Super Duper Extra Hard Brainteaser

[Patrick del Poker Grande]

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Seems to me that this is only a brainteaser because it induces people to waste a bunch of time with the Sarah red herring when the answer is still 1/3.

Think about intuitively: If you gathered a population of mothers with two children, at least one of which is a girl, and divided them into subpopulations based on whether one they had named a child Sarah, would you expect the subpopulations to have different estimated means on the #ofGirls variable. Basically, the Sarah thing just identifies a subpopulation but shouldn't change any other characteristics.

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Think about it like this, it might change your mind: knowing that a women has two girls increases the likelihood that she has a child named Sarah. Conversely, knowning she has a child named Sarah....

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...means she has a 1/3 chance that her other child is also a girl, provided she has exactly 2 children.

[Patrick del Poker Grande]

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What is the probability of girl named sarah, girl?

1/200 * 1/2 which is 100/40,000

Where am I going wrong?

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1/200 * (99/100) * 1/2. They can't both be sarah.

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Hmm. The probability of the first child being a girl called Sarah is just 1/200 and then the probability of the next one being a girl is 1/2 n'est pas?

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Non. Read this. All of it.

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Please explain which part of what I said was wrong.

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I gave you that link implying that this has been explained several times in this thread and that you should read the thread.

[partygirluk]

That is not what your link did.

[DMBFan23]

jason,

I don't really think your code checks what we want it to check. you only update the check if both are girls, which leaves out all possibilities of boy, sarah and sarah, boy.

you are checking the ratio of all pairs to pairs with one sarah. we know that this particular pair does NOT have AT LEAST one sarah, it has EXACTLY one.

the correct pseudo code is

c1 = girl (this is sarah)
c2 = rand (1 - 199)

if c2 = 1-100, c2 = boy
if c2 = 101-199, c2 = girl

then take the fraction of c1, c2 pairs that are both girl in relation to the whole. it will be approximately .31

I hate this cliche, but,

do you see why?

[gaming_mouse]

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jason, by that logic the answer to the first, easier, problem is 1/2, which is wrong.

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No it's not. Let's say she tells us the name of her girl, Sara.

Sara, G
G, Sara
B, Sara
Sara, B

The chance of "Sara, G" is not 1/4. It's 1/8. Do you see why?

[DMBFan23]

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Seems to me that this is only a brainteaser because it induces people to waste a bunch of time with the Sarah red herring when the answer is still 1/3.

Think about intuitively: If you gathered a population of mothers with two children, at least one of which is a girl, and divided them into subpopulations based on whether one they had named a child Sarah, would you expect the subpopulations to have different estimated means on the #ofGirls variable. Basically, the Sarah thing just identifies a subpopulation but shouldn't change any other characteristics.

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Think about it like this, it might change your mind: knowing that a women has two girls increases the likelihood that she has a child named Sarah. Conversely, knowing she has a child named Sarah....

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decreases the odds of her having two girls given that

a) some % of girls are named sarah
b) boys and girls are equal
c) not both girls can be sarah.

I will bet money that I am right.

[partygirluk]

How much are you willing to bet?

[gaming_mouse]

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I will bet money that I am right.

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I will bet up to $100 that my answer of 1/2 is correct.

[DMBFan23]

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jason, by that logic the answer to the first, easier, problem is 1/2, which is wrong.

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No it's not. Let's say she tells us the name of her girl, Sara.

Sara, G
G, Sara
B, Sara
Sara, B

The chance of "Sara, G" is not 1/4. It's 1/8. Do you see why?

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gaming mouse,

there was an original distribution of boys to girls. in that distribution, having a boy and a girl was twice as likely as having two girls. the fact that we now know she has one girl who happens to be named sarah does not change anything - the girl named sarah can be either of the two outcomes of girl, it doesn't add a permutation like property to it, which seems to be what you and jason are doing. if you stipulate that her OLDEST daughter is sarah, then order matters, and the answer is slightly less than 1/2 due to the fact that she can't have two sarahs. but as it is, the answer is slightly less than 1/3.


knowing that this first girl is named sarah means nothing except that it means the other girl cannot be sarah, which makes it slightly less likely that the other person is a girl. so the sarah clause, is, like patrick said, SOMEwhat of a red herring. it still provides useful information though.

[Patrick del Poker Grande]

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I will bet money that I am right.

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I will bet up to $100 that my answer of 1/2 is correct.

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I'll take it.