3 ms·

This is incorrect. There's no reason to assume that the stranger will say, "at least one of them is a girl" only in the case of GG. In the case of BB, the stra

by ggreer 13y ago

This is incorrect. There's no reason to assume that the stranger will say, "at least one of them is a girl" only in the case of GG. In the case of BB, the stranger will always say, "At least one is a boy." Given GB or BG, they have even odds of saying "at least one is a boy" vs "at least one is a girl." Given GG, the stranger will always say, "At least one is a girl." The correct diagram would be this: http://abughrai.be/pics/boy_probability.png http://abughrai.be/pics/boy_probability.png For the probabilities in that post to apply, the stranger would say, "I have two children." Then you ask, "Is at least one of them a boy?" and they answer, "Yes." Edit: Of course, the answer to, "Given there are two siblings and one is a boy, what are the chances both are boys?" is 1/3. But the phrasing of the puzzle does not reflect that question.

javert 13y ago
The post is not about the probability of what a person with a given pair of children will say. Rather, what they said is given, and the question is: what pair of children do they have? You are either going over my head or under my head, so this will probably make one of us temporarily look a bit foolish :).
- ggreer 13y ago
  The blog post[1] that the author based his short explanation on goes into this. Peter Winkler, a math professor at Dartmouth, is quoted in it[2]: Suppose the puzzle is phrased this way: X says “I have two children and at least one is a boy.” What is the probability that the other is a boy? Put this way, the puzzle is highly ambiguous. Computer scientists, cryptologists and others who must deal carefully with message-passing know that what counts is not what a person says (even if she is known never to lie) but under what circumstances would she have said it. The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl." 1. http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday-birthday-problem/ http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday... 2. http://blog.tanyakhovanova.com/?p=234 http://blog.tanyakhovanova.com/?p=234
  - javert 13y ago
    Thanks for the clarification, though I think my point stands. > The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl." Right. But that is assumed, because the puzzle states that the parent says, "At least one of them is a boy." In other words, "Put this way, the puzzle is highly ambiguous." No, not it is not. It is absolutely not ambiguous. I made this point in a different comment for a different blog post with the same example, but: both things you link to contain a fallacy, as follows. "Suppose we know that a certain man has two children and we also know that the older one is a boy" IMPLIES "Now suppose we know simply that a man has two children and that one of them is a son. This time we would reason that there is no possibility that the person has two girls. It follows that the sexes of his two children, ordered from oldest to youngest, are either BB, BG or GB...." You can't conclude, as these authors do, from the first case, that the other child's gender is 1/2, and from the second, it is 1/3. It's not that the two cases are different (in any important way); they are not; one implies the other. It's that in the 1/2 case, the person is naive, and in the 1/3 case, the person takes into account the known fact that there is a 1/2 chance per child of getting a certain gender, thus allowing the BB/BG/GB/GG analysis.
Confusion 13y ago
Fine, then I pose to you the same problem description, with the added fact that the stranger is proud of having a boy, will only draw attention to a boy and will thus say "at least one is a boy", independent of whether the other is a girl, while he would never say "at least one is a boy". You can't solve a logic puzzle by arguing the psychology of a persona depicted in the puzzle.
- stephencanon 13y ago
  You can’t solve a logic puzzle that way, but you can explain the pragmatics of why the “logical” answer is contrary to our expectations based on interaction with real people.

by ggreer 13y ago

javert 13y ago

The post is not about the probability of what a person with a given pair of children will say. Rather, what they said is given, and the question is: what pair of children do they have? You are either going over my head or under my head, so this will probably make one of us temporarily look a bit foolish :).

ggreer 13y ago
The blog post[1] that the author based his short explanation on goes into this. Peter Winkler, a math professor at Dartmouth, is quoted in it[2]: Suppose the puzzle is phrased this way: X says “I have two children and at least one is a boy.” What is the probability that the other is a boy? Put this way, the puzzle is highly ambiguous. Computer scientists, cryptologists and others who must deal carefully with message-passing know that what counts is not what a person says (even if she is known never to lie) but under what circumstances would she have said it. The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl." 1. http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday-birthday-problem/ http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday... 2. http://blog.tanyakhovanova.com/?p=234 http://blog.tanyakhovanova.com/?p=234
- javert 13y ago
  Thanks for the clarification, though I think my point stands. > The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl." Right. But that is assumed, because the puzzle states that the parent says, "At least one of them is a boy." In other words, "Put this way, the puzzle is highly ambiguous." No, not it is not. It is absolutely not ambiguous. I made this point in a different comment for a different blog post with the same example, but: both things you link to contain a fallacy, as follows. "Suppose we know that a certain man has two children and we also know that the older one is a boy" IMPLIES "Now suppose we know simply that a man has two children and that one of them is a son. This time we would reason that there is no possibility that the person has two girls. It follows that the sexes of his two children, ordered from oldest to youngest, are either BB, BG or GB...." You can't conclude, as these authors do, from the first case, that the other child's gender is 1/2, and from the second, it is 1/3. It's not that the two cases are different (in any important way); they are not; one implies the other. It's that in the 1/2 case, the person is naive, and in the 1/3 case, the person takes into account the known fact that there is a 1/2 chance per child of getting a certain gender, thus allowing the BB/BG/GB/GG analysis.

Confusion 13y ago

Fine, then I pose to you the same problem description, with the added fact that the stranger is proud of having a boy, will only draw attention to a boy and will thus say "at least one is a boy", independent of whether the other is a girl, while he would never say "at least one is a boy". You can't solve a logic puzzle by arguing the psychology of a persona depicted in the puzzle.

stephencanon 13y ago
You can’t solve a logic puzzle that way, but you can explain the pragmatics of why the “logical” answer is contrary to our expectations based on interaction with real people.