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This question, has me stumped. Looking at the first version, how there's two lines of thought that go around in my head, both (seemingly) justifiable.

1 : 'One of them is a girl' implies 'Both aren't boys', which implies three posibilites, each with 1/3 possibilites. Thus the possibility of both being girls is 1/3.

2 : 'One of them is a girl'. Let it be the one playing in the garden (without loss of generality) then the one shrieking like the Bee Gees in the room above is either a boy or a girl. 50-50. Thus 1/2.

Followup : Why does the order matter? I will really appreciate a clear, detailed, answer to derail my wrong-logic train of thought. Thanks.

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Fullmetal
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  • "Why does the order matter?" What order? I'm not sure what you're talking about. –  Apr 28 '15 at 05:28
  • {Girl-Boy and Boy-Girl} as opposed, hey one of 'em is a girl, so the other must be a boy or a girl and there's equal chance of that. I hope you can see the linked question. – Fullmetal Apr 28 '15 at 05:31
  • Oh, I know what the question is talking about. –  Apr 28 '15 at 05:32
  • Okay, okay. Do you get my question? One of those two trains of thought is incorrect. I'm told its the second, but can't see why. – Fullmetal Apr 28 '15 at 05:33
  • I've posted an answer below. –  Apr 28 '15 at 05:34
  • Did you not read my answer to the other question? The phrase "one is a girl" is ambiguous and both your interpretations could apply. More clear would be if she said "none are boys". Or are you asking why one is 1/3 and the other is 1/2? – JS1 May 08 '15 at 02:38
  • I don't understand why the two questions, when yielding the same answers, yields to different probabilites. I get the first bit, given one child is in the kitchen and the other is behind the curtains, and their mother knows who's who. She thinks of one of 'em and says it's a girl. There's equal likeliness of either of the kids being a girl and equal likeliness of the other's sex being female. Thus 1/21/2 + 1/21/2 = 1/2. But the second one baffles me. I mean, how is 'one is a girl' different from 'atleast one is a girl', because in both cases couldn't we have made the same argument? – Fullmetal May 08 '15 at 03:32
  • Because "at least one is a girl" was a direct response to the question "is at least one of your children a girl?" which is the same question as "are both your children boys?" (except the opposite answer). Your confusion is that you think that a woman who has one boy and one girl could answer no to the question "is at least one of your children a girl?" if she is thinking about the boy. But I say that she cannot answer no in that situation. – JS1 May 08 '15 at 10:02
  • Why don't you think, 'alright, so atleast one is a girl, now move her aside, the other could either be a girl or a boy. And there's equal likeliness of that. XX or XY. So, there: 1/2' ? – Fullmetal May 08 '15 at 13:54

2 Answers2

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While there are enough explanations in the answers/comments, the OP here still has Doubts, so let me try to answer with a simple approach and hopefully clarify the problem to the OP.

A family has two kids; What can they be ? BB BG GB GG , where the order XY is required because they are born in sequence, X before Y. All four possibilities are equally likely.

When the mother says "One of them is a girl", she is eliminating the first possibility. So three equally-likely possibilities remain. In one such possibility, both are girls, hence probability is 1/3.

When the mother says "Elder kid is a girl", she is eliminating the first two possibilities. So two equally-likely possibilities remain. In one such possibility, both are girls, hence probability is 1/2.

Prem
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  • Wouldn't it be that there is only 1 possible scenario to be chosen of 3 possible outcomes, but the probability of 1 girl and 1 boy (BG, GB), and 2 girls (GG) is still 50% since it is not related to the number of possible scenarios? i.e if she said she had 49 girls (of 50 kids), then there is 1 of 50 possible outcomes of all girls, but still only a 50% chance that the other kid is a girl or boy? – Mark N May 07 '15 at 17:07
  • The critical part is equally-likely possibilities. (BG , GB) & (GG) are not equally likely. – Prem May 07 '15 at 17:19
  • Sorry, it's the ambiguity in the question which causes the argument of whether order is intended to matter or not. I have been taught to assume order doesn't matter unless stated (which it wasn't explicitly in the question). I understand the proper solutions now – Mark N May 07 '15 at 17:51
  • Thank you for answering and sensing my doubt. I've thought about this a lot. I was good at solving probability problems as high school kid, but now when I question my basics, it only seems I got 'used' to the math that was involved in probability theory. Don't kill me. What if they were born at the same time? – Fullmetal May 08 '15 at 02:12
  • Also, could you please read the comments under Joe Z's answer, especially the very last one made by me, please? Bringing order into play yields a set of 4 elements, and a constraint which removes one element (not both boys. so no BB. As you've said too.) But the other way of thinking (and this is wrong somewhere, supposedly) is this: Okay, so one of 'em is a girl? Okay. Kill her. Now there's another one left. There's 50% chance of it being a girl. In which case you've got both girls, by unkilling the first. – Fullmetal May 08 '15 at 02:21
  • Only Joe would be the right person to reply to comments about his answer, with his way of explanation, which was why I added this simple answer to explain the situation. – Prem May 09 '15 at 16:44
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Like most of these conundrums, this "paradox" is just a trick of language.

The only sense in which "one of them is a girl" means "at least one of them is a girl" is a strictly mathematical, logical one, and the sense in which "what's the probability both of them are girls" gives an answer of 1/3 is again a strictly mathematical construction which means "out of a randomized trial consisting of selecting between the scenarios where at least one child of mine is a girl with equal probability, what is the expected frequency of the scenario where my children are both girls?".

Nobody really talks like that in real life, hence when you hear it first, your brain automatically assumes the mother means your second meaning when she really means the first. When you say "one of them", you're usually thinking of a specific one, such as "the eldest one" or "the one playing in the garden". But using the strictly logical sense as above, the answer differs.

Your follow-up question asks, "why does the order matter?"

We say that each child has a 50% probability of being a boy or a girl. In order to identify which child "each" child is to calculated the combined probabilities, we need to give them separating labels, which we apply an order to for convenience. Child A being male and Child B being female is a different case from Child A being female and Child B being male, even if they're both cases of "one boy and one girl".

  • are you saying that there are two valid interpretations, thus leading to two different probabiities ? – Fullmetal Apr 28 '15 at 05:49
  • My fundamental - now rephrased perhaps - question would be : When told that there's atleast one girl amongst the two, mathematical constructors write their event set as {GB, BG, GG}. Why can't one say, "Oh well. There's a girl, so let's put her on the left side and decide what takes stand in the right, the unknown. And thus: {GB, GG.} This is where peraps my 'why does order matter?' part shoots in. – Fullmetal Apr 28 '15 at 05:52
  • Yes, there are two valid interpretations. –  Apr 28 '15 at 05:57
  • @Fullmetal That's a more fundamental probability problem - now you're simply asking why the probability is 1/3 at all in the first case. This requires the mathematical reasoning we saw before where you select among cases where at least one child is a girl - order matters because otherwise when you rearrange the events, you get {GB, GB, GG}. GB occurs twice. –  Apr 28 '15 at 05:59
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    "order matters because. . .{GB, GB, GG}." : But aren't we writing the set that way because we think order matters? My question is why does it? Say the mother says one is a girl. Without lose of generality let it be her first, clearly the sex of the second is independent of the first's. So 50% ain't it? (I'm sorry if you're losing your nerve on me for this. I won't ask again.) – Fullmetal Apr 28 '15 at 06:10