Boy or Girl?

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Eduk
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Re: Boy or Girl?

Post by Eduk »

As I said if there was any ambiguity in my opening scenario I would be happy to resolve it. You initially said
To be clear, the ambiguity is not the result of a mistake on your part.
So my intention was to formulate a scenario which represents A. I am sorry I did not achieve this.

Out of interest do you agree with the Wiki article? As in formulation A gives an answer of 1/3?
Unknown means unknown.
Fooloso4
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Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
So my intention was to formulate a scenario which represents A. I am sorry I did not achieve this.
But I think this is much more interesting and, if I’m right (I can’t calculate the probability that I am, but it is not high enough that I would bet on it), yields the paradoxical result that when we select a family the probability is ⅓ but when we select the boy the probability is ½. Hence the title "boy-girl paradox".
Out of interest do you agree with the Wiki article? As in formulation A gives an answer of 1/3?
I defer to the mathematicians and statisticians cited in the article.
As I understand it (or don’t) the difference is that when calculating A we must consider all three possible combinations because we are looking at one random family with with at least one boy, that is, families might have an older boy and families might have a younger boy, but with B all we are looking at one boy and need only to consider whether the other child is either a boy or a girl. If the boy is older then there is a 50/50 chance the younger child is a boy. If the boy is younger then there is a 50/50 chance the older child is a boy. We do not have to consider all three possible combinations as in formulation A.
Eduk
Posts: 2466
Joined: December 8th, 2016, 7:08 am
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Re: Boy or Girl?

Post by Eduk »

the paradoxical result that when we select a family the probability is ⅓ but when we select the boy the probability is ½. Hence the title "boy-girl paradox".
Well it only feels like a paradox, like many paradox it isn't actually a paradox.
For example if you and me were in my kitchen (and I had two children) and my children were playing outside but you couldn't see them, and you asked if at least one of my children was a boy (odd question but lets role with it) and I said yes. Then in this scenario the chance of both being boys would be 1/3. Note the important thing here is that I didn't select a random true statement, if both my children were girls I would have answered no to your question. To explain this I like to think of cards A, B, C and D. A = BB, B = BG, C = GB, D = GG. Your question has eliminated any chance of D. This leaves A, B and C. The chance of any one of those cards being the right answer is 1/3.
However in the same scenario, you and me are sitting in my kitchen, children playing outside. And you can see one of the children (and it's a boy). Then in this scenario the chance of the other being a boy is 1/2. This feels like a paradox, what is the difference between seeing one is a boy and knowing one is a boy? The difference in this case is that looking at a boy is testing one boy not both boys. For example with the card example of A, B, C and D. We obviously remove D. But as we have seen the boy then either B or C is also removed, depending on if you saw my older or younger child (note it doesn't matter if you know which is older or younger). So we are left choosing between A and (B or C) which gives the answer of 1/2.
The results are counter intuitive and confusing. We simply aren't well equipped for this to make sense. Some people simply can't accept the answer, which I find interesting. You (Fooloso4) even if intuitively you find it hard to believe do defer to the wiki article, which personally I find quite refreshing :)
Unknown means unknown.
Fooloso4
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Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
Well it only feels like a paradox …
After trying to figure this problem out I did some research and found an extensive literature. Each time I think I have it figured out I stubble.

In your examples it makes no difference whether I see that at least one child is boy or I am told that at least one child is a boy. We do not begin by selecting from among a group of two child families with at least one boy. If we were then we would have to look at all possible combinations - BG, GB, and BB. We begin with your child who is identified as a boy and ask about his sibling. There are only two possibilities, the sibling is either a girl or a boy.
We simply aren't well equipped for this to make sense.
As I understand it, it does not make sense only when there is some ambiguity as to how the problem is set up. It has to do with how the family and child are selected, whether we select from a pool of families or select a child from a given family. In both your examples we start with the latter and so the answer should be the same.
Eduk
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Re: Boy or Girl?

Post by Eduk »

In your examples it makes no difference whether I see that at least one child is boy or I am told that at least one child is a boy. We do not begin by selecting from among a group of two child families with at least one boy. If we were then we would have to look at all possible combinations - BG, GB, and BB. We begin with your child who is identified as a boy and ask about his sibling. There are only two possibilities, the sibling is either a girl or a boy.
Let me put it this way. If we had 3 families in 3 houses, made up of BB, BG and GB. And you didn't know which house was which and I asked you to select a house. What are the chances that you select the house with BB? This is clearly 1/3 right?
Unknown means unknown.
Fooloso4
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Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
Let me put it this way. If we had 3 families in 3 houses, made up of BB, BG and GB. And you didn't know which house was which and I asked you to select a house. What are the chances that you select the house with BB? This is clearly 1/3 right?
You need to be a bit more precise. If you mean that one house is BB, the second is BG, and the third GB, then yes.

If John lives in one of those house then the probability of John having a brother is ⅓.

John meets a boy Jim, who just moved into the neighborhood. Jim tells John that he has a sibling but does not say whether it is a boy or girl. John, who is a lot smarter than me, calculates the probabilities: Jim has an older brother = ¼, a younger brother =¼, an older sister = ¼, a younger sister = ¼. The probability that Jim has a brother is ½.

If we have 100 two child families each with at least one boy, the probability of BB for any randomly chosen family is 1/3.

If I flip two coins and one came up heads the probability that the other also came up heads is ½ because there is only two possibilities for the other coin. If I take a group of 100 flips in which at least one of the coins came up heads and then randomly selected one of those flips the probability the other coin also came up heads is ⅓ because there are now three possibilities: HH, HT,TH.
Eduk
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Re: Boy or Girl?

Post by Eduk »

If I flip two coins and one came up heads the probability that the other also came up heads is ½ because there is only two possibilities for the other coin.
Everything you said is right except this :) Although this does display how ambiguity can change things as how you came to your knowledge is key.
If you flipped two coins, without seeing what the two results were, but I had. Then you asked me if at least one of the flips had been heads and I replied yes, then the probability of the other being heads is 1/3. Remember I'm not making up random true statements if both flips had been tails I would have replied no to your question. The number of times you flip two coins does not change the odds.
Unknown means unknown.
Fooloso4
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Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
If you flipped two coins, without seeing what the two results were …
You have changed the problem. I did not state it as clearly as I should have. When I said one came up heads I meant that I knew that it was heads and was asking about the other coin. Given that one was heads, what was the other? It was meant to be analogous to Jim’s sibling.

Here is a question about the Monty Hall problem:
The standard answer is that it is better to switch. Suppose that both of us were playing but neither of us was aware that there was another player. I pick door 1 and you pick door 3. Door 2 reveals a goat. Now if we both switch then we both increase our odds of winning, but how can that be?
Eduk
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Joined: December 8th, 2016, 7:08 am
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Re: Boy or Girl?

Post by Eduk »

You have changed the problem. I did not state it as clearly as I should have. When I said one came up heads I meant that I knew that it was heads and was asking about the other coin.
Ah I understand. That's the main problem with this boy/girl paradox to my mind, the phrasing to remove ambiguity is quite tricky :)

Regarding Monty Hall. In your scenario we both initially select a door (1 and 3) which looks like there is a 1/3 chance of winning each. But the host can only open door 2 and door 2 must have a goat behind it, in your scenario. The only way door 2 can have a goat behind is if either you or me selected the door with the prize. This means we both have a 1/2 chance of being right, switching or not switching in this case would make no difference.
Unknown means unknown.
Fooloso4
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Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
This means we both have a 1/2 chance of being right, switching or not switching in this case would make no difference.

You are right. Thinking about it a bit more, the two player problem invalidates the rules. According to the rules Monty must always open a door that has not been picked and the door must contain a goat.
Eduk
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Re: Boy or Girl?

Post by Eduk »

Yup exactly. The fact that Monty has to pick a door with a goat and can only pick a door which you haven't is the factor which means switching gives you a 2/3 chance of winning. The same as the boy/girl paradox the intuitive feeling that the answer must be 1/2 (not 2/3) is more caused by a failure of language rather than of an inherent misunderstanding of maths and probabilities. The actual maths itself is very simple.
Unknown means unknown.
Fooloso4
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Joined: February 28th, 2014, 4:50 pm

Re: Boy or Girl?

Post by Fooloso4 »

Eduk:
… more caused by a failure of language rather than of an inherent misunderstanding of maths and probabilities.
I don’t agree. I think there are two related problems. One stems from an ambiguity in how the problems are posed, and this can lead to two different but correct answers. If, however, this problem is eliminated and the questions are posed in a way that there is only one correct answer, the second problem stems from having not learned how to properly think about probabilities, and this requires overcoming certain assumptions, such as if two doors remain the probability of one having the car is equal to the other. It is the assumptions rather than the language that is the stumbling book. In some cases these assumptions are only rejected when one tests them against repeated simulated games.

The problem of assumptions extends to many areas of our lives, including some of the philosophical issues raised in this forum. While one side accuses the other of not being rational, the problem is that given their assumptions what follows is rational.
Eduk
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Joined: December 8th, 2016, 7:08 am
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Re: Boy or Girl?

Post by Eduk »

Well people just aren't always rational. Even if they are mostly rational most people have some irrational beliefs.

Although I agree there are a lot of problems caused by assumptions.
Unknown means unknown.
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Aristocles
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Re: Boy or Girl?

Post by Aristocles »

The ambiguity claims appeared ambiguous, as does the distinction of boy versus girl. As I understand it, some social media sites are up to about 50 types of genders. Gender versus sex, I would agree there is excessive ambiguity in that distinction also... Klinefelter's syndrome, XYY syndrome, Turner's syndrome, XYZ? Jenner, etc, make probabilities with numerical precision more variable then we tend to be comfortable with...
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