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Author Topic: Annoying probability question of the day
Badvok
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quote:
Originally posted by Jordan:
How would you go about answering this?

You're having a laugh aren't you?
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Pyrtolin
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quote:
Originally posted by Badvok:
Please! Someone please tell me that my English is not that bad that no one can understand me!
Why do people like Pyrtolin keep quoting me but then not understanding what I actually said?
There is no field from which a selection was made in the original statement:
quote:
John Doe comes up to you and says: "I have two kids. One of them is a boy. He was born on a Tuesday." What are the odds that he has two boys?
Where is the field? I can't see it! All I see is one father, one son and one other child.
Please open my eyes! Or are we talking about an assumed field? As we all know Assume makes an ...

The field is that of all possible fathers. You're not in a magical world that only has this father or only fathers whose first child was a boy. Until he opened his mouth he could have had 100 daughters, for all you knew. When he does talk, he reveals that he's from the specific set of fathers with two children, and then narrows that to the set of fathers with two children, one of which is a boy. But all of those conditions were established randomly before he said a word- the only thing that has changed is the information that you have to work with.

He didn't say We had a boy, and then another child. So you have to count in the situations where he had a girl first and then a boy as well.

So if you really must have a situation where you put a known sex into the room, you have to count both ways: Put a boy in the room, and then have a child of unknown sex, you'll get a boy 50% of the time in that case. Then put girl in the room and count the cases where the unknown, from that perspective, happened to be a boy, as the statement about having a boy applies to that situation as well.

In the first situation, you've got a 50% chance of getting two boys, but in the second, your chances are 0%, but only half of the case in the second scenario are valid. So out of 3 possible valid situations, only one results in two boys.

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Badvok
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quote:
Originally posted by Pyrtolin:
The field is that of all possible fathers. You're not in a magical world that only has this father or only fathers whose first child was a boy.

Where did I ever say that the first child was a boy? Where does order come into it?
Where did I say he was the only father in the world? He is simply the only father in the problem space.

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DonaldD
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Now, as to your tangent.

I'm going to simplify it a bit, because your 'scratch' bit is not equivalent to 'Tuesday', but we're not anywhere near discussing Tuesday yet.

quote:
I have a box that contains two chess pieces.
Full stop. What are the chances that the box contains two white pieces? Two black? A mix?

The answer: 25%/25%/50%. Agreed?

Now, someone looks at one of the pieces. He does not touch them. He does not say anything - yet.

By looking, has he changed the above probabilities?

Think about it this way - there are (yes) 100 boxes, 100 men, and an infinite number of chess pieces in a barrel (with an equal number of white and black pieces.)

The 100 men each grab 2 pieces and put them in a box. How many boxes would you expect to contain 2 black pieces? 2 white pieces? A mix?

The answer (on average) would be 25/25/50, right? In fact, you look into each of the 100 boxes and verify this fact. you then leave the room and the men trade around their boxes.

Now, one of these men looks into his box at the pieces. By looking, he has not changed the pieces. He tells you that one of the pieces is white. What can you immediately, unequivocally state about the actual contents of the box? That there are not 2 black pieces. Agreed? Anything else?

If the only additional fact you have now is that there are not two black pieces, you can logically exclude the possibility that your box is one of the twenty five boxes with both black, but nothing else.

In which case, you know that your box is one of the 75. You also know the contents of these 75 boxes, and that only 25 of them have two white pieces.

Now, seriously, answer my previous post [Smile]

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Badvok
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quote:
Originally posted by DonaldD:
Now, one of these men looks into his box at the pieces. By looking, he has not changed the pieces. He tells you that one of the pieces is white.

Noooo! You are changing the problem space again! The set of possibilities is only those where one of the pieces is white. There are no boxes where there is not at least one white piece because this is a stated fact! (Plus there is actually only one box.)
quote:
Now, seriously, answer my previous post [Smile]
I don't think there is much point do you?

[ July 28, 2010, 12:02 PM: Message edited by: Badvok ]

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Pyrtolin
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quote:
Originally posted by Badvok:
quote:
Originally posted by Pyrtolin:
The field is that of all possible fathers. You're not in a magical world that only has this father or only fathers whose first child was a boy.

Where did I ever say that the first child was a boy? Where does order come into it?
Order comes into it because the order of the children introduces additional possibilities. The ambiguity of the order is exactly what's at key here and shifts the overall probability. The problem in all of your attempts to model the situation is explicitly that you don't account for the possibility of different orders or that he may have had two girls until he revealed order ambiguous information.

quote:
Where did I say he was the only father in the world? He is simply the only father in the problem space.
No- all fathers are in the problem space. He just happens to be the one that stepped forward and asked to be analyzed.
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Badvok
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quote:
Originally posted by Pyrtolin:
He just happens to be the one that stepped forward and asked to be analyzed.

Now that is a very big assumption! What did I say about assume?

[ July 28, 2010, 12:07 PM: Message edited by: Badvok ]

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TomDavidson
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I'd also like to present my previous example.

You meet a man who says he has four children. He then tells you "at least three of them are boys."

What are the odds that he has four boys? (As I said before, it is not 50%.)

----------

The problem you're having is that you're looking at the remaining child in question and saying "there is a 50% chance that this child is a boy." That's correct.

But you don't know which child in the set is the remaining child. Let's say he has four boys. Which one of the four boys was the one he didn't mention? Now let's say that he has three boys and one girl. Of the four possible combinations of three boys and a girl, which combination is actually the correct one? Based on the information provided, we don't know. Ergo, we see one possible valid combination of four boys and four possible valid combinations of three boys and a girl.

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Badvok
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quote:
Originally posted by TomDavidson:
Ergo, we see one possible valid combination of four boys and four possible valid combinations of three boys and a girl.

If you want to take order into account then you need to treat each child as a distinct entity and then there is a lot more than one combination for four boys - I'm too tired to work it all out now (time for me to head home) but I'm sure you can.
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Pyrtolin
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quote:
Originally posted by Badvok:
quote:
Originally posted by Pyrtolin:
He just happens to be the one that stepped forward and asked to be analyzed.

Now that is a very big assumption! What did I say about assume?
That's not an assumption. That's axiomatic. Using anything less than the general population when such a limit isn't explicitly stated in the problem is making an assumption.
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Aris Katsaris
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quote:
Originally posted by Badvok:
quote:
Originally posted by Pyrtolin:
He just happens to be the one that stepped forward and asked to be analyzed.

Now that is a very big assumption! What did I say about assume?
A big assumption? It is indeed a big ASSUMPTION as I detailed in the first page of this thread, which you don't seem to have read.

However, you've repeatedly refused to acknowledge that this assumption affects the probabilities in any way. The quiz was rephrased to you as following:

"John Doe: I have two children.
Me: Is at least one a boy?
John Doe: Yes."

And you KEPT insisting that the probability for two sons is 1/2 even with this rephrasing.

So yes, with the initial phrasing it's very ambiguous, but you were given the unambiguous phrasing in which the dad volunteers no other information except "two children" -- and you still believed it was 1/2.

The new rephrasing with the probing question is unambiguous: and therefore you're unambiguously wrong in calculating the probabilities as 50%. They're absolutely 33.3%.

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TomDavidson
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quote:
If you want to take order into account then you need to treat each child as a distinct entity and then there is a lot more than one combination for four boys...
This may be the source of your confusion. We don't particularly want to take order into account, but we have to take order into account. However, while there are a lot of possible permutations of four boys, we're only interested in the single combination of four boys; those permutations all collapse into one combination, because without any additional detail we can't tell any of the boys apart. They're all boys. The possible permuations, as written, are BBBB, BBBB, BBBB, and BBBB (etc.); they're all the same. The situation is different with one girl in the mix, since we can distinguish the permutation of BGBB from BBGB, BBBG, and GBBB.

[ July 28, 2010, 12:25 PM: Message edited by: TomDavidson ]

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threads
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Badvok:

There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?

[ July 28, 2010, 12:26 PM: Message edited by: threads ]

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DonaldD
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quote:
I don't think there is much point do you?
Actually, yes. Your problem is one of semantics, and clarifying the initial statement is the only way to agree on what the puzzle actually means. You keep saying things like "Noooo! You are changing the problem space again!" but you refuse to clarify what you think the puzzle meant by "I have two kids. One of them is a boy".

So once again:

"I have two kids. The first born is a boy"
"I have two kids. Either the first or the second born (or both) is a boy"
"I have two kids. My child John is a boy"
"I have two kids. At least one of them is a boy"
"I have two kids. Exactly one of them is a boy"
"I have two kids. They are not both girls"
"I have two kids. The child I am currently thinking of is a boy"

Choose as many as you think are correct, as there is some overlap.

Tom - Aris actually gave the same example on the previous page, but with 7 children I think.

Badvok, if you play Tom and Aris' game, it will become quite clear to you.

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TomDavidson
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It's important to note that this is why the Tuesday detail matters, because it becomes possible to distinguish a boy born on a Tuesday from another boy except if the second boy was also born on Tuesday.
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Ciasiab
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If the information is volunteered the odds of the second kid being a boy is 50/50.


quote:
Let's see exactly how you're approaching this. I'm going to try and reword the question a little more clearly:


quote:
In front of you are two screens, standing side by side. You are told that there is one person behind both of them. You are further told that at least one of the two people behind the screeens is male. What is the probability that both of the people behind the screens are male?
How would you go about answering this?

You are assuming that if at least one of the people is a male the announcer will tell you that there is a male. Without further information this is a bad assumption. If there is one male and one female, 50% of the time the announcer will choose male and 50% of the time the announcer will choose female.

So we have two people behind the screens, and have been told that one is male. It is tempting to say that there is a 2 in 3 chance of the other person being female, but this is not the case since if the other person is female, the odds of being told about a male have been reduced by half.

If your logic is right, once we know the gender of one contestant, we can conclude that 2/3 of the time the gender of the other contestant is the opposite. This cannot be true.

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Aris Katsaris
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Badvok, because each child (of a given father) has an independent 50% chance of being born girl or a boy, that's why we have to take "order" into account. We don't have to take the order of their births: we can take alphabetical order, or order of prettiness, or order of weight, or any other order we seek.

But the point is treating them as each having a distinct 50% chance of having been born boy or girl.

The problem is the following:
Father: I have two children.
Me: Is atleast one of them a boy?
Father: Yes.

Given the above, the probability of two boys is 33%. UNAMBIGUOUSLY.

FFS, we're not being either 'malicious' or 'mischievous', we're trying to make you see how logic works.

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Ciasiab
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quote:
Originally posted by threads:
Badvok:

There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?

Of the 50 families with a boy and a girl, what are the odds of a father choosing to tell you he has a boy?
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Ciasiab
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quote:
So yes, with the initial phrasing it's very ambiguous, but you were given the unambiguous phrasing in which the dad volunteers no other information except "two children" -- and you still believed it was 1/2.

The intial problem is not ambiguous. The answer is 1/2. In order for the answer 1/3 (or 13/27) we have to assume the father would only tell us the gender if he had a boy and only tell us the day of the week if it happened to be Tuesday.

[ July 28, 2010, 12:36 PM: Message edited by: Ciasiab ]

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Aris Katsaris
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quote:
If the information is volunteered the odds of the second kid being a boy is 50/50
Sorta. It depends on the guy's reason for volunteering the information. He may have volunteered it for reasons that resolve to the same 33% possibility (e.g. asking army recruitment information, so all the fathers with atleast one boy would have to go there, while fathers with only girls wouldn't).

The following phrasing:
"Father: I have two children.
Me: Is atleast one of them a boy?
Father: Yes."

is unambiguous, though, and I wish more people from both sides used it, instead of insisting that the phrasing doesn't matter. It does matter, very much.

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threads
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quote:
Originally posted by Ciasiab:
quote:
Originally posted by threads:
Badvok:

There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?

Of the 50 families with a boy and a girl, what are the odds of a father choosing to tell you he has a boy?
You should read Aris' discussion of this point. Relevant post:
quote:
Originally posted by Aris Katsaris:
Let me put it in another way.
Possibilities for two children:
G/G 1/4
G/B 1/4
B/G 1/4
B/B 1/4
--
If guy decides to reveal one of these genders randomly, possibilities now become
G/G (reveals G) 1/4
G/B (reveals 1st: G) 1/8
G/B (reveals 2nd: B) 1/8
B/G (reveals 1st: B) 1/8
B/G (reveals 2nd: G) 1/8
B/B (reveals B) 1/4

Now if we know "B" was revealed, this corresponds to 1/8 + 1/8 + 1/4 = 50%. Out of these, it's even odds that the other one was a boy, or that the other one was a girl.

So if the guy randomly selected the kid whose gender he'd reveal, it's even odds that the other kid is either gender. 50% says common sense, and 50% it indeed is.
--
HOWEVER if the guy thinks: I will NOT mention there's a girl, but I will only reveal if there exists a boy. The possibilities become:
G/G (mentions no information) 1/4
G/B (mentions there's a B) 1/4
B/G (mentions there's a B) 1/4
B/B (mentions there's a B) 1/4

Now, knowing he revealed it was a boy, there only 33.3% chances that the other kid is a boy too, and 66.7% chances that the other kid is a girl.
--
AND if the guy thinks: I will mention ALL my boys, but none of my girls. The possibilities become:
G/G (mentions no information) 1/4
G/B (mentions there's a B) 1/4
B/G (mentions there's a B) 1/4
B/B (mentions there's two boys) 1/4

Now, with the knowledge he mentioned only *one* B for certain, we can be 100% sure that the other kid is a girl -- because he'd have mentioned two boys if B/B was the reality.
--
That's what I mean when I say motivation matters. WHY did he reveal the particular gender? Was he randomly picking a kid, or was he choosing that gender for some reason?



[ July 28, 2010, 12:38 PM: Message edited by: threads ]

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Ciasiab
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quote:
Sorta. It depends on the guy's reason for volunteering the information. He may have volunteered it for reasons that resolve to the same 33% possibility (e.g. asking army recruitment information, so all the fathers with atleast one boy would have to go there, while fathers with only girls wouldn't).

You have to project additional reasons to get there. You have to add an assumption about army recruitment or a sexist dad.
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Aris Katsaris
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quote:
"The initial problem is not ambiguous. The answer is 1/2. In order for the answer 1/3 (or 13/32) we have to assume the father would only tell us the gender if he had a boy and only tell us the day of the week if it happened to be Tuesday. "
The original phrasing is ambiguous exactly because we don't know if these assumptions are right or wrong. A Spartan soldier volunteering to be among the 300 would want at least one son to carry the family line. The existence or non-existence of girls wouldn't matter to him.
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threads
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Ciasiab, you are projecting just as much by assuming that the father is randomly choosing which kid to talk about. That's not stated anywhere in the initial problem. Just agree that it's ambiguous and let it be.
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Ciasiab
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quote:
You should read Aris' discussion of this point. Relevant post:

Aris' post proves my point. Unless you assume the father is not providing random infomation (i.e. he's at an army recruiting station), the problem as stated resolves to 1/2.
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Aris Katsaris
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quote:
"You have to project additional reasons to get there."
Sure, but "I've randomly picked a kid of mine whose gender I'm telling you" is also an assumption, as you don't know that.

That's why I say the phrasing is ambiguous.

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Ciasiab
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quote:
Originally posted by threads:
Ciasiab, you are projecting just as much by assuming that the father is randomly choosing which kid to talk about. That's not stated anywhere in the initial problem. Just agree that it's ambiguous and let it be.

Most problems can be ambiguous. You can read it as random, or you can add assumptions in your head. The only way to solve the problem is to assume it is random. If you add another assumption you are adding to the problem (like saying you happen to be at a military recruitment center). If you assume (another assumption I know) that all the information is contained in the problem statement, then I don't see how not to conclude he is choosing a kid at random.
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Ciasiab
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quote:
Originally posted by Ciasiab:
quote:
Originally posted by threads:
Ciasiab, you are projecting just as much by assuming that the father is randomly choosing which kid to talk about. That's not stated anywhere in the initial problem. Just agree that it's ambiguous and let it be.

Most problems can be ambiguous. You can read it as random, or you can add assumptions in your head. The only way to solve the problem is to assume it is random. If you add another assumption you are adding to the problem (like saying you happen to be at a military recruitment center). If you assume (another assumption I know) that all the information is contained in the problem statement, then I don't see how not to conclude he is choosing a kid at random.
Edit to add: How about we call it ambiguous and leave it at that. [Embarrassed]
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threads
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Agreed!
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Aris Katsaris
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quote:
You can read it as random, or you can add assumptions in your head. The only way to solve the problem is to assume it is random.
Eh. What if the randomness is that he randomly chooses a gender, and then mentions its existence or non-existence?

In that case from an original set of 200 dads:
GG 50
GB 50
BG 50
BB 50
we go to
GG 25 (mentions absence of B)
GG 25 (mentions existence of G)
GB 25 (mentions existence of B)
GB 25 (mentions existence of G)
BG 25 (mentions existence of B)
BG 25 (mentions existence of G)
BB 25 (mentions existence of B)
BB 25 (mentions absence of G)

So you have 75 dads that randomly chose to mention existence of B. Out of them only 25 have a second B. It reverts to 33%

The original phrasing is just ambiguous. Did he randomly choose a child whose gender he revealed (50% chance of two boys), or did he randomly choose a gender whose absense/presence he revealed (33% chance of two boys)?

[ July 28, 2010, 12:54 PM: Message edited by: Aris Katsaris ]

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Ciasiab
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quote:
Eh. What if the randomness is that he randomly chooses a gender, and then mentions its existence or non-existence?

You make a very excellent point.
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Jordan
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quote:
Badvok:
You're having a laugh aren't you?

No, Badvok; I'm trying to make an opening—and something of a break from the prior discussion—with what I hope is a somewhat cleaner (and clearer) example. [Smile]

So that you know where I'm going with this, I'm asking you to start by explaining how you would approach it with the intention of asking you some more questions to follow up.

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Aris Katsaris
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Badvok, can I ask another question? Here's the following narrative, please examine it:

Father: I have exactly two children.

(Probabilities now stand at:
25% chance of two girls.
25% chance of two boys.
50% chance of one boy and one girl)

DO YOU AGREE WITH ME SO FAR? Please answer this question first: If you disagree there's no point in continuing further, but if you agree with me, please continue reading.

Me: Are they both girls?
Father: No.

(The probabilities as I calculate them now stand at:
0% chance of two girls.
33% chance of two boys.
66% chance of one boy and one girl)

----------------

When you argue that the chance of two boys is now at 50% you're essentially saying that by excluding the probability of two daughters, we are ONLY increasing the probability of two sons, but we're not increasing the probability of one son and one daughter.

Does that seem logical to you? Isn't it common sense that all other possibilities must increase in probability (at least slightly) by the exclusion of *one* previously possible scenario?

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edgmatt
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quote:
There are 100 two-child families in the room. The world is naturally perfect so 25 of the families have 2 boys, 25 of the families have 2 girls, and 50 of the families have a boy and a girl. A father comes up to you and says "at least one of my children is a boy". What is the probability that he has two boys?
Badvok - This post by Threads seems to be the best way to phrase the question so as to be understood. Does it make more sense when it is put this way?
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Badvok
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quote:
Originally posted by Aris Katsaris:
"John Doe: I have two children.
Me: Is at least one a boy?
John Doe: Yes."

And you KEPT insisting that the probability for two sons is 1/2 even with this rephrasing.

LOL, no I don't think I KEPT insisting that phrasing meant the probability was 1/2. I admit that I did mistakenly respond that way once though.

I see that I really don't get what this thread is all about (as I said in my first post to the thread). I don't understand why the assumption that John Doe comes from a population of random families was made. To me he is one man, he has one son and there is one child we do not know the gender of.

You are obviously not talking about the original statement and I am just confusing things by not accepting your assumptions about it.

I'll bow out in disgrace now.

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TomDavidson
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quote:
To me he is one man, he has one son and there is one child we do not know the gender of.

Even still, his children come in the following permutations:

Boy / Girl
Girl / Boy
Boy / Boy

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Jordan
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quote:
Badvok:
You are obviously not talking about the original statement and I am just confusing things by not accepting your assumptions about it.

No drama, Badvok. [Smile] I think everyone here accepts that there are two very different interpretations of the question floating about—whether or not they accept that either one of them is a more natural or correct way of reading it!

The main thing that people are confused about is, simply, if you a) understand the logic behind the 1/3 answer, and b) agree that if the scenario is constructed carefully enough, 1/3 will definitely be the correct answer.

(By the way, I did exactly what I didn't want to do and read back through all your posts so far—and even after that, I honestly can't work out what your position is on b)! [Big Grin] )

[ July 29, 2010, 08:09 AM: Message edited by: Jordan ]

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Aris Katsaris
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quote:
You are obviously not talking about the original statement and I am just confusing things by not accepting your assumptions about it.
*I*, for one, wasn't making assumptions, I rephrased the question so that no assumptions needed be made, and I detailed in the first page of this thread how different assumptions (that other people were making) affect the probability.

I'm disappointed that you didn't respond in my last post. That you bowed out and said this isn't the original situations, doesn't mean you couldn't have taken half a min to say "yes" or "no" to my question. It's frustrating that even with the different rephrasing you can't bring yourself to actually state unambigously "Yes, in this case it's 33%".

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Aris Katsaris
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quote:
"To me he is one man, he has one son and there is one child we do not know the gender of."
No, he is one man, and he has two children whose gender we don't know of -- with the knowledge that atleast one of them is a son affecting the corresponding probabilities for each child being a son.

With this bit of knowledge, any given child has 66% chance of being a son, and the situation for both being sons either way reverts to:
(if first child is a boy, second child has 50% possibility) (2/3)*(1/2) = 1/3
(if first child is a girl, second child has 100% chance of being a boy) (1/3)*(1) = 1/3

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Badvok
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quote:
Originally posted by Badvok:
Your numbers are not wrong in that sense.
1/3 of the 75 fathers (from a population of 100) who have at least one son have two sons.


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