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I've been thinking about this for an hour, and I'm now convinced that the author is wrong. The fact that we found out about one of the children from the father

I've been thinking about this for an hour, and I'm now convinced that the author is wrong. The fact that we found out about one of the children from the father means that all probabilities are not equal, even though they're treated here like they are. The difference is between the information being offered, and determined independantly. I'll do this with the boy/girl problem, for simplicities sake. If we ask a man if he has at least one boy, and he says yes, we can work out the chance the other child is a boy like so: Assume all four possibilities are equally likely: BB BG GB GG If we ask him if he has at least one boy, and he says yes, we effectively filter off GG, which brings the list down to: BB BG GB Therefore, the chance of the other child being a boy is 1/3. Pretty straight forward. However! Because the father offered the information on his own, it effectively turns it into the author's other problem, where the older child is a boy, find out the gender of the younger child. The trick is that he's equally likely to give information about either of his children, therefore there are eight possibilities: He gives information about child A: BB - B BG - B GB - G GG - G He gives information about child B: BB - B BG - G GB - B GG - G There are still 3 possibilities, but BB is twice as likely as the others, because if both children are a boy he's definitely going to reveal the gender of one of them as a boy; whereas if one is a girl and one is a boy, there's only a 50% chance he will. BB - 50% GB - 25% BG - 25% So, the answer to this: You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.” What is the probability that the other child is also a son? Is 1/2 The answer to this: A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son? Is 13/27 It's nitpicky, but I think the author should be very exact about this kind of thing, since he's trying to clear things up.

jfager 15y ago
You're making the exact mistake the author is cautioning against, which is assuming the day doesn't matter. Write out all the possibilities (see my other comment in this thread), eliminate the dupe, and you get 13/27.
- reitzensteinm 15y ago
  Did you read my post carefully? I do get 13/27, when presented with the information from a neutral third party - i.e. filter for all 2 child families with one son male/Tuesday, what is the chance the other is also male/Tuesday. But the fact that the father voluntarily offered up the information changes the probability distribution. We can assume he's selecting one of his children at random, and revealing their birthday and gender. If only one of his children is a male/Tuesday, there's a 50% chance he'll say male/Tuesday. If both are, there's a 100% chance. So I'm not counting the possibility twice; I'm saying that given that the father reveals male/Tuesday, it's disproportionately likely that's as a result of having two male children born on a Tuesday compared to any other possibility.
  - bermanoid 15y ago
    We can assume he's selecting one of his children at random, and revealing their birthday and gender. This is the entire point of the article, IMO: we have to make some assumption about how we selected this guy to talk to, and how he chose what to tell us. It's not pinned down by the statement of the problem, and what you might consider a natural assumption is not necessarily what other people might assume. Which is why these problems tend to suck...
    - reitzensteinm 15y ago
      I definitely agree - I hate problems like this, because the difficulty is caused by ambiguity of English, not the problem itself. But even given other assumptions as to why the father selects the child he does - sort by date, males first etc, the author's answer of 13/27 is still almost certainly wrong - he should have just taken the father out of the equation completely.
      - brlewis 15y ago
        I agreee -- this problem should not be discussed in English. Python seems better: http://news.ycombinator.com/item?id=3290313 http://news.ycombinator.com/item?id=3290313
NickPollard 15y ago
The issue here is with assumptions - you have made a different set of assumptions from the author, and hence are getting a different result. A lot of people here are having similar issues, by misreading exactly what the initial proposition means. Your reasoning above relies on the 'likeliness' of a man giving you the information, which is something that is not meant to be a part of the problem. Although it is phrased as a man 'telling' you something, that statement is really a metaphor for 'you determine the following piece of information, 100% truthfully'. In particular, your explanation assigns agency to the man - that if he does in fact have a son, he may or may not choose to reveal the truth 'I have a son'. However it makes no allowance for the man lying - so you are assuming if he answers it will be truthfully, but you are allowing him the lie of omission. Whilst there is nothing specific that rules out your interpretation, it is not what is intended. Read it instead as: ---- There exists a man, A. A has exactly two children. The statement 'A has at least one Son, B' is true What is the chance that the statement 'The Non-B child of A, is a son' is true? ----
- reitzensteinm 15y ago
  I agree that that is what the author intended to communicate. But I actually think that's a bigger stretch than my own interpretation - it changes how the information was determined, which has a definite impact on the outcome. The reason why I posted was to suggest that the author should have worded it the second way, i.e. A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son? Which leaves no doubt. I guess I didn't really make that clear enough with my original post.
- jerf 15y ago
  I'd also observe that carefully read, this article is really about how important assumptions are, and not about the problem per se. The Peter Winkler quote is key.
bh42222 15y ago
0.48 ~ 0.5 Close enough for statistical purposes!
zerostar07 15y ago
That last comparison is false. In both cases p=13/27 .
monorailkitty 15y ago
Yeah, what matters is the contents of the initial set of families over which we determine probability. "A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?" If the man is randomly chosen from the set of all families the answer is 1/2. If the man is randomly chosen from the set of all families with a son born on a Tuesday then the answer is 13/27. The reason for the difference is that a boy/girl family has a 1/7 chance that the boy was born on a Tuesday whereas the boy/boy family has only a 13/49 chance. B G (7/49 probability of a Tuesday boy) G B (7/49 probability of a Tuesday boy) B B (13/49 probability of a Tuesday boy) 13 / (7 + 7 + 13) = 13/27
- anorwell 15y ago
  >If the man is randomly chosen from the set of all families the answer is 1/2. By assumption, the man has a son born on Tuesday, so this is hardly relevant. If A is a subset of B, then choosing x uniformly at random from A given that x is in B is the same as choosing uniformly at random from B.
  - monorailkitty 15y ago
    I agree but set B is not a uniformly chosen subset of A in this case. That is the core of the trick. The rule for choosing B is intuitively uniform but actually slightly favours families with a girl and a boy over those with two boys.
tripzilch 15y ago
The distinction you're making is in your first example both the man is arbitrary, as well as the day of the week. In the second example, only the man is arbitrary. With those assumptions what you say is correct. However, the question is "You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.”" and not "You meet a man on the street and he tells you he has two children, that one is a son and he tells you what day of the week he was born". So the day is not arbitrary, it's specifically Tuesday.

jfager 15y ago

You're making the exact mistake the author is cautioning against, which is assuming the day doesn't matter. Write out all the possibilities (see my other comment in this thread), eliminate the dupe, and you get 13/27.

reitzensteinm 15y ago
Did you read my post carefully? I do get 13/27, when presented with the information from a neutral third party - i.e. filter for all 2 child families with one son male/Tuesday, what is the chance the other is also male/Tuesday. But the fact that the father voluntarily offered up the information changes the probability distribution. We can assume he's selecting one of his children at random, and revealing their birthday and gender. If only one of his children is a male/Tuesday, there's a 50% chance he'll say male/Tuesday. If both are, there's a 100% chance. So I'm not counting the possibility twice; I'm saying that given that the father reveals male/Tuesday, it's disproportionately likely that's as a result of having two male children born on a Tuesday compared to any other possibility.
- bermanoid 15y ago
  We can assume he's selecting one of his children at random, and revealing their birthday and gender. This is the entire point of the article, IMO: we have to make some assumption about how we selected this guy to talk to, and how he chose what to tell us. It's not pinned down by the statement of the problem, and what you might consider a natural assumption is not necessarily what other people might assume. Which is why these problems tend to suck...
  - reitzensteinm 15y ago
    I definitely agree - I hate problems like this, because the difficulty is caused by ambiguity of English, not the problem itself. But even given other assumptions as to why the father selects the child he does - sort by date, males first etc, the author's answer of 13/27 is still almost certainly wrong - he should have just taken the father out of the equation completely.
    - brlewis 15y ago
      I agreee -- this problem should not be discussed in English. Python seems better: http://news.ycombinator.com/item?id=3290313 http://news.ycombinator.com/item?id=3290313

NickPollard 15y ago

The issue here is with assumptions - you have made a different set of assumptions from the author, and hence are getting a different result. A lot of people here are having similar issues, by misreading exactly what the initial proposition means. Your reasoning above relies on the 'likeliness' of a man giving you the information, which is something that is not meant to be a part of the problem. Although it is phrased as a man 'telling' you something, that statement is really a metaphor for 'you determine the following piece of information, 100% truthfully'. In particular, your explanation assigns agency to the man - that if he does in fact have a son, he may or may not choose to reveal the truth 'I have a son'. However it makes no allowance for the man lying - so you are assuming if he answers it will be truthfully, but you are allowing him the lie of omission. Whilst there is nothing specific that rules out your interpretation, it is not what is intended. Read it instead as: ---- There exists a man, A. A has exactly two children. The statement 'A has at least one Son, B' is true What is the chance that the statement 'The Non-B child of A, is a son' is true? ----

reitzensteinm 15y ago
I agree that that is what the author intended to communicate. But I actually think that's a bigger stretch than my own interpretation - it changes how the information was determined, which has a definite impact on the outcome. The reason why I posted was to suggest that the author should have worded it the second way, i.e. A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son? Which leaves no doubt. I guess I didn't really make that clear enough with my original post.
jerf 15y ago
I'd also observe that carefully read, this article is really about how important assumptions are, and not about the problem per se. The Peter Winkler quote is key.

bh42222 15y ago

0.48 ~ 0.5 Close enough for statistical purposes!

zerostar07 15y ago

That last comparison is false. In both cases p=13/27 .

monorailkitty 15y ago

Yeah, what matters is the contents of the initial set of families over which we determine probability. "A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?" If the man is randomly chosen from the set of all families the answer is 1/2. If the man is randomly chosen from the set of all families with a son born on a Tuesday then the answer is 13/27. The reason for the difference is that a boy/girl family has a 1/7 chance that the boy was born on a Tuesday whereas the boy/boy family has only a 13/49 chance. B G (7/49 probability of a Tuesday boy) G B (7/49 probability of a Tuesday boy) B B (13/49 probability of a Tuesday boy) 13 / (7 + 7 + 13) = 13/27

anorwell 15y ago
>If the man is randomly chosen from the set of all families the answer is 1/2. By assumption, the man has a son born on Tuesday, so this is hardly relevant. If A is a subset of B, then choosing x uniformly at random from A given that x is in B is the same as choosing uniformly at random from B.
- monorailkitty 15y ago
  I agree but set B is not a uniformly chosen subset of A in this case. That is the core of the trick. The rule for choosing B is intuitively uniform but actually slightly favours families with a girl and a boy over those with two boys.

tripzilch 15y ago

The distinction you're making is in your first example both the man is arbitrary, as well as the day of the week. In the second example, only the man is arbitrary. With those assumptions what you say is correct. However, the question is "You meet a man on the street and he says, “I have two children and one is a son born on a Tuesday.”" and not "You meet a man on the street and he tells you he has two children, that one is a son and he tells you what day of the week he was born". So the day is not arbitrary, it's specifically Tuesday.