There is no sisters paradox. The trick is how the question is weirdly framed and has to be interpreted. What people think about when they hear the question would effectively lead to a probability of 0.5: if you see a family in the street with a girl and know they have two kids, the probability of the other kid being a girl is indeed 0.5.

The trick of the so-called "paradox" is turning the question into the Monty Hall but with an ambitious enough formulation that you might be confused it’s not.

▲ 6gvONxR4sf7o 4 days ago | parent | next [-]

The way to see this is bayes rule. p(answer | data) = p(data | answer) * p(answer) / (sum_{all possible answers'} p(data | answer') * p(answer')). So for this question, that's expands to:

    p(both are girls | you're told at least one is a girl)
     = p(you're told at least one is a girl | both are girls) * p(both are girls) / (
            p(you're told at least one is a girl | both are girls) * p(both are girls)
            +
            p(you're told at least one is a girl | they aren't both girls) * p(they aren't both girls)
        )

The problem is that we don't know p(you're told at least one is a girl | they aren't both girls). Clearly if both are boys, then you won't be told at least one is a girl (or at least it's implied that you're told the truth). But that still leaves us p(you're told at least one is a girl | one boy and one girl).

This is the crux of the thing. Different readings of the setup imply different answers to p(what you're told | the unknowns).

It's also a great case of where bayes rule shorthands can be slippery. You'll usually abbreviate it out (hell, it was tedious to write this way even with copy-paste). But if you abbreviate "you're told there's at least one girl" to "there's at least one girl", then you've stopped modeling a crucial part of the setup. p(there's at least one girl | they aren't both girls) has an unambiguous answer.

▲ lqet 4 days ago | parent | prev | next [-]

This. A less confusing way to ask the question with the 1/3 answer would be:

  What is the probability that a family with 2 children has exactly 2 daughters *if you know that the family does not have 2 boys*?

The reasons why the original problem is so confusing is the same reason why the Monty Hall is so confusing: people have different understandings of the question, and don't realize it in discussions. As I have written a few years ago [0]:

Because most people don't talk formal probabilities, your explanations will be so vague that the other person will not realize your different understanding. You will discuss forever, you will both be right, and you will part ways with the strange feeling that maybe the other person was right, when all along you were talking about different problems. This is why this problem is so notorious.

[0] https://news.ycombinator.com/item?id=24707305

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JeffJor 11 hours ago | parent | next [-]

> The reasons why the original problem is so confusing is the same reason why the Monty Hall is so confusing: people have different understandings of the question, and don't realize it in discussions.

Almost everybody understands the same problem, AND STILL GET DIFFERENT ANSWERS. If they don't understand it, they make pedantic arguments about Monty's motivations. All of which make the puzzle impossible to answer.

What they don't understand is probability. Probability is a measure of the information you lack about what causes a certain result to occur. That includes the physical details (where the prize is, what the genders are) but also the choices made for hidden reasons.

In the Monty Hall Problem, to reduce complexity, label the doors C (the contestant's original door), R (the door to its right, wrapping around if necessary), and L (the door to its left). What leads up to the game state at the time the decision to switch is made are (A) Where the prize was placed and (B) How Monty Hall chooses a door to open if the prize is behind C.

The naive answer is based on only (A). The two unopened doors (C and R, or C and L) started with the same probability. So they must now have the same probability, 1/2, right? No, wrong, because we need to take (B) into account. If the prize is behind R then the host had to open L. If the prize is behind L then the host had to open R. But if the prize is behind C then the host had to choose. Since we don't know how, we have to assume there was a 50% chance that he would choose R, and 50% for L. Once we see him open, say, R? This 50:50 reduces the probability that the prize is behind C, so switching becomes twice as likely to win.

The Two Child Problem works exactly the same way. What leads up to the point where we are asked for a probability is (A) the gender makeup of the family and (B) how the information came to us if there is a boy and a girl.

The naive answer is based on only (A). A mixed family is twice as likely as either two-of-a-kind family. So the probability of two-of-a-kind is 1/3, right? Wrong, unless we know WITH CERTAINTY that we could not have learned about the other gender. If we do not have that certainty, then just like with Monty Hall we have to assume that half of the time in a mixed we would have learned the other gender. This makes a mixed family half as likely as (A) alone would suggest; in other words, the same as two-of-a-kind.The answer is 1/2.

Joseph Bertrand pointed out, in 1889, why we need to take (B) into account. Martin Gardner, who originated the Two Child Problem, repeated it in 1959. In the same article where he introduced the predecessor to Monty Hall (called the Three Prisoners Problem), and explained why (B) is important. It should be embarrassing to anyone who thinks that the "Tuesday" variation's answer is 13/27. Because it was first mentioned at a puzzle convention named in honor of Martin Gardner and forgot his warning. Adding irrelevant information can't change the answer, and if you take (B) into account the answer doesn't change.

	▲	kgwgk 9 hours ago \| parent [-]
		> What they don't understand is probability. Exactly. As the first reply to the first comment explains “The problem is that we don't know p(you're told at least one is a girl \| they aren't both girls).” It’s funny that the same commenter who writes that “to get the right answer you must be careful about conditional probabilities” finds that doing so is “splitting hairs about something boring and irritating.”

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edanm 4 days ago | parent | prev [-]

I think this is true of the "children" question, but I actually disagree that this is what makes the Monty Hall question so confusing.

For one thing, I vaguely recall this being asked directly, and even after people agree on all the definitions explicitly, they still consider the answer wrong. (See e.g. some mathematicians like Erdos refusing to believe the correct answer without actually running simulations on computers... by that point you clearly have a real definition.)

For another, when I personally talk to people about Monty Hall, even after I explain the correct answer, and explain all the nuances, people tend to still have a hard time accepting the correct solution and claim to find it counterintuitive (as did I!).

	▲	chatmasta 3 days ago \| parent [-]
		The typical Monty Hall formulation has similar ambiguity because it’s not clear whether or not the host knows the right door. Just like in this question, it’s not clear if the “narrator” knows the first child is a girl. (Also, this problem has an additional layer of ambiguity where “birth order is irrelevant,” but MF and FM are treated as distinct items in the probability set. Is the order irrelevant to the probability, or is it impossible to distinguish the age of the two children? It would be clearer to simply say “each birth is an independent event.” One of the comments on the blog explains this better than I can.)

▲ js2 4 days ago | parent | prev | next [-]

So it's "what is the probability both are girls?" vs. "what is the probability the other is a girl?" and most people will hear the latter and answer 1/2 whereas the question is the former and its answer is 1/3. Do I have that right?

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AnotherGoodName 4 days ago | parent | next [-]

"The question writer took all sets of two child families and ruled out the bb case. Then they asked the exact question above" This is 1/3 chance - select gg from [gg,bg,gb]'

"The question writer came across a girl from a two child family, then they asked the exact question above". This is 1/2 chance - select gg from [gg, gg, bg, gb] with gg listed twice since there's two ways to select a girl from that set; ie. coming across a girl is twice as likely to occur from the gg case than it is either gb or bg.

I think that's the clearest wording to get the message across. Either way it's the exact same question but it reasonably has a completely different answer. There's no way to resolve this ambiguity with the question as written.

	▲	LegionMammal978 4 days ago \| parent [-]
		That's a good framing. It's similar to the fact that the chance of a given star being in a multiple system (~47% in our vicinity [0]) is significantly higher than the chance of a given system having multiple stars (~30%), because counting by individual stars gives more weight to the multiple systems. [0] https://astronomy.stackexchange.com/a/55505

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LudwigNagasena 4 days ago | parent | prev | next [-]

Those questions are equivalent. What is important is the conditional “… given that I looked at a random child and it was a girl” / “… given that I looked at both children and at least one of them was a girl”.

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SkiFire13 4 days ago | parent | prev [-]

The issue with asking whether the other is a girl is how you choose the first one.

If you look at one random child, see it's a boy and exclude the family, even though the other child may be a girl, then you get the 1/2 probability. If however in that case you also look at the second child, see that's a girl and consider the family anyway, then you get the 1/3

▲ lIl-IIIl 4 days ago | parent | prev | next [-]

The other interpretation that leads to 1/3 probably is also pretty intuitive. That's the fun part of this question is that it leaves crucial information unspecified.

I think this is a reasonable interpretation:

You meet a family at a party. They say "We have two children". You ask "Do you have any girls"? They say "yes!"

This will give you 1/3 probability that the other child is also a girl.

I think this interpretation is more intuitive because it doesn't make any assumptions about how you get your information. Usually in probability questions you assume any information you have is given to you from on high. For example, you just "know" that the family has two children, you don't somehow deduce it. Therefore I assume the same for "one child is a girl" information.

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kgwgk 4 days ago | parent [-]

> I think this interpretation is more intuitive because it doesn't make any assumptions about how you get your information.

Do you mean “interpretation” or “alternative problem”.

Because if it’s an “interpretation” of the original problem you’re indeed making assumptions to fill the unspecified information.

If you mean that it’s an alternative problem which has a definite solution I agree. It’s a different problem and its relevance to the original one is to illustrate that additional assumptions were required.

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lIl-IIIl 4 days ago | parent [-]

I mean the original problem can be interpreted in both ways. You have to ask the questioner for more information to remove the ambiguity.

	▲	kgwgk 4 days ago \| parent [-]
		I think we agree then. The original problem cannot be answered without making additional assumptions about how you get your information. Different interpretations may reach different answers by making different assumptions.

▲ taeric 4 days ago | parent | prev | next [-]

I'd hazard that people also typically hear "what are the odds of this from the outset?" Effectively, "you flip two quarters, and see one land heads; what were your odds to flip two tails?"

▲ zahlman 4 days ago | parent | prev | next [-]

>ambitious

ambiguous?

	▲	4 days ago \| parent [-]
		[deleted]

▲ guy2345 4 days ago | parent | prev [-]

[dead]