Super Duper Extra Hard Brainteaser

[hoyaboy1]

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I think this is what I am getting at - it only works with an infinite population. If we are talking about earth, for example, 99% would change the answer a TON, as the odds of having a second girl would be waaaaaaaaaaay lower (of course, this whole thing would only work if people were killing off their second kid once they had a Sarah).

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No, no. It doesn't change the calculation at all. I wasn't suggesting that. It's just that you are making up these numbers, and then trying to imagine realistic situations (ie, population distributins like ours here on earth) where they occur.

But look, I can get around your objection by just having 1 or 2 two person families with a girl named sara. All the rest have 1 kid or 3+ kids.

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Yea, I mentioned the possibility of that earlier.

But you do acknowledge that with a limited population with a normal distribution of family sizes (let's say our planet), the answer if 1% or 99% of girls is Sarah is not the same, right?

The real problem here, which I fully acknowledge, is that as the % of Sarahs gets higher people will have to artificially adjust to keep naming 99% of girls Sarah and never have 2 girls named Sarah (like killing babies) if they have 2 kids.

Edit - anything above 2/3 and the chance of a second girl starts dropping due to some sort of articial measure needing to be introduced.

[gaming_mouse]

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But you do acknowledge that with a limited population with a normal distribution of family sizes (let's say our planet), the answer if 1% or 99% of girls is Sarah is not the same, right?

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No, I don't. What I acknowledge is that the following three things are inconsistent logically:

1. normal distribution of family sizes like on our planet.
2. the assumptions of the puzzle.
3. 99% of girls names sara

This does not mean that probability is breaking down. It means that your question is flawed. It's like saying, "There are three marbles. Two of them are black. Two are white. You draw one at random. What is the chance it's white?"

[Keats13]

My first reply was not thought out well enough.

HOWEVER:
Because there can't be more than 1 Sarah per family, the probability of naming your daugher Sarah must increase if you already have a daughter not named Sarah. This counteracts the fact that the probability decreases (to 0) if you already have a daughter who IS named Sarah.

If you have 2 girls,
P(2nd Sarah) = P(2nd Sarah|1st Sarah)*P(1st Sarah) + P(2nd Sarah|1st not Sarah) * P(1st not Sarah)
= 0 + P(2nd Sarah|1st Not Sarah) * P(1st not Sarah)
= P(2nd Sarah|1st Not Sarah) * 99/100
(because there can't be 2 Sarahs)

If 1/100 first daughters are named Sarah, then 1/100 second daughters must also be named Sarah for the stated assumption (1/100 of all girls are named Sarah) to be true.

Therefore,
P(2nd Sarah) = 1/100

Plugging in above,
1/100 = P(2nd Sarah|1st not Sarah) * 99/100
P(2nd Sarah|1st not Sarah) = 1/99

So if you have 2 girls, the probability that one of them is named Sarah is

P(1st Sarah) + P(2nd Sarah|1st Not Sarah) * P(1st Not Sarah)
1/100 + 1/99 * 99/100 = 2/100

Plugging this back in to the original equations gives the answer of 1/2.

Unless I screwed something up again, which is entirely possible.

[hoyaboy1]

Actually, it isn't like that at all. That example is impossible. 99% of girls being named Sarah and never having 2 girls named Sarah could never happen naturally (as I've said a bunch of times) but could happen if, say, the government set up a program to do so.

So, again, assuming the question was talking about earth, and I have to assume it was, we have a a choice: put a cap on the number of Sarahs at 2/3, or accept that above that the odds of a second girl falls below 50%, even if the whole situation is being maintained artificially.

So, once more, if we are talking about earth, changing the number of Sarahs to 99% changes the answer. Please accept this so I can go to sleep, it is 3:40 here.

This isn't an important point or anything, I just felt like pointing this out.

[gaming_mouse]

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So, once more, if we are talking about earth, changing the number of Sarahs to 99% changes the answer. Please accept this so I can go to sleep, it is 3:40 here.

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This is incorrect. The answer is the answer, period. I'm just saying that on earth, where most families are 2 children, it won't be the case that 99% of girls are named Sara and you never have 2 sisters names sara. That is the way in which the assumptions are like the marble problem.

[mostsmooth]

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So, once more, if we are talking about earth, changing the number of Sarahs to 99% changes the answer. Please accept this so I can go to sleep, it is 3:40 here.

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This is incorrect. The answer is the answer, period. I'm just saying that on earth, where most families are 2 children, it won't be the case that 99% of girls are named Sara and you never have 2 sisters names sara. That is the way in which the assumptions are like the marble problem.

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its just a hypothetical
if the problem was "bill can run 80mph, how far can he run in 2 hours"?, we know nobody can run that fast, but we solve it anyway. no debating how fast people can actually run

[jason_t]

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So, once more, if we are talking about earth, changing the number of Sarahs to 99% changes the answer.

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If this were the case, math would be useless. I assure you that it is not. The answer is independent of whether or not we are talking about Earth.

[hoyaboy1]

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So, once more, if we are talking about earth, changing the number of Sarahs to 99% changes the answer. Please accept this so I can go to sleep, it is 3:40 here.

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This is incorrect. The answer is the answer, period. I'm just saying that on earth, where most families are 2 children, it won't be the case that 99% of girls are named Sara and you never have 2 sisters names sara. That is the way in which the assumptions are like the marble problem.

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Fine, never mind, even though I said many times the 99% would only work if some sort of program was set up to maintain it, you still won't acknowledge the obvious for some odd reason. This has nothing to do with the validity of your argument with Patrick, so I don't quite get your problem. Anyway, I'm going to bed now.

[hoyaboy1]

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So, once more, if we are talking about earth, changing the number of Sarahs to 99% changes the answer.

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If this were the case, math would be useless. I assure you that it is not. The answer is independent of whether or not we are talking about Earth.

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Wait, before I do go to bed - you think that if 99% of girls on earth were named Sarah, those in 2 children families would still have a 50% chance of having a sister not named Sarah? Because that is wrong. Even mouse admitted this - the answer would either be some much lower %(I haven't bothered calculating it), or you'd have to say it would be impossible (the cheap and pointless answer).

[gaming_mouse]

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Wait, before I do go to bed - you think that if 99% of girls on earth were named Sarah, those in 2 children families would still have a 50% chance of having a sister not named Sarah? Because that is wrong. Even mouse admitted this - the answer would either be some much lower %(I haven't bothered calculating it), or you'd have to say it would be impossible (the cheap and pointless answer).

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mouse didn't admit this. also, i don't know why you're getting mad -- I know this has nothing to do with the original problem, but your understanding is flawed and I'm trying to help.

The point is that if 99% of girls were named sara, and no girl in a two person family could have a sister named sara, there could only be a small percentage of two person familites. but the probability in the puzzle would not change.