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Author Topic: Annoying probability question of the day
DonaldD
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quote:
Well yes, but turning the question around doesn't change the number of unknowns.
Correct. Bear with me.

In the case of a 2-child family, do you agree the the following four cases are the only possibilities?
first child born is Girl, second born is Girl
first born is Girl, second born is Boy
first born is Boy, second born is Girl
first born is Boy, second born is Boy

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Badvok
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Yep, I think I said that earlier.
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DonaldD
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I assume you agree that each case has an equal chance of occuring. I also assume you agree that the structure of the puzzle implies that all these births predate the father's statements and as such the father's statements can have no effect on the probability of each combination.

Now, given that you are presented with a father of two who claims that he has no daughters, what are the remaining possible family structures available?

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Aris Katsaris
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quote:
'The combination "one girl and one boy" is just as like to occur in a population (of fathers with at least one son) as the combination "two boys"'.
Except, that it's really really not.

Think about it like this. A population of fathers with 7 children, and at least 6 daughters. What's the chance they'll have 6 daughters and 1 son?

The possibilities are the following:
GGGGGGG
GGGGGGB
GGGGGBG
GGGGBGG
GGGBGGG
GGBGGGG
GBGGGGG
BGGGGGG

So 6 daughters and 1 son are SEVEN TIMES MORE LIKELY than 7 daughters, if it is given that the collective number of daughters is at least 6.

If however the six FIRST children are daughters, then the possible populations are only these:
GGGGGGG
GGGGGGB

So, we're back at 50% chance for a boy existing.
--
In short, not knowing *which* children the father is referring to, increases the number of possibilities for the non-matching children.

If the daughters' locations are perfectly pinned down, there's only possible position for the boy, which reduces his chance to a mere 50%.

[ July 27, 2010, 11:30 AM: Message edited by: Aris Katsaris ]

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Badvok
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quote:
Originally posted by DonaldD:
I assume you agree that each case has an equal chance of occuring. I also assume you agree that the structure of the puzzle implies that all these births predate the father's statements and as such the father's statements can have no effect on the probability of each combination.

Yes.

quote:
Now, given that you are presented with a father of two who claims that he has no daughters, what are the remaining possible family structures available?
Do you want to rephrase that - I don't think it is asking what you wanted to ask. If he has no daughters then he can have only two sons.
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DonaldD
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oops - "who claims he does not have two daughters."
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Badvok
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quote:
Originally posted by DonaldD:
Now, given that you are presented with a father of two who claims he does not have two daughters, what are the remaining possible family structures available?

Either two boys or a boy and a girl.

[ July 27, 2010, 11:37 AM: Message edited by: Badvok ]

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DonaldD
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Uh uh. Go by family structure. You did accept that there were initially 4 possibilities, and each had an equal chance of occuring already...

Now that you know the father's family is not a two-daughter family, what is left?

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Badvok
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Yeah, I guessed where you were going [Smile]

The statement "does not have two daughters" gives the following mutually exclusive possibilities:

first born is Boy or Girl, second born is Boy
or
first born is Boy, second born is Boy or Girl

Only the gender of one child can be indeterminate/unknown and that gender can only be one of two values.

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DonaldD
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I think you are trying to obfuscate now. You do see how you repeated yourself above, correct? [Smile]

Let me answer the question, and you tel me how this is wrong. Of the 4 possible family structures:

first child born is Girl, second born is Girl (25%)
first born is Girl, second born is Boy (25%)
first born is Boy, second born is Girl (25%)
first born is Boy, second born is Boy (25%)

The first family structure is excluded since the father does not have two daughters. That leaves you with:
first born is Girl, second born is Boy (25%)
first born is Boy, second born is Girl (25%)
first born is Boy, second born is Boy (25%)

Do you disagree?

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Pete at Home
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c, don't you mean changes from 1/4 to 1/3?
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Pyrtolin
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quote:
Originally posted by Badvok:
Yep. But that is the wrong question.
The answer to the original question is 'The combination "one girl and one boy" is just as like to occur in a population (of fathers with at least one son) as the combination "two boys"'.



But he wasn't picked out of the population of fathers with one son. He was picked out of the population of all fathers with two children.

And even in the population of fathers with two kids, one of who is a boy, twice as many have a boy and a girl than do two boys.


quote:
quote:
Again, man: out of 100% dads with two boys: 75% will have at least one boy. But only 25% of the dads will have two boys.
No I think out of 100% dads with two boys 100% will have at least one [Smile]
Let's get even simpler here since you missed that the 25% was a subset of the original 75%

You have 100 fathers with two kids.
How many have at least one boy?
How many have only one boy?
How many have two boys?

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Ciasiab
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Pete:

No. I was making the distinction between the case where we are asking the father questions (i.e. do you have at least one son?) to the case where the Dad declares information about one of his two children. In the first case, we ask the dad if he has at least one son. If he answers in the affirmative, then the odds are 1/3 that he has two sons.

In the second case (this is how I think the problem is stated), the father chooses one of his kids and starts giving you information about that child (i.e. he is a boy, he was born on Tuesday, he likes the color green and votes Republican). In this case the odds that the other child is a boy is 1/2. Since the father arbitrarily chooses one child to give random information about, it does not affect the probability of the other child's gender.

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Pyrtolin
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Ciasiab:
That doesn't change the fact that _you_ don't know which child he's giving information about, until he narrows it down for you. You can only assess the situation from the information that you have, not what he's got up his sleeve, until he lets you know it. You don't know for sure that he's chosen a specific one to think about until he goes on to give you specifics (And then you don't know for sure that those specifics don't apply to both of them and that he's still talking in the general sense)

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Ciasiab
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Pyrtolin,

It doesn't matter that I don't know which child he is giving information about. The key is that he does. The way the problem is stated, the man walks up and tells you information about one of his kids.
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."
John is clearly giving information about one of his two kids. Regardless of how much information John gives you about this particular kid, the odds of the other one being a boy or a girl is 50-50.

Now if the problem was stated as an interview, then things change. That's when we don't know for sure he's chosen a specific one to think about. If we ask do you have a boy that was born on a Tuesday, and he answers yes, the odds are 13/27 that the other one is a boy.

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Pyrtolin
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quote:
John is clearly giving information about one of his two kids. Regardless of how much information John gives you about this particular kid, the odds of the other one being a boy or a girl is 50-50.
Unless both of his kids were born on a Tuesday. From your perspective you cannot be clear that he's talking about a specific one until he tells you that he's talking about a specific one. Intuition and common sense don't apply here; if the information isn't explicitly given, you can't imagine that you have it.

Even if he's volunteering the information, he still came out of the full distribution, where it is twice as likely that a parent will have a boy and a girl than two boys.

He may know which one he's talking about, but you don't have the information to say whether he's talking about one or the other until he provides it.

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Ciasiab
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Pyrtolin,

It absolutely does matter how we come across the information. Say we have 100 dads with two kids. 25 have two girls, 25 have two boys and 50 have one of each. Now we ask each father in the room to tell us information about at least one of their kids. 50 dads will likely choose to tell us about a daughter. 50 dads will likely choose to tell us about a son. If I understand your logic correctly you predict that of the 50 dads of girls, 2/3 will have a boy. Of the 50 dads of boys 2/3 will have a girl. This just doesn't add up. Where am I going wrong?

[ July 27, 2010, 02:01 PM: Message edited by: Ciasiab ]

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Ciasiab
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In the problem statement John Doe volunteers that he has at least one boy. We know he comes from the general population, and 2/3 of the population with at least one boy also have one girl. 1/3 of the population would have 2 boys given that at least one is a boy. (We agree to this point).

Since he is volunteering information, if he has one girl and one boy, there is only a 50% chance he would tell us about a boy. If he has two boys there is a 100% chance he would tell us about a boy. Hence the final odds of a boy and a girl can be calculated as follows: ((2/3)*(1/2))/(2/3)=1/2

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Jordan
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quote:
JoshuaD:
Jordan: I was up all night thinking about this, and I'm now nearly certain that you're partially wrong. :-)

After some more careful parsing, I think you're right—I was wrong. But I think you may be partially wrong about why I was wrong. [Wink]

Aris identified where I flubbed: I asked a meaningless question. There isn't any way that you can ask that question without forcing the father to select one of his sons. In fact, no matter how I think of it, the closest to a meaningful means of asking the question is: "Go and stand over there if you have at least one son who was born on a Tuesday." The fathers who go where you indicated have a 13/27 probability of having two sons.

Incidentally, Aris has shown possibly the clearest understanding of the problem throughout, though his patience obviously got a bit frayed towards the end! [Smile]

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Ciasiab
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Jordan,

How about: "Do you have at least one son who was born on Tuesday?"

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Brian
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Pyrtolin:
quote:
Intuition and common sense don't apply here; if the information isn't explicitly given, you can't imagine that you have it.
Where the heck were you 3 pages ago?!?

I'm almost certain I asked if deduction is the same as information.

This is why I suck at some forms of math. I don't "understand" it, I intuitively know how it works. When intuition doesn't work (I'm looking at you, quantum mechanics!) then the only result I get is a pounding headache.

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Jordan
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quote:
Ciasiab:
How about: "Do you have at least one son who was born on Tuesday?"

Basically what I was doing, but addressed to Joshua's hypothetical million fathers instead of just one. It's kind of hard to know who's saying "yes"! [Wink]
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Wayward Son
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quote:
When intuition doesn't work (I'm looking at you, quantum mechanics!) then the only result I get is a pounding headache.
Intuition works with quantum mechanics.

Just not human intuition. [Smile]

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Badvok
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I think both Mariner and Aris have attempted to complicate this and confuse the issue to a very high degree but I do not know whether their motives for doing so are malicious or mischievous.

Unfortunately some have fallen into the trap cunningly presented to them.

If we look at this from a pure mathematical perspective and not from a motivational psycho-analysis perspective then:

The original proposition contained three statements of fact made by one person (i.e. not a person selected from a population or a randomly selected person being interviewed):

I have two kids.
(We now have two unknown genders.)
One of them is a boy.
(We now have only one unknown gender.)
He was born on a Tuesday.
(We still have only one unknown gender.)

The second statement does not remove one possibility from the two unknown gender probability space. It alters the probability space to be that of only a single unknown gender.

I don't know if you have ever seen this one:

"One day 3 women each obtained £10 from work making £30 in total. They located a TV priced £30 and decided to buy it. They went in the shop and gave the manager the £30 and started to carry the TV home. The manager realized that the TV was on sale giving £5 off. The manager told his assistant to take £5 from the till and return it to the women. However the assistant decided to keep £2 for him self leaving only £3 to give back to the women. So each lady paid £9 pounds each instead of £10 each. So adding the 3 lots of £9 gives £27 and including the £2 the assistant kept gives £29. So what happened to the missing pound?"

But the answer is the same - if you do the maths wrong you get the wrong answer!

[ July 28, 2010, 06:54 AM: Message edited by: Badvok ]

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TomDavidson
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quote:
The second statement does not remove one possibility from the two unknown gender probability space. It alters the probability space to be that of only a single unknown gender.
No, it does not. You still do not know which of his children is a boy. Saying, "I have two children, at least one of whom is a boy," leaves you with a 33% chance of having two boys.

Seriously. I have explained to you several times why this is the case. But I'll do it again:

quote:

"I have two children."
Result set:
Boy/Boy
Girl/Girl
Boy/Girl
Girl/Boy

quote:

"At least one of them is a boy."
Result set:
Boy/Boy
Boy/Girl
Girl/Boy

quote:

"The boy I mentioned was born on a Tuesday."
Result set:
Boy Tuesday/Boy Sunday
Boy Tuesday/Boy Monday
Boy Tuesday/Boy Tuesday *collapsed
Boy Tuesday/Boy Wednesday
Boy Tuesday/Boy Thursday
Boy Tuesday/Boy Friday
Boy Tuesday/Boy Saturday
Boy Tuesday/Girl Sunday
Boy Tuesday/Girl Monday
Boy Tuesday/Girl Tuesday
Boy Tuesday/Girl Wednesday
Boy Tuesday/Girl Thursday
Boy Tuesday/Girl Friday
Boy Tuesday/Girl Saturday



[ July 28, 2010, 09:10 AM: Message edited by: TomDavidson ]

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Badvok
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quote:
Originally posted by TomDavidson:
You still do not know which of his children is a boy.
Seriously. I have explained to you several times why this is the case.

And I have been trying (and obviously failing) to point out that it doesn't matter which one is a boy just that one is a boy and therefore the other is either a boy or a girl. Two options only = 50:50 chance of each.
It wouldn't make the slightest difference if he said he had 50 kids and that at least 49 of them were boys. The probability that they are all boys still comes down to only one single gender question.

[ July 28, 2010, 09:12 AM: Message edited by: Badvok ]

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Badvok
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Try thinking of it a different way:

I put a child in a room and ask you to guess the gender. What are your chances of being right?

I now say that I'm going to put a boy in the room too. Now what are your chances of being right about the genders?

[ July 28, 2010, 09:19 AM: Message edited by: Badvok ]

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TomDavidson
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quote:
And I have been trying (and obviously failing) to point out that it doesn't matter which one is a boy just that one is a boy and therefore the other is either a boy or a girl.
Except that you're wrong. Seriously. You really are.

The question is not "what is the probability that my other child is a boy." It is "what is the probability that both my children are boys."

I know it seems ridiculous. It seems like these should be the same answer, because on the face of it these appear to be the same question. But they're not. And this is because you have no mechanism that would permit you to "fix" in place one of the children and say "this particular child, of two, is a boy, and therefore we only need to solve for the other child." As the question is framed, there is no way to tell which child is a boy.

If it would help to imagine this problem with four children, go ahead. You have four children. Three of them are boys. What are the odds that they are all boys? (Hint: it is not 50%.)

[ July 28, 2010, 09:23 AM: Message edited by: TomDavidson ]

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Aris Katsaris
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We have given you the numbers, Badvok.

100 fathers with 2 children: With random even distribution 75 of these have atleast one son, but only 25 of them have two boys.

This means that only 1/3 of fathers with atleast one son have two sons.

Can you pinpoint to us which one of these numbers you don't accept?

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Badvok
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quote:
Originally posted by Aris Katsaris:
We have given you the numbers, Badvok.

100 fathers with 2 children: With random even distribution 75 of these have atleast one son, but only 25 of them have two boys.

This means that only 1/3 of fathers with atleast one son have two sons.

Can you pinpoint to us which one of these numbers you don't accept?

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.
But you are just solving the wrong problem and hence using the wrong numbers for the problem. We have ONE father, ONE son, and ONE other child who's gender we do not know.

[ July 28, 2010, 09:41 AM: Message edited by: Badvok ]

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DonaldD
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I told you all it was semantics [Smile]

Badvok, why did you quit on me above. You were soooo close [Wink]

Here is the original statement of fact, Badvok: "I have two kids. One of them is a boy"

Which of the following is/are equivalent to the above?

"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.

[ July 28, 2010, 10:21 AM: Message edited by: DonaldD ]

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Pyrtolin
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quote:
But you are just solving the wrong problem and hence using the wrong numbers for the problem. We have ONE father, ONE son, and ONE other child who's gender we do not know.
Right- and given the overall distribution of choices, there are two chances that that unknown child will be a girl and one that it will be a boy, because boy-girl families are twice as common as boy-boy families.

We have one father and one boy, sure. but they were picked from the full set of families, some of which had no boys.

quote:
I put a child in a room and ask you to guess the gender. What are your chances of being right?

I now say that I'm going to put a boy in the room too. Now what are your chances of being right about the genders?

That's not a comparable situation. When you put two kids into the room, the possibility existed that you put two girls in as well. It's not till _after_ you randomly picked them that you checked one at random and found out that it was a boy. The process of selecting both kids was completely random.

If I put my hand into a bag with an equal number of red and blue marbles and grab two at the same time, what's the chance that I pull out two blue ones together?

Does this change if, after I've pulled the two out, I randomly pick one to look at and see that it's blue? It's twice as likely that, in the initial grab, I got one of each than it is that I got either specific case of having two of the same. Knowing what one of them is after the random selection doesn't change that initial likelihood.

This is the point your missing here. We're not locking the parent in as having a boy then asking what the next/other kid will be. We're assigning two kids randomly up front, then discovering that one of the assignments happened to be a boy.

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Badvok
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quote:
Originally posted by DonaldD:
Badvok, why did you quit on me above. You were soooo close [Wink]

LOL, I had to head off to catch my commuter train home.
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Badvok
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quote:
Originally posted by Pyrtolin:
We have one father and one boy, sure. but they were picked from the full set of families, some of which had no boys.

Nope that is not what was stated!
quote:

When you put two kids into the room, the possibility existed that you put two girls in as well. It's not till _after_ you randomly picked them that you checked one at random and found out that it was a boy. The process of selecting both kids was completely random.

Nope, I put one child in the room then added a boy. I didn't select two random children.

quote:
If I put my hand into a bag with an equal number of red and blue marbles and grab two at the same time, what's the chance that I pull out two blue ones together?

If you want to use balls then it is one blue ball and one ball that might be either blue or red in the bag. What is the probability there are two blue balls in the bag?

Or how about:

I put a stuffed toy and a child in a room and ask you to guess the child's gender. What are your chances?

I magically change the stuffed toy into a boy and tell you that is what I have done. What are your chances now?

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DonaldD
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Badvok, you are chasing your tail.

You are arguing that the puzzle involves only a single unknown, and then trying to show us the math.

Here is the thing: everybody on this thread, if they were to agree that there is only a single unknown and that unknown has two possibilities of equal probability, would agree that the chance is 50/50. But nobody is arguing against that math.

But that is not how the vast majority of people here interpret the puzzle in question. Go back to the puzzle statement, the words themselves, and we can discuss what they mean first.

You could start with my post from 10:18 as a shortcut...

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Badvok
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quote:
Originally posted by DonaldD:
You are arguing that the puzzle involves only a single unknown

Yep! The only math I have tried to correct is the erroneous elimination of an option from a probability space. This is, I think, the key cause of most people here misinterpreting the problem.
Like the nonsense math problem I posted earlier it is mangled maths that leads to the 1/3 probability.
The use of children seems to cloud the issue with the introduction of birth order and birth day.
If I take 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?
and rephrase it without using children:
quote:
I have a box that contains two chess pieces. One of them is white. It is also scratched. What are the odds that they are both white?
Does this help at all?

[ July 28, 2010, 11:02 AM: Message edited by: Badvok ]

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DonaldD
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OK, so you are now working with the original puzzle (less the Tuesday information, but OK for now).

But before going off on your tangent, let's finish with mine, alright?

Which of the following is/are equivalent to puzzle statement above?

"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.

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Pyrtolin
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quote:
Originally posted by Badvok:
quote:
Originally posted by Pyrtolin:
We have one father and one boy, sure. but they were picked from the full set of families, some of which had no boys.

Nope that is not what was stated!


It very explicitly is. This isn't a world where everyone starts with one boy and then has children after that, it's a world that includes the initial possibility that the man could have had two girls before he explicitly eliminated that possibility.

quote:
quote:

When you put two kids into the room, the possibility existed that you put two girls in as well. It's not till _after_ you randomly picked them that you checked one at random and found out that it was a boy. The process of selecting both kids was completely random.

Nope, I put one child in the room then added a boy. I didn't select two random children.


And that's not a parallel situation. The parent in question had both children randomly before he offered you any information. The fact that there was a boy to reveal in the first place was part of the random generation of the possibilities. Letting you know after the fact doesn't change the initial field from which the possibilities were drawn.

quote:
quote:
If I put my hand into a bag with an equal number of red and blue marbles and grab two at the same time, what's the chance that I pull out two blue ones together?

If you want to use balls then it is one blue ball and one ball that might be either blue or red in the bag. What is the probability there are two blue balls in the bag?

I put a stuffed toy and a child in a room and ask you to guess the child's gender. What are your chances?

I magically change the stuffed toy into a boy and tell you that is what I have done. What are your chances now?

[/qb][/quote]

Those aren't comparable to the initial statement. He didn't say "I have a boy, what will my next child be" He said "I have to randomly selected kids. One of them turned out to be a boy." Both were picked at random before the information was given. Revealing the information after the fact doesn't alter the initial odds; until he said something, there was still a 25% chance that he had two girls.

Go back to the room with 100 people. 75 of them will have one boy. If you pick on of those 75 at random, 66% of them will have a girl as well and only 33% will have two boys.

The parent in the initial question was one from that pool of 100, not one from a more limited pool of only ones whose first child was a boy. So revealing that they have a boy only eliminates the possibility that they were one 25 who have no boys, none of the rest. You still have 75 possibilities, only 25 of which have two boys.

The information was only fixed after both selection were made randomly.

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Badvok
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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 ...

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Jordan
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quote:
Badvok:
Unfortunately some have fallen into the trap cunningly presented to them.

I can assure you it's not a trap. The question is badly worded for the conclusion that you're meant to draw from it, but it is absolutely not a trap.

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?

[ July 28, 2010, 11:27 AM: Message edited by: Jordan ]

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