3 ms·

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 comme

by 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
    - jsdalton 15y ago
      I don't think this has anything to do with the ambiguity of language or English. It does, however, like the Monty Hall problem, require you to make assumptions about how/why a speaker presents certain information. Think of it like this. What would the father say if he did in fact have two boys who were both born on Tuesday. Would he really say, "I have two children and one is a son born on a Tuesday"? Wouldn't he instead say, "I have two children and both are sons born on a Tuesday."? I mean he could say it the first way, but such a comment would be borderline misleading. To say you have one son born on Tuesday when in fact you have two is technically correct, but I think the problem assumes that the man is speaking somewhat plainly. So I do agree with you that communication intent is ambiguous, but I agree with some others that this is sort of the whole point of the problem, and it's not always immediately obvious that statistical information is hiding in seemingly irrelevant data.
- jfager 15y ago
  The probability that he would make that statement given what his children are is a different question than the probability that the other child is also a boy given that he made that statement. Your conditional probabilities are correct, but your conclusion about what they mean for the original question is flawed. N = Total number of ways to have 2 children over 7 days = 14^2 = 196 B = Two sons born on Tuesday. O = Exactly one son born on Tuesday. A = At least one son born on Tuesday. T = Two sons. S = The statement. Priors: P(B) = 1/N = 1/196 = 0.005 P(O) = 26/N = 26/196 = 0.133 P(A) = P(B) + P(O) = 27/N = 27/196 = 0.138 Your conditional probabilities: P(S|O) = .5 P(S|B) = 1 An interesting number we can infer from your conditionals is the probability that a father selected at random would make the statement: P(S) = P(S|B)P(B) + P(S|O)P(O) = 1.0 * 0.005 + 0.5 * 0.133 = 0.0715 But the question we asked about the other child already takes into account the fact that he did make that statement, meaning we're back to only caring about those 27 cases: P(B|S) = 1/27 = 0.037 P(O|S) = 26/27 = 0.963 P(A|S) = 27/27 = 1.0 P(T|S) = 13/27 = .481 Another interesting number we can infer from your conditionals is the probability that a father would make the statement given that at least one of his children was a boy born on Tuesday: P(S|A) = P(S|B)P(B|A) + P(S|O)P(O|A) = 1.0 * 0.037 + 0.5 * 0.963 = 0.519 If you still think this is incorrect, can you point to exactly which number is wrong and explain why?
  - reitzensteinm 15y ago
    The problem I have is that you're assuming that each of the 27 outcomes has equal probability, but the chance that we received the information in the way we did makes that a flawed assumption. The best analogous problem is the German Tank Problem: http://en.wikipedia.org/wiki/German_tank_problem http://en.wikipedia.org/wiki/German_tank_problem If we have destroyed a single German tank with a serial number 100, we can at least to begin to make an estimate on the size of the German force, by basically asking the question: "If they have 200 tanks, what was the chance one we randomly killed was this serial number? 500? 1000?" And then combining n=100->infinity to form a probability distribution. You can then say that there is an x% chance that Germany has 500 tanks, and a y% chance that Germany has 10,000 tanks. However - if instead, we asked 'does there exist a German tank with a serial number 100', and the answer is yes, this does NOT tell us anything past the fact that their tanks are >= 100 in number. We have the exact same information, but how it was determined changes the outcome drastically. Does that make sense?

by jfager 15y ago

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
  - jsdalton 15y ago
    I don't think this has anything to do with the ambiguity of language or English. It does, however, like the Monty Hall problem, require you to make assumptions about how/why a speaker presents certain information. Think of it like this. What would the father say if he did in fact have two boys who were both born on Tuesday. Would he really say, "I have two children and one is a son born on a Tuesday"? Wouldn't he instead say, "I have two children and both are sons born on a Tuesday."? I mean he could say it the first way, but such a comment would be borderline misleading. To say you have one son born on Tuesday when in fact you have two is technically correct, but I think the problem assumes that the man is speaking somewhat plainly. So I do agree with you that communication intent is ambiguous, but I agree with some others that this is sort of the whole point of the problem, and it's not always immediately obvious that statistical information is hiding in seemingly irrelevant data.
jfager 15y ago
The probability that he would make that statement given what his children are is a different question than the probability that the other child is also a boy given that he made that statement. Your conditional probabilities are correct, but your conclusion about what they mean for the original question is flawed. N = Total number of ways to have 2 children over 7 days = 14^2 = 196 B = Two sons born on Tuesday. O = Exactly one son born on Tuesday. A = At least one son born on Tuesday. T = Two sons. S = The statement. Priors: P(B) = 1/N = 1/196 = 0.005 P(O) = 26/N = 26/196 = 0.133 P(A) = P(B) + P(O) = 27/N = 27/196 = 0.138 Your conditional probabilities: P(S|O) = .5 P(S|B) = 1 An interesting number we can infer from your conditionals is the probability that a father selected at random would make the statement: P(S) = P(S|B)P(B) + P(S|O)P(O) = 1.0 * 0.005 + 0.5 * 0.133 = 0.0715 But the question we asked about the other child already takes into account the fact that he did make that statement, meaning we're back to only caring about those 27 cases: P(B|S) = 1/27 = 0.037 P(O|S) = 26/27 = 0.963 P(A|S) = 27/27 = 1.0 P(T|S) = 13/27 = .481 Another interesting number we can infer from your conditionals is the probability that a father would make the statement given that at least one of his children was a boy born on Tuesday: P(S|A) = P(S|B)P(B|A) + P(S|O)P(O|A) = 1.0 * 0.037 + 0.5 * 0.963 = 0.519 If you still think this is incorrect, can you point to exactly which number is wrong and explain why?
- reitzensteinm 15y ago
  The problem I have is that you're assuming that each of the 27 outcomes has equal probability, but the chance that we received the information in the way we did makes that a flawed assumption. The best analogous problem is the German Tank Problem: http://en.wikipedia.org/wiki/German_tank_problem http://en.wikipedia.org/wiki/German_tank_problem If we have destroyed a single German tank with a serial number 100, we can at least to begin to make an estimate on the size of the German force, by basically asking the question: "If they have 200 tanks, what was the chance one we randomly killed was this serial number? 500? 1000?" And then combining n=100->infinity to form a probability distribution. You can then say that there is an x% chance that Germany has 500 tanks, and a y% chance that Germany has 10,000 tanks. However - if instead, we asked 'does there exist a German tank with a serial number 100', and the answer is yes, this does NOT tell us anything past the fact that their tanks are >= 100 in number. We have the exact same information, but how it was determined changes the outcome drastically. Does that make sense?