my cards and tell you, "at least one of them is black" that really means nothing other than I decided to tell you that one of the cards was black. I could have said one of the cards was red. I think a better wording of the classic problem is:A man has two kids. If you ask him if at least one is a boy and he says "yes," what are the odds he also has a daughter.

I don't see a difference between the two. Both are 2/3.

It is getting down to the real nitty gritty, and I'm still struggling with it myself. But since we beat this horse for this long we might as well go a bit further.I would contend that as long as I have an option to tell you about either my boy or my girl (or my red card or my black card), then we still have a 50/50 situation. So if I look at two cards and have the CHOICE to tell you about either one, and I say one of them is black, the the odds are 50/50 that the other one is red. This assumes that I was just as likely to tell you about a girl as I was a boy (or my red card as opposed to my black card).If however, I do NOT have a choice to reveal whichever one I want, and I am instead asked if I have a black card (or a son) and I must answer correctly, then the odds of the other being a girl (or a red card) is 2/3.So if someone says that a guy has two kids and at least one of them is a boy, he peeked at both kids (or cards) and decided to tell me about the boy. I would say this results in a 50/50 situation.

I don't disagree with you on principle, but I think you have it ###-backwards. Satre's question would be 2/3. Yours (might be) 50%, based on what you say. There is no choice in satre's classical version.

So in Satre's version, let me ask it this way.A guy looks at your family and sees you have two kids. He then walks up and tells me, "at least one kid is a boy." I would say the odds are still 50/50 the other one is a girl.Because if one was a girl, the man could have just as easily revealed to me that "at least one kids is a girl." I think when you have the choice to reveal either the boy or the girl, you are still in a 50/50 situation.Just as in this situation. A guy looks at two cards and knows both of them. He then lays them both face down and picks one and tells me, "at least one is 'the suit I picked.'" What are the odds the other card is the opposite. It will be 50/50.Just knowing both cards doesn't matter. HOW you choose to reveal one of the cards is what is important.

I agree that a "ramdom" decision to tell you about a boy or girl is 50%, but I disagree that saying "one of the kids is a boy" can be interpreted as a "random selection of the gender."I think the easiest way to conceptualize it is to put it in terms of the "reveal" about "hypothetical families."Suppose we have a "reveal" of 100 hypothetical two person families. If we put it in terms of "A man has two children. At least one is a boy," then according to the construct of our statement, we will have revealed that this person has a boy 50 out of 75 times, for 2/3. Of course, this also means that we will have "revealed" that the person would have a girl 50/75 times, if we were going to talk about a "girl." So, I agree that the person needs to reveal that "one of the children is a boy" every time there is a boy. And that is how I interpret sarte's statement. Simply put, I don't think it is a reasonable interpretation of the statement "one of the children is a boy" to say "I have randomly decided to tell you one of the children is a boy."We could in good faith declare that the OP's initial question is vague, because a "reasonable person" could interpret "a man tells you he has a son" to mean that he randomly decides to tell you he has a son. However, I don't think it would be a reasonable argument to say "one of the children is a boy" as meaning "I have randomly decided to tell you one of the children is a boy."

 

[kutta]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem. The author of the problem could have just as easily said, "A man has two children. At least one is a girl..." Again, there was a choice made, in this case by the author of the problem, and once a choice is made, the odds go to 50/50.Let's use cards. I look at both cards and say, "At least one is black." No one asked if one was black. I VOLUNTEERED the info. This means I had a choice, which implies a I could have told you I had a red card. As soon as I am able to make a choice of which card (or child) to reveal, it becomes 50/50. If someone asks if I have a black card and I say yes, the odds go to 2/3.

 

[rick6668]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem. The author of the problem could have just as easily said, "A man has two children. At least one is a girl..." Again, there was a choice made, in this case by the author of the problem, and once a choice is made, the odds go to 50/50.Let's use cards. I look at both cards and say, "At least one is black." No one asked if one was black. I VOLUNTEERED the info. This means I had a choice, which implies a I could have told you I had a red card. As soon as I am able to make a choice of which card (or child) to reveal, it becomes 50/50. If someone asks if I have a black card and I say yes, the odds go to 2/3.

I disagree with this. It does not mean you had a choice, I have no idea of your motivation of why you choose to say "At least one is black". Maybe everytime you look at the two cards and at least one is black, you say, "At least one is black" and when they are both red, you say, "At least one is red".

 

[kutta]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem. The author of the problem could have just as easily said, "A man has two children. At least one is a girl..." Again, there was a choice made, in this case by the author of the problem, and once a choice is made, the odds go to 50/50.Let's use cards. I look at both cards and say, "At least one is black." No one asked if one was black. I VOLUNTEERED the info. This means I had a choice, which implies a I could have told you I had a red card. As soon as I am able to make a choice of which card (or child) to reveal, it becomes 50/50. If someone asks if I have a black card and I say yes, the odds go to 2/3.

I disagree with this. It does not mean you had a choice, I have no idea of your motivation of why you choose to say "At least one is black". Maybe everytime you look at the two cards and at least one is black, you say, "At least one is black" and when they are both red, you say, "At least one is red".

Right. But the fact is that there is no guarantee that if at least one card is black you will be privy to that information. I could choose to tell you, or I could choose not to tell you.I would contend that the answer is 2/3 if, and only if, on every occasion a black card (or a boy) is in the stack (or family), the person answering the question is aware of it. I can see it could be argued that stating the problem the way Satre said it the answer could be 2/3.But I think if you change "a man" to "satre" and phrase it the way I did, the answer is not 2/3.So to be clear:1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?2. I walk into Satre's house and see Satre has two children and then tell you: Satre has two children. At least one of them is a boy. What are the odds the other one is a girl?3. General statement: In all the cases of two child families with one boy, what are the odds the second child is a girl?I would contend the odds on #2 is 50/50. The odds on #1 can be argued to be either 50/50 or 2/3, but if we agree they are the same statement, then #1 is 50/50. If we agree that #1 is closer to #3, then I would agree #1 is 2/3. But I would contend #1 is closer to #2 than it is to #3.

 

[sartre]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"

Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?

If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem.

Disagree. There is a difference between saying "one of them is a boy" in which case you are assigning a property to an individual, and "at least one of them is a boy" in which case you've assigned a property to the {set} of the man's children.1) A man has two kids. What are the odds that he has a daughter?

BB, BG, GB, GG = 3/4

2) A man has two kids. At least one of them is a boy. What are the odds that he has a daughter?

BB, BG, GB = 2/3

3) A man has two kids. One of them is a boy. What are the odds that he has a daughter?

(B)B, (B)G = 1/2

When we say "at least one of them is" then we are offering knowledge of the set -- not the individuals. When we say "one of them is" we are offering knowledge of an individual, but no information about the others in the set. I admit this concept is hard to wrap one's mind around.

 

[sartre]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.ETA: so is this:

2. Satre has two children. At least one of them is a boy. What are the odds the other one is a girl

Even choice #3 seems like it needs rephrasing. The second child?

 

[kutta]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

 

[sartre]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

It's your use of the phrase "the other one", which implies we've already identified an individual.See this post

 

[kutta]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"

Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?

If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem.

Disagree. There is a difference between saying "one of them is a boy" in which case you are assigning a property to an individual, and "at least one of them is a boy" in which case you've assigned a property to the {set} of the man's children.1) A man has two kids. What are the odds that he has a daughter?

BB, BG, GB, GG = 3/4

2) A man has two kids. At least one of them is a boy. What are the odds that he has a daughter?

BB, BG, GB = 2/3

3) A man has two kids. One of them is a boy. What are the odds that he has a daughter?

(B)B, (B)G = 1/2

When we say "at least one of them is" then we are offering knowledge of the set -- not the individuals. When we say "one of them is" we are offering knowledge of an individual, but no information about the others in the set. I admit this concept is hard to wrap one's mind around.

I disagree with your definition.I have two cards and look at one of them and it is black. It is perfectly OK for me to say "at least one of these cards is black." I look out my window and see a Ford. It is perfectly acceptable for me to say, "There is at least one Ford in the parking lot." I do not need to know anything about any of the other cars in the parking lot.

You do not need knowledge of the complete set to use the phrase "at least."

 

[kutta]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

It's your use of the phrase "the other one", which implies we've already identified an individual.See this post

But we have. I think this will come out in the "at least one" debate in the other post, so I'll let this go for now.ETA: I don't necessarily agree with the post you linked either. Because I don't think you need knowledge of the complete set to say "at least."

 

[Sweet J]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

He is really, really, splitting hairs. He is saying that when you say "one of the children," that you mean a specific child. Ie, the older OR the younger, but not "one of the two."He is saying there is a difference between stating "one is a boy, is there at least a girl in the family?" and "one is a boy, what is the other." Because the term "the other" implies some specificity. I understand, but still think it's splitting hairs on something that is meant to reflect a simple mathmatical probability. It's not a grammar test.

 

[kutta]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

He is really, really, splitting hairs. He is saying that when you say "one of the children," that you mean a specific child. Ie, the older OR the younger, but not "one of the two."He is saying there is a difference between stating "one is a boy, is there at least a girl in the family?" and "one is a boy, what is the other." Because the term "the other" implies some specificity. I understand, but still think it's splitting hairs on something that is meant to reflect a simple mathmatical probability. It's not a grammar test.

But why take the fun out of it? We've beat this thing to absolute hell. I'd like to get resolution on all variations of the problem.I actually think we are understanding this problem better than most of the "math and logic" sites that have published it.

 

[sartre]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"

Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?

If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem.

Disagree. There is a difference between saying "one of them is a boy" in which case you are assigning a property to an individual, and "at least one of them is a boy" in which case you've assigned a property to the {set} of the man's children.1) A man has two kids. What are the odds that he has a daughter?

BB, BG, GB, GG = 3/4

2) A man has two kids. At least one of them is a boy. What are the odds that he has a daughter?

BB, BG, GB = 2/3

3) A man has two kids. One of them is a boy. What are the odds that he has a daughter?

(B)B, (B)G = 1/2

When we say "at least one of them is" then we are offering knowledge of the set -- not the individuals. When we say "one of them is" we are offering knowledge of an individual, but no information about the others in the set. I admit this concept is hard to wrap one's mind around.

I disagree with your definition.I have two cards and look at one of them and it is black. It is perfectly OK for me to say "at least one of these cards is black." I look out my window and see a Ford. It is perfectly acceptable for me to say, "There is at least one Ford in the parking lot." I do not need to know anything about any of the other cars in the parking lot.

You do not need knowledge of the complete set to use the phrase "at least."

You could certainly do that, but you would be throwing information away. You know that specific card or that specific Ford exists. I could say x = 25, or I could say x > 0. Neither are wrong, but the first contains more information.

 

[sartre]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

It's your use of the phrase "the other one", which implies we've already identified an individual.See this post

But we have. I think this will come out in the "at least one" debate in the other post, so I'll let this go for now.ETA: I don't necessarily agree with the post you linked either. Because I don't think you need knowledge of the complete set to say "at least."

Like I said, the problem is saying "at least one of" and then COMBINING it with "the other one". When you say "at least one of" you have not identified any individual from which to distinguish "the other one"Joe has two children. Both are Cajun. What are the odds that the other one is Cajun?

Does that question make any sense?

 

[sartre]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

He is really, really, splitting hairs. He is saying that when you say "one of the children," that you mean a specific child. Ie, the older OR the younger, but not "one of the two."He is saying there is a difference between stating "one is a boy, is there at least a girl in the family?" and "one is a boy, what is the other." Because the term "the other" implies some specificity. I understand, but still think it's splitting hairs on something that is meant to reflect a simple mathmatical probability. It's not a grammar test.

It's the language of logic. And IMO, it is really important that we use it correctly if we want to keep these hypos straight.

 

[kutta]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"

Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?

If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem.

Disagree. There is a difference between saying "one of them is a boy" in which case you are assigning a property to an individual, and "at least one of them is a boy" in which case you've assigned a property to the {set} of the man's children.1) A man has two kids. What are the odds that he has a daughter?

BB, BG, GB, GG = 3/4

2) A man has two kids. At least one of them is a boy. What are the odds that he has a daughter?

BB, BG, GB = 2/3

3) A man has two kids. One of them is a boy. What are the odds that he has a daughter?

(B)B, (B)G = 1/2

When we say "at least one of them is" then we are offering knowledge of the set -- not the individuals. When we say "one of them is" we are offering knowledge of an individual, but no information about the others in the set. I admit this concept is hard to wrap one's mind around.

I disagree with your definition.I have two cards and look at one of them and it is black. It is perfectly OK for me to say "at least one of these cards is black." I look out my window and see a Ford. It is perfectly acceptable for me to say, "There is at least one Ford in the parking lot." I do not need to know anything about any of the other cars in the parking lot.

You do not need knowledge of the complete set to use the phrase "at least."

You could certainly do that, but you would be throwing information away. You know that specific card or that specific Ford exists. I could say x = 25, or I could say x > 0. Neither are wrong, but the first contains more information.

I'm not quite sure what you mean by this.If I have two cards and know one of them is black, there is absolutely nothing wrong with me saying "at least one is black." Nothing at all. Nothing in that statement requires that I know the sex of the other card.

Let's stick with this example and discuss it, because I really would like to know why you are insistent that "at least one" implies knowledge of both.

So say a few more words about this if you could.

 

[kutta]

1. General statement: A man has two children. At least one of them is a boy. What are the odds the other one is a girl?

Ack! This is unanswerable.

Now I'm getting confused.Here was your statement:

A man has two children. At least one is a boy. What are the odds that he has a daughter?

I'm not sure how my statement and your statement differ.

He is really, really, splitting hairs. He is saying that when you say "one of the children," that you mean a specific child. Ie, the older OR the younger, but not "one of the two."He is saying there is a difference between stating "one is a boy, is there at least a girl in the family?" and "one is a boy, what is the other." Because the term "the other" implies some specificity. I understand, but still think it's splitting hairs on something that is meant to reflect a simple mathmatical probability. It's not a grammar test.

It's the language of logic. And IMO, it is really important that we use it correctly if we want to keep these hypos straight.

OK. Let's go with that. I don't remember that from my logic class. I don't remember learning that you must have complete knowledge of the set in order to use "at least." Let's find a good "logic link" or source for this. I'll start looking.

 

[sartre]

"A man has two children. At least one is a boy. What are the odds that he has a daughter?"

Nobody offers anything here. It is just known information.

I walk into your house and see you have two kids.I walk out and say, "Satre has two kids. At least one is a boy." In this case, I would contend it is 50/50 that he has a girl. Agree or disagree?

If you agree, then the statement, "A man has two children..." is very very very similar. I would grant there is a slight difference, but I am not convinced that difference matters in this problem.

Disagree. There is a difference between saying "one of them is a boy" in which case you are assigning a property to an individual, and "at least one of them is a boy" in which case you've assigned a property to the {set} of the man's children.1) A man has two kids. What are the odds that he has a daughter?

BB, BG, GB, GG = 3/4

2) A man has two kids. At least one of them is a boy. What are the odds that he has a daughter?

BB, BG, GB = 2/3

3) A man has two kids. One of them is a boy. What are the odds that he has a daughter?

(B)B, (B)G = 1/2

When we say "at least one of them is" then we are offering knowledge of the set -- not the individuals. When we say "one of them is" we are offering knowledge of an individual, but no information about the others in the set. I admit this concept is hard to wrap one's mind around.

I disagree with your definition.I have two cards and look at one of them and it is black. It is perfectly OK for me to say "at least one of these cards is black." I look out my window and see a Ford. It is perfectly acceptable for me to say, "There is at least one Ford in the parking lot." I do not need to know anything about any of the other cars in the parking lot.

You do not need knowledge of the complete set to use the phrase "at least."

You could certainly do that, but you would be throwing information away. You know that specific card or that specific Ford exists. I could say x = 25, or I could say x > 0. Neither are wrong, but the first contains more information.

I'm not quite sure what you mean by this.If I have two cards and know one of them is black, there is absolutely nothing wrong with me saying "at least one is black." Nothing at all. Nothing in that statement requires that I know the sex of the other card.

Let's stick with this example and discuss it, because I really would like to know why you are insistent that "at least one" implies knowledge of both.

So say a few more words about this if you could.

You could absolutely say "at least one of these cards is black". I'm not saying you need to see the other card to make that statement. But do you see how "this card is black" is more informative? All you are doing is reducing the amount of information in your statement.I have to get going now but will try to expand later.

 

[sartre]

Like I said, the problem is saying "at least one of" and then COMBINING it with "the other one". When you say "at least one of" you have not identified any individual from which to distinguish "the other one"

Joe has two children. Both are Cajun. What are the odds that the other one is Cajun?

Does that question make any sense?

kutta, I'd really like you to consider this.

 

[kutta]

Like I said, the problem is saying "at least one of" and then COMBINING it with "the other one". When you say "at least one of" you have not identified any individual from which to distinguish "the other one"

Joe has two children. Both are Cajun. What are the odds that the other one is Cajun?

Does that question make any sense?

kutta, I'd really like you to consider this.

OK. I see your point on the wording. I'm totally cool with that.But I don't (yet) agree that the problem changes with this.

If I look at two cards and see both of them and I say, "at least one is black, what are the odds there is a red card under here?" we still have the same issue of having a choice to reveal one of the "at leasts."

I also have to run so I'll pick this up later too.

But I'm cool on the wording as explained above.

 

[kutta]

3 Members: kutta, Maurile Tremblay, sartre

 

[Maurile Tremblay]

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)

As phrased, the answer is probably 2/3, but it depends on exactly what the guy said.He could have said, "One of my children is in the military and the other isn't. I won't tell you which is older, but I will tell you that the one in the military is male." In that case, the chance that the other child is female is 50%.Or he could have said, "At least one of my children is male. That's all I'm telling you." In that case, the chance that the other child is female is 66.67%.

 

[kutta]

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)

As phased, the answer is probably 2/3, but it depends on exactly what the guy said.He could have said, "One of my children is in the military and the other isn't. I won't tell you which is older, but I will tell you that the one in the military is male." In that case, the chance that the other child is female is 50%.Or he could have said, "At least one of my children is male. That's all I'm telling you." In that case, the chance that the other child is female is 66.67%.

To get you totally caught up on the thread:Using cards instead of kids, if I have black and red cards:BBRBBRRRAnd I randomly select a pair, and then randomly select a card, what are the odds the other card is of an opposite suit?

 

[Maurile Tremblay]

The answer is 2/3. If a guy has two children, there are four equally-likely possibilities for their sequence of birth and genders:

boy-boy

boy-girl

girl-boy

girl-girl

We know the last one didn't happen, but the remaining three are all still equally likely, so 2/3 of the time his other kid is a girl. And there's a probability of 1 that she can't drive as well as most men.

What if the question was phrased as:A man and his son walk up to you and say hello. The man says he has another child at home. What are the odds the other child is a girl?

Wouldn't it be 50/50 in that case?

No, it's still 2/3.

This is not correct. In this case, it's 50-50.Saying, "the child currently with him is male" is different from saying "at least one of his children is male."

It's a Bayes' theorem problem.

P(A|B) = P(B|A) * P(A) / P(B)

Let's solve for the probability that both children are male, given that the the child currently with him is male.

A = both children are male

B = the child currently with him is male

P(A|B) = 100% * 25% / 50% = 50%

On the other hand, if we solve for the probability that both children are male given that at least one child is male . . .

A = both children are male

B = at least one child is male

P(A|B) = 100% * 25% / 75% = 33.33%

(Which means that the probability that the other child is female is 66.67%)

We've got four a priori possibilities, where the child currently with the father is in the left column and the other is in the right colum:

BB

BG

GB

GG

"The child currently with the father is male" is true for 50% of the cases while "at least one child is male" is true for 75% of the cases. That's the difference.

 

[Maurile Tremblay]

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)

As phased, the answer is probably 2/3, but it depends on exactly what the guy said.He could have said, "One of my children is in the military and the other isn't. I won't tell you which is older, but I will tell you that the one in the military is male." In that case, the chance that the other child is female is 50%.Or he could have said, "At least one of my children is male. That's all I'm telling you." In that case, the chance that the other child is female is 66.67%.

To get you totally caught up on the thread:Using cards instead of kids, if I have black and red cards:BBRBBRRRAnd I randomly select a pair, and then randomly select a card, what are the odds the other card is of an opposite suit?

Fifty percent. See the Bayesian analysis in my last post. "The card we look at first" will be in the left column while "the card we don't look at first" will be in the right column.BBRBBRRR"The card we look at first will be black" will be true 50% of the time (not 75% of the time, as would be the case with "at least one card is black") -- so the other card will be red 50% (not 67%) of the time.

 

[FatMax]

About time Maurile showed up.

 

[kutta]

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)

As phased, the answer is probably 2/3, but it depends on exactly what the guy said.He could have said, "One of my children is in the military and the other isn't. I won't tell you which is older, but I will tell you that the one in the military is male." In that case, the chance that the other child is female is 50%.Or he could have said, "At least one of my children is male. That's all I'm telling you." In that case, the chance that the other child is female is 66.67%.

To get you totally caught up on the thread:Using cards instead of kids, if I have black and red cards:BBRBBRRRAnd I randomly select a pair, and then randomly select a card, what are the odds the other card is of an opposite suit?

Fifty percent. See the Bayesian analysis in my last post. "The card we look at first" will be in the left column while "the card we don't look at first" will be in the right column.BBRBBRRR"The card we look at first will be black" will be true 50% of the time (not 75% of the time, as would be the case with "at least one card is black") -- so the other card will be red 50% (not 67% of the time).

Thank you. Where were you 23 pages ago?

 

[Maurile Tremblay]

Holy crap. We're on post #1228?

I just read the first four pages before realizing I still had 37 more to go . . . .

 

[kutta]

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)

As phased, the answer is probably 2/3, but it depends on exactly what the guy said.He could have said, "One of my children is in the military and the other isn't. I won't tell you which is older, but I will tell you that the one in the military is male." In that case, the chance that the other child is female is 50%.Or he could have said, "At least one of my children is male. That's all I'm telling you." In that case, the chance that the other child is female is 66.67%.

To get you totally caught up on the thread:Using cards instead of kids, if I have black and red cards:BBRBBRRRAnd I randomly select a pair, and then randomly select a card, what are the odds the other card is of an opposite suit?

Fifty percent. See the Bayesian analysis in my last post. "The card we look at first" will be in the left column while "the card we don't look at first" will be in the right column.BBRBBRRR"The card we look at first will be black" will be true 50% of the time (not 75% of the time, as would be the case with "at least one card is black") -- so the other card will be red 50% (not 67%) of the time.

So what I'm struggling with a little bit is the wording of "at least one card is black." What is the difference between the scenario above, and the scenario where I look at one of the cards and then tell you "at least one of the cards is black. What are the odds there is also a red card?"Or the scenario where I look at both cards and make a choice to tell you that "at least one of the cards is black."I completely understand the situation where I look at both cards and I am asked if I have a black card. I must answer yes 100% of the time I have a black card in my hand, so the odds of the other being red is 2/3. But if I have a choice of which color to reveal, (i.e. at least one is black, or at least one is red), it would seem to me the odds would be 50/50 on also having the other color.

 

[kutta]

Holy crap. We're on post #1228?I just read the first four pages before realizing I still had 37 more to go . . . .

Yeah, it was pretty rough in here for awhile.

 

[Sweet J]

Maurile, we are stuck on this:

You lay out 400 pairs (100 bb, 100 br, 100 rb, 100 rr). You flip them over randomly (either first or second card), so there should be 200 black facing up. If it is black, you say to me "there is at least one black card, what are the odds that the other is red" This is 50%, because there are only 200 "reveals," (i.e., 200 of the 400 cards that are turned up are black). You told me it was black in the 100 times it was a BB pair (so 100 times the other card is black), 50 times it was a BR pair, and 50 times RB pair (So 100 times the other card was red). So, 100 out of 200 times the other card is Red, for 50/50. We all seem to have agreed on this. Do you agree?

If you picked up the cards, look at both of them, then put them down, and then you will then tell me that a card is black whenever a card is black. This is a 2/3 chance that the other card is red, because 100 out of 100 times we have a BB pile we say it's black, 100/100 times we have a RB pile we'll say it's black, and 100/100 times we have a BR pile we'll say it's black. So 200 out of every 300 times there was a black card in the pile, there is also a red card. So - 66%.

What about this, though: What if in the second scenario, you look at both cards, then decide in your head that you are going to tell me: "one of the cards is X (either red or black)." Does it follow that the other card is opposite 66% of the time, or 50%? Why? I'm can't seem to accept 50%, but it looks more like scenario 1. Help?

Similarly, When you tell me "At least one of the children is a boy," does it matter that you randomly decided to tell me it was a boy or a girl in the Boy/Girl families?

 

[Maurile Tremblay]

So what I'm struggling with a little bit is the wording of "at least one card is black." What is the difference between the scenario above, and the scenario where I look at one of the cards and then tell you "at least one of the cards is black. What are the odds there is also a red card?"

If you look at only one card and then say "at least one card is black," you're not giving me all the information you have. You should have told me, "I know that at least one card is black, and I know this from looking only at one card, not both."If you look at two cards, you'll tell me "at least one is black" 75% of the time. If you look at just one card, you can tell me "at least one card is black" only 50% of the time. When you tell me "at least one card is black," you're eliminating only 25% of the cases -- but if in fact you've only looked at one card, you could properly eliminate 50% of the cases, so you're not giving me all of the relevant information.

Or the scenario where I look at both cards and make a choice to tell you that "at least one of the cards is black."

Yes, this case is different from looking at just one card.

I completely understand the situation where I look at both cards and I am asked if I have a black card. I must answer yes 100% of the time I have a black card in my hand, so the odds of the other being red is 2/3. But if I have a choice of which color to reveal, (i.e. at least one is black, or at least one is red), it would seem to me the odds would be 50/50 on also having the other color.

Right. "At least one is black" is different from "the one on the left is black" or "the one I looked at first is black" or whatever. In the first case ("at least one is black"), the cards are opposite colors 67% of the time, while in the other cases (e.g., "the one I looked at first is black"), the cards are opposite colors 50% of the time.

 

[Sweet J]

So what I'm struggling with a little bit is the wording of "at least one card is black." What is the difference between the scenario above, and the scenario where I look at one of the cards and then tell you "at least one of the cards is black. What are the odds there is also a red card?"

If you look at only one card and then say "at least one card is black," you're not giving me all the information you have. You should have told me, "I know that at least one card is black, and I know this from looking only at one card, not both."If you look at two cards, you'll tell me "at least one is black" 75% of the time. If you look at just one card, you can tell me "at least one card is black" only 50% of the time. When you tell me "at least one card is black," you're eliminating only 25% of the cases -- but if in fact you've only looked at one card, you could properly eliminate 50% of the cases, so you're not giving me all of the relevant information.

Or the scenario where I look at both cards and make a choice to tell you that "at least one of the cards is black."

Yes, this case is different from looking at just one card.

I completely understand the situation where I look at both cards and I am asked if I have a black card. I must answer yes 100% of the time I have a black card in my hand, so the odds of the other being red is 2/3. But if I have a choice of which color to reveal, (i.e. at least one is black, or at least one is red), it would seem to me the odds would be 50/50 on also having the other color.

Right. "At least one is black" is different from "the one on the left is black" or "the one I looked at first is black" or whatever. In the first case ("at least one is black"), the cards are opposite colors 67% of the time, while in the other cases (the one I looked at first is black"), the cards are opposite colors 50% of the time.

Maurile, the problem is that when he picks up both cards and looks at them, and they are a RB or BR pair, he must make a decision on whether to tell you "at least one of the cards is black," or "at least one of the cards is red." He can't do both. So the question is, by "randomly" deciding what color to tell you in RB and BR situations, does that make different results than if he ALWAYS tells you "at least one card is black" in RB BR situations.

 

[Sweet J]

If you look at two cards, you'll tell me "at least one is black" 75% of the time.

The problem with this conclusion is that it assumes that every time you look at two cards and you have RB or BR, you will choose black. If you randomly decide to tell me "red" or "black," does that change the analysis?In a nutshell, it seems like saying the statement "at least one of the cards is black" appears to be significantly different than responding affirmatively to the question "Is at least one card black."Is that right?

 

[Maurile Tremblay]

Maurile, we are stuck on this: You lay out 400 pairs (100 bb, 100 br, 100 rb, 100 rr).

Why not just four pairs? But either way . . .

You flip them over randomly (either first or second card), so there should be 200 black facing up.

Yes.

If it is black, you say to me "there is at least one black card, what are the odds that the other is red"

Actually, I would say "The one I flipped up first is black." Otherwise, I'm withholding relevant information, thus preventing you from updating your probability estimates as well as you otherwise could.

This is 50%, because there are only 200 "reveals," (i.e., 200 of the 400 cards that are turned up are black). You told me it was black in the 100 times it was a BB pair (so 100 times the other card is black), 50 times it was a BR pair, and 50 times RB pair (So 100 times the other card was red).

You can label them like that, but you're losing information that way. A better way to label them is that the letter on the left is the first card you turn over and the letter on the right is the second card you turn over.So out of the 400 pairs, I've turned over 200 blacks: the 100 in the BB pairs and the 100 in the BR pairs. I've turned over zero blacks in the the RR pairs and zero blacks in the RB pairs.

So, 100 out of 200 times the other card is Red, for 50/50. We all seem to have agreed on this. Do you agree?

Yes. Specifically, it's red in each of the 100 BR pairs.

If you picked up the cards, look at both of them, then put them down, and then you will then tell me that a card is black whenever a card is black. This is a 2/3 chance that the other card is red, because 100 out of 100 times we have a BB pile we say it's black, 100/100 times we have a RB pile we'll say it's black, and 100/100 times we have a BR pile we'll say it's black. So 200 out of every 300 times there was a black card in the pile, there is also a red card. So - 66%.

Yes.

What about this, though: What if in the second scenario, you look at both cards, then decide in your head that you are going to tell me: "one of the cards is X (either red or black)." Does it follow that the other card is opposite 66% of the time, or 50%? Why? I'm can't seem to accept 50%, but it looks more like scenario 1. Help?

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

Similarly, When you tell me "At least one of the children is a boy," does it matter that you randomly decided to tell me it was a boy or a girl in the Boy/Girl families?

It doesn't matter whether it's random or non-random. But what you can't do is change what category of information you're going to give me depending on what you see (unless you tell me you're going to do so).For example:RR - "there's at least one red"RB - "there's at least one red"BR - "there's at least one red"BB - "there's at least one black"If that's the legend you always stick to, then there will be mixed colors zero percent (not 67%) of the time that you tell me that there's at least one black.

 

[Sweet J]

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

Kutta, are you following? Because I'm not.Maurile, how is this not 50/50?You look at the cards 100 times. 25 times you will look at BB and say "at least one card is black." 25 times you will look at RR and say "at least one card is Red."However, 12.5 times you will look at BR and say "at least one card is Black" and 12.5 times you will look at BR and say "at least one card is red."Simlarly, 12.5 times you will look at RB and say "at least one card is Black," and 12.5 times you will look at RB and say "at least one card is red."So, 100 looks. 50 times you've looked at a pair with a black card and said "at least one card is black." Of those times, the other card will be red 25 times, for 50%. And 50 times you've looked at a pair with a red card and said "at least one card is Red." Of those times, the other card will be black 25 times, for 50%.So it sounds like the statement "at least one card is black" is incomplete. You need to know that every time a black card is present, you will say "at least one is black." Which is equivelent to answering the affirmative to the quesiton "is at least one card black"?

 

[kutta]

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

Kutta, are you following? Because I'm not.Maurile, how is this not 50/50?You look at the cards 100 times. 25 times you will look at BB and say "at least one card is black." 25 times you will look at RR and say "at least one card is Red."However, 12.5 times you will look at BR and say "at least one card is Black" and 12.5 times you will look at BR and say "at least one card is red."Simlarly, 12.5 times you will look at RB and say "at least one card is Black," and 12.5 times you will look at RB and say "at least one card is red."So, 100 looks. 50 times you've looked at a pair with a black card and said "at least one card is black." Of those times, the other card will be red 25 times, for 50%. And 50 times you've looked at a pair with a red card and said "at least one card is Red." Of those times, the other card will be black 25 times, for 50%.So it sounds like the statement "at least one card is black" is incomplete. You need to know that every time a black card is present, you will say "at least one is black." Which is equivelent to answering the affirmative to the quesiton "is at least one card black"?

I agree with Sweet J here.I think.

 

[kutta]

So what I'm struggling with a little bit is the wording of "at least one card is black." What is the difference between the scenario above, and the scenario where I look at one of the cards and then tell you "at least one of the cards is black. What are the odds there is also a red card?"

If you look at only one card and then say "at least one card is black," you're not giving me all the information you have. You should have told me, "I know that at least one card is black, and I know this from looking only at one card, not both."If you look at two cards, you'll tell me "at least one is black" 75% of the time. If you look at just one card, you can tell me "at least one card is black" only 50% of the time. When you tell me "at least one card is black," you're eliminating only 25% of the cases -- but if in fact you've only looked at one card, you could properly eliminate 50% of the cases, so you're not giving me all of the relevant information.

Or the scenario where I look at both cards and make a choice to tell you that "at least one of the cards is black."

Yes, this case is different from looking at just one card.

I completely understand the situation where I look at both cards and I am asked if I have a black card. I must answer yes 100% of the time I have a black card in my hand, so the odds of the other being red is 2/3. But if I have a choice of which color to reveal, (i.e. at least one is black, or at least one is red), it would seem to me the odds would be 50/50 on also having the other color.

Right. "At least one is black" is different from "the one on the left is black" or "the one I looked at first is black" or whatever. In the first case ("at least one is black"), the cards are opposite colors 67% of the time, while in the other cases (e.g., "the one I looked at first is black"), the cards are opposite colors 50% of the time.

The bolded part above is what I don't agree with. If I have a BR or a RB pair, I may not tell you "at least one card is black." I might say, "at least one card is red." That is where the confusion is coming in. If I am ASKED if I have a black card, I MUST tell you yes in ALL RB/BR cases. But if I am looking at both, I don't understand why I would be required to say "at least one is black" in all those cases.ETA: or what Sweet J said above. Same thing.

 

[kutta]

For example:RR - "there's at least one red"RB - "there's at least one red"BR - "there's at least one red"BB - "there's at least one black"If that's the legend you always stick to, then there will be mixed colors zero percent (not 67%) of the time that you tell me that there's at least one black.

I (and Sweet J, I think) would contend we would have the following:RR - "there's at least one red" 100% of the timeRB - "there's at least one red" 50% of the time - "there is at least one black" 50% of the timeBR - "there's at least one red" 50% of the time - "there is at least one black" 50% of the timeBB - "there's at least one black" 100% of the timeWithout any prompting (i.e. "Do you have a black card?") how do we know the moderator will tell us about the black or the red card in the RB and BR situations?

 

[sartre]

He then starts talking about his son.

After hours in this thread I think it all boils down to this.To me, the above statement can mean only one thing. An individual male child is specified. It doesn't matter if the father says

* "One of my kids broke his arm"

* "That boy there is my son!"

* "My son loves pickles"

* "Mr. Pickles is my son"

Or he could have said, "At least one of my children is male. That's all I'm telling you."

If he says only "At least one of my children is a male" then he is making a statement about his children collectively. He is not "talking about his son".Further, if we allow "He then starts talking about his son" to be translated as "at least one of my children is a son", then question that follows "What is the probability that his other child is a girl" can have no answer. Because "the other child" has NO meaning if an individual isn't previously distinguished in the set.

If we can agree on that, then the puzzle can be simplified:

A man has two children. One child is a son. What is the probability that his other child is a girl?

As such, the answer can only be 50%.

 

[Maurile Tremblay]

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

Kutta, are you following? Because I'm not.Maurile, how is this not 50/50?You look at the cards 100 times. 25 times you will look at BB and say "at least one card is black." 25 times you will look at RR and say "at least one card is Red."

No you won't.You'll look at BB 25 times. Of those, half the time you'll have gotten heads, so you'll say "It's not the case that at least one is red." And the other half you'll have gotten tails and say, "At least one is red."

However, 12.5 times you will look at BR and say "at least one card is Black" and 12.5 times you will look at BR and say "at least one card is red."Simlarly, 12.5 times you will look at RB and say "at least one card is Black," and 12.5 times you will look at RB and say "at least one card is red."So, 100 looks. 50 times you've looked at a pair with a black card and said "at least one card is black." Of those times, the other card will be red 25 times, for 50%. And 50 times you've looked at a pair with a red card and said "at least one card is Red." Of those times, the other card will be black 25 times, for 50%.

100 looks. 50 times you'll have flipped heads. Of those, 37.5 times you'll look at a pair with a black card and say "there's at least one black card." And 25 of those times the other card will be red. 25 out of 37.5 is 67%.

 

[Maurile Tremblay]

He then starts talking about his son.

After hours in this thread I think it all boils down to this.To me, the above statement can mean only one thing. An individual male child is specified. It doesn't matter if the father says

* "One of my kids broke his arm"

* "That boy there is my son!"

* "My son loves pickles"

* "Mr. Pickles is my son"

Or he could have said, "At least one of my children is male. That's all I'm telling you."

If he says only "At least one of my children is a male" then he is making a statement about his children collectively. He is not "talking about his son".Further, if we allow "He then starts talking about his son" to be translated as "at least one of my children is a son", then question that follows "What is the probability that his other child is a girl" can have no answer. Because "the other child" has NO meaning if an individual isn't previously distinguished in the set.

If we can agree on that, then the puzzle can be simplified:

A man has two children. One child is a son. What is the probability that his other child is a girl?

As such, the answer can only be 50%.

Agreed. The phrase "the other child" doesn't have a precise meaning -- although I think we informally know what it means in context. (It means "If I have a boy and a girl, it's the girl; if I have two sons, it's one of them.") But strictly speaking, we really can't talk about the "other" child, so the original question should be re-phrased.

 

[Sweet J]

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

Kutta, are you following? Because I'm not.Maurile, how is this not 50/50?You look at the cards 100 times. 25 times you will look at BB and say "at least one card is black." 25 times you will look at RR and say "at least one card is Red."

No you won't.You'll look at BB 25 times. Of those, half the time you'll have gotten heads, so you'll say "It's not the case that at least one is red." And the other half you'll have gotten tails and say, "At least one is red."

However, 12.5 times you will look at BR and say "at least one card is Black" and 12.5 times you will look at BR and say "at least one card is red."Simlarly, 12.5 times you will look at RB and say "at least one card is Black," and 12.5 times you will look at RB and say "at least one card is red."So, 100 looks. 50 times you've looked at a pair with a black card and said "at least one card is black." Of those times, the other card will be red 25 times, for 50%. And 50 times you've looked at a pair with a red card and said "at least one card is Red." Of those times, the other card will be black 25 times, for 50%.

100 looks. 50 times you'll have flipped heads. Of those, 37.5 times you'll look at a pair with a black card and say "there's at least one black card." And 25 of those times the other card will be red. 25 out of 37.5 is 67%.

It seems I am still not phrasing my basic "problem" correctly. My appologies for the confusion. The cruxt of the problem that the moderator (for lack of a better term) has a "choice" to say "there is at least one red vs. "there is at least one black card" only arises when you have RB or BR. Otherwise, when he looks at BB he will always say "there is at least one black card" and when he looks at RR he will say "there is at least one red card." Assumptions:1. BB - Moderator looks at both cards. He puts them down. He will always say to you, "there is at least one Black."2. BR - Moderator looks at both cards. He puts them down. He "randomly" (i.e., flips a coin). Half the time he will tell you "there is at least one black."3. RB - ditto.4. RR. Moderator looks at both cards. He puts them down. He will always tell you "there is at least one Red."Kutta has posited that saying "there is at least one black card" is never enough information, because it is unclear if the moderator has used the above way of determining what to tell you (in contrast to the situation where the moderator will ALWAYS tell you "there is at least one black card" in each of the BB, BR, and RB situations).Is my problem being communicated properly?

 

[Sweet J]

100 looks. 50 times you'll have flipped heads. Of those, 37.5 times you'll look at a pair with a black card and say "there's at least one black card." And 25 of those times the other card will be red. 25 out of 37.5 is 67%.

I guess I'm not following, because I was under the impression that you were only "flipping a coin" (I.e., randomly deciding to reveal red or black cards) when faced with the RB or BR situations. When you have BB or RR, there is no "flip" necessary, because there is no choice. If you turn over BB, you can't say "there is at least one red," because, well, there isn't. I just assumed you say "there is at least one black" every time you turn over BB. What am I missing?

 

[Maurile Tremblay]

Is this the scenario you're asking about?

1. You look at both cards.

2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.

If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

Kutta, are you following? Because I'm not.Maurile, how is this not 50/50?

You look at the cards 100 times. 25 times you will look at BB and say "at least one card is black." 25 times you will look at RR and say "at least one card is Red."

No you won't.You'll look at BB 25 times. Of those, half the time you'll have gotten heads, so you'll say "It's not the case that at least one is red." And the other half you'll have gotten tails and say, "At least one is red."

However, 12.5 times you will look at BR and say "at least one card is Black" and 12.5 times you will look at BR and say "at least one card is red."

Simlarly, 12.5 times you will look at RB and say "at least one card is Black," and 12.5 times you will look at RB and say "at least one card is red."

So, 100 looks. 50 times you've looked at a pair with a black card and said "at least one card is black." Of those times, the other card will be red 25 times, for 50%. And 50 times you've looked at a pair with a red card and said "at least one card is Red." Of those times, the other card will be black 25 times, for 50%.

100 looks. 50 times you'll have flipped heads. Of those, 37.5 times you'll look at a pair with a black card and say "there's at least one black card." And 25 of those times the other card will be red. 25 out of 37.5 is 67%.

It seems I am still not phrasing my basic "problem" correctly. My appologies for the confusion. The cruxt of the problem that the moderator (for lack of a better term) has a "choice" to say "there is at least one red vs. "there is at least one black card" only arises when you have RB or BR. Otherwise, when he looks at BB he will always say "there is at least one black card" and when he looks at RR he will say "there is at least one red card."

OK, that's different from any of the hypos I'd contemplated before.

Assumptions:

1. BB - Moderator looks at both cards. He puts them down. He will always say to you, "there is at least one Black."

2. BR - Moderator looks at both cards. He puts them down. He "randomly" (i.e., flips a coin). Half the time he will tell you "there is at least one black."

3. RB - ditto.

4. RR. Moderator looks at both cards. He puts them down. He will always tell you "there is at least one Red."

OK, back to Bayes' theorem.P(A|B) = P(B|A) * P(A) / P(B)

What is the probability that the cards are of mixed color, given that the moderator told us that "at least one is black"?

A = the cards are of mixed color

B = the moderator tells us that at least one is black

Here, the moderator will tell us that at least one is black 25% of the time (BB) plus 12.5% of the time (RB) plus 12.5% of the time (BR) = 50%. Half of the time that he tells us that at least one is black, they will be mixed. (That is, 25% of the time, he will tell us that at least one is black and in fact they'll both be black; and another 25% of the time, he will tell us that at least one is black and in fact they'll be mixed. The other 50% of the time, he will not tell us that at least one is black.)

P(A) = 50% [i.e., the cards will be mixed 50 times out of 100]

P(B) = 50% [i.e., the moderator will say "at least one is black" 50 times out of 100]

So P(B|A) = 50% [i.e., the moderator will say "at least one is black" 25 times out of the 50 that they are mixed]

P(A|B) = 50% * 50% / 50% = 50%

Kutta has posited that saying "there is at least one black card" is never enough information, because it is unclear if the moderator has used the above way of determining what to tell you (in contrast to the situation where the moderator will ALWAYS tell you "there is at least one black card" in each of the BB, BR, and RB situations).

Is my problem being communicated properly?

If we don't know the moderator's rules, we can't really adjust our probabilities beyond the priors -- BB, RB, BR, RR each being 25%. I mean, the moderator could be lying and just always say "both are red" no matter what they really are. If we don't know what makes the moderator tick, we can't use any "information" he gives us.

 

[Maurile Tremblay]

100 looks. 50 times you'll have flipped heads. Of those, 37.5 times you'll look at a pair with a black card and say "there's at least one black card." And 25 of those times the other card will be red. 25 out of 37.5 is 67%.

I guess I'm not following, because I was under the impression that you were only "flipping a coin" (I.e., randomly deciding to reveal red or black cards) when faced with the RB or BR situations. When you have BB or RR, there is no "flip" necessary, because there is no choice. If you turn over BB, you can't say "there is at least one red," because, well, there isn't. I just assumed you say "there is at least one black" every time you turn over BB. What am I missing?

Here's the way I had it:

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

If the moderator flips heads, he tells us whether there's at least one red. If there isn't, he says so. He says "It's not the case that there's at least one red" (which gives us a lot more information than just saying "there is at least one black").I see now that you were envisioning a different scenario. (Which is why I asked.) But I think we're on the same page now.

 

[Texas Football]

I'll vote for anything with 666 in it!

It's obvious they are both girls and he wished he had a son. The answer is in the Onslaught thread.

 

[kutta]

100 looks. 50 times you'll have flipped heads. Of those, 37.5 times you'll look at a pair with a black card and say "there's at least one black card." And 25 of those times the other card will be red. 25 out of 37.5 is 67%.

I guess I'm not following, because I was under the impression that you were only "flipping a coin" (I.e., randomly deciding to reveal red or black cards) when faced with the RB or BR situations. When you have BB or RR, there is no "flip" necessary, because there is no choice. If you turn over BB, you can't say "there is at least one red," because, well, there isn't. I just assumed you say "there is at least one black" every time you turn over BB. What am I missing?

Here's the way I had it:

Is this the scenario you're asking about?1. You look at both cards.2. You flip a coin. Heads = you tell me whether there's at least one red. Tails = you tell me whether there's at least one black.If that's the scenario, they'll be mixed colors 67% of the time that you told me "there's at least one ___" (instead of "there is not at least one ___").

If the moderator flips heads, he tells us whether there's at least one red. If there isn't, he says so. He says "It's not the case that there's at least one red" (which gives us a lot more information than just saying "there is at least one black").I see now that you were envisioning a different scenario. (Which is why I asked.) But I think we're on the same page now.

Perfect! I think we are all on the same page now.So would agree with these statements:The cards are dealt in stacks of BB, BR, RB, RR. A moderator picks up one set of cards and looks at both of them. He lays the cards down and states, "at least one of the cards is red." In this scenario, he is not looking for either red or black cards. He is choosing which color to tell you about. We assume the method used for determining which color to mention is random. The odds of there being a black card along with the red card is 50%.If however, the moderator begins by saying that he is looking for Red cards on this trial and will tell you if he has at least one Red card and repeats the experiment above, the odds of their being a black card along with the red card is 2/3.