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Boys Puzzle

  1. Jul 29, 2010 #1
    Your new mathematics-obsessed friend says to you, "I have two children.
    One is a boy born on Tuesday. What is the probability I have two boys?"

    The first thing you think to ask him is, "What the heck does Tuesday have to do with it?"
    "Everything!", he replies...

    So what is the probability?
     
  2. jcsd
  3. Jul 30, 2010 #2
    Tuesday does indeed have nothing to do with it, assuming he's truly anal, or without further information. Hence, the probability, assuming his statements are true, is 1/3.

    However, using intuition, I might infer from the inclusion of "Tuesday", that if it's important, that he uses it as a way of distinguishing his children. If so, further inferring that the day of the week of a birth is generally a less useful statistic than gender, I might conclude that he has twin boys, one born on Tuesday, and the other born on either Monday or Wednesday. Hence, the day of the week becomes more relevant as a distinguishing factor, whereas if the children were years apart or were of different genders, one probably wouldn't use the day of the week as a method of distinguishing them. Hence, the probability of him having 2 boys would increase to some higher number, depending on how sure of myself I am in inferring my friend's likely precedence on distinguishing his children.

    DaveE
     
  4. Jul 30, 2010 #3
    The day is important, and the answer is not 1/3. Keep trying.....

    Pete B
     
  5. Jul 30, 2010 #4

    Jonathan Scott

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    My attempt:

    Firstly, from the point of view of the mathematician, the answer is that the probability is either 0 or 1, so that's one possible answer.

    If "Tuesday" was just a piece of information that was provided randomly about one child, it would provide no information about the other child, so "Tuesday" could not have any significance and the probability of both being boys would simply be 1/2.

    The only way that "Tuesday" can have significance is if there is an unstated assumption that it provides some information about the other child, presumably that the other child is NOT "a boy born on a Tuesday" (so the other child is either a girl or a boy born on some other day of the week). As this is unstated, I consider it cheating, but in that case I think the probability of the other one also being a boy, and hence two boys, would be 6/13.

    This is not the same as knowing that there are two children and obtaining the answer to "is at least one of them a boy", in which case the chances of two boys would of course be 1/3.
     
  6. Jul 30, 2010 #5
    I don't follow here-- Why not? The probability given 2 children (assuming 50% chance of having either gender in an individual case, and no hermaphrodites or otherwise) is 1/4 of 2 boys, 1/4 of 2 girls, and 1/2 of having one of each gender. Knowing that 1 is a boy eliminates the possibility of having 2 girls, meaning that there's a 1/3 chance of having two boys.

    As stated, if you wantonly assume that one of them being a "boy born on a Tuesday" means the other child is NOT a "boy born on Tuesday", then you could get some other outcome. But that's an assumption. The other one could just as well be another boy born on Tuesday, a girl born on Tuesday, a boy born on Friday, or a girl born on Monday.

    DaveE
     
  7. Jul 30, 2010 #6

    Ygggdrasil

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    Two cases: 1) the elder child is a boy born on Tuesday, 2) the younger child is a boy born on Tuesday.

    case 1) There are 7 possible scenarios for the younger child to be a girl (girl born on any day of the week) and 7 possible scenarios for the younger child to be a boy.

    case 2) There are 7 possible scenarios for the older child to be a girl, but only six possible scenarios for the older child to be a boy (the case where both children are boys born on Tuesday was already covered above).

    Therefore, the probability of having two sons is 13/27 or just under 50%. See the link below for a useful graphical explanation.

    http://sciencenews.org/view/generic/id/60598/title/When_intuition_and_math_probably_look_wrong
     
  8. Jul 30, 2010 #7

    D H

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    Everything indeed. There is a big difference between
    • "I have two children. One is a boy."
    • "I have two children. The older of the two is a boy."
    • "I have two children. One is a boy born on Tuesday."

    Most of us are pretty lousy at dealing with conditional probabilities. Many people, even many otherwise very intelligent people get the Monty Hall problem wrong, and many apparently got this one wrong as well.

    A paper on why we suck at this kind of reasoning: http://fox-lab.org/papers/Fox&Levav(2004).pdf [Broken]
     
    Last edited by a moderator: May 4, 2017
  9. Jul 30, 2010 #8

    Jonathan Scott

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    As usual, it depends on the implicit question, which defines the space of other possible answers, as described in the sciencenews article. If you asked someone "is your first child a boy or a girl?" and "on what day of the week was the child born?", then neither answer would tell you anything at all about the second child.
     
  10. Jul 30, 2010 #9
    Ygggdrasil has the correct answer and solution. It is worth noting that the item of interest here is not unique in that the boy was born on Tuesday. Any other specified day would produce the same probability. Also consider that other conditionals, such as a statement that the boy was "born in February", would also produce a probability that differs from the unconditional problem statement. Thus the intuitive component is found in recognizing that it is the presence of any conditional of this type that alters the result, not just the one specific conditional mentioned in this example.


    Pete B
     
    Last edited: Jul 30, 2010
  11. Jul 30, 2010 #10

    D H

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    Correct. This oldie but goodie admits a lot of variations, each of which is an interesting case study regarding how/why we are so bad at computing probabilities -- and such easy marks for con men.
     
  12. Jul 30, 2010 #11
    lessee... the possibilities originally were 14 for child1 (boy or girl times 7 days of week) and 14 for child2. and 14x14 gives a space of 196 combinations. 1 in 14 have child1 as boy-tuesday. 1 in 14 have child2 as boy-tuesday. exactly 1 has both child1 and child2 as boy-tuesday. so... it should be (1/14 +1/14)*196 - 1 = 28-1 = 27 possibilities. so, a chance of 1/27 that both are boys with at least 1 born on tuesday.

    edit:^yeah, i did that wrong i think. 27/196 maybe? i need more coffee. and got something else i need to work on, too. i'd like to spend some more time thinking about that solution in the article, but i'm still feeling like it's answering a question that's not being asked at this point.

    but... the way the question is worded, knowing that one child was born on tuesday does not reduce the space you're looking at. it is extraneous information. you could also say that one has blue eyes, webbed feet, and a freckle on his nose. it does not matter. all we care about is the sex. and regarding sex, we know that one is a boy (ergo, not both girls). that leaves three possibilities, of which only one is BB. so 1/3.
     
    Last edited: Jul 30, 2010
  13. Jul 30, 2010 #12
    Suppose instead of the original question, the friend asks, "I have two children.
    One is a boy born on Tuesday. What is the probability the other is a boy?"

    Or perhaps "I have one child who is a boy born on Tuesday and my wife is pregnant with another child. What are the chances that child will be a boy?"
     
  14. Jul 30, 2010 #13
    the first doesn't specify which child is a boy. so the question is really, what are the odds that both are boys, given that i've excluded the possibility of girl-girl. 1/3.

    the second specifies the first is a boy. and that gives you no information about the future event. each new birth has the same odds: 1/2. this is also why schemes to pick lottery numbers do not work. each new lottery is an independent event.
     
  15. Jul 30, 2010 #14
    That's... quite odd, but just to prove it to myself, I wrote a program to simulate it, and indeed, it's true. I'm not quite sure I've quite wrapped my head around why specifying the date of birth ought to affect anything, but indeed, it does. As apparently would specifying any other equally arbitrary statistic (time of day, day of the year, etc).

    DaveE
     
  16. Aug 12, 2010 #15
    Suppose: "I have two children. One is a boy born on some day. What is the probability I have two boys?" Does this change the answer?
     
  17. Aug 12, 2010 #16
    I think that's the part that trips people up:

    "I have two children".

    Probability of two boys is currently 1/4.

    "One of my children is a boy".

    Probability of two boys is now 1/3.

    "The boy in question was born on a particular day of the week. I haven't told you which day that is, and it goes without saying that of course he was born on a particular day. But in just a second, I'm going to tell you which day that was, but I haven't yet."

    Probability of two boys is still 1/3.

    "The boy was born on a Tuesday".

    Probability of two boys is now 13/27.

    Effectively, the way I'm imagining it, by specifying that one of the boys was born on a Tuesday, you've just now eliminated ALL of the possibilities of having NO boys born on Tuesdays, which is a hefty portion of options-- hence bringing the probability a lot closer to 1/2.

    DaveE
     
  18. Aug 12, 2010 #17
    Suppose you meet 196 mathematicians, each one of them picks one of his kids at random and asks you the question.

    98 of them will say "one of them is a girl".

    84 of them will say "one of them is a boy born on Monday / Wednesday / ..."

    Of the remaining 14, 7 will have two boys and 7 will have a boy and a girl.

    The answer is 1/2, as to be expected.
     
  19. Feb 9, 2011 #18
    It's 1/2. And no, I am not missing a subtle point about probability. You are.
    And the answer to the problem you reduced it to, by omitting "Tuesday" from peteb's question, is also 1/2.
    You are right in your assessment at the end, but you need to add more to your list.
    1. A: "I have two children." B:"Is one of them is a boy?" A: "Yes."
    2. A: "I have two children." B:"Is one of them is a boy born on a Tuesday?" A: "Yes."
    The answer, when these two are the condition, is 1/3 and 13/27, respectively. The answer when any of yours is the condition is 1/2. The reason they are different, is that my condition #1 is different than your condition #1, and my #2 is different than your #3.

    There could be some people who would respond "Yes" to my #1, but who, for whatever unspecified reason they are telling you what they told you in your list, would not say "One is a boy." They would say "One is a girl." Since you provided no reason for why they would choose one statement over the other, we can only assume both are equally likely when both are possible. So, while it is true that there are twice as many parents of two who have a boy and a girl, as those who have two boys; the same number in each group would tell you "One is a boy." The same arguments apply to the Tuesday version.

    And the really ironic thing, is that the people who get the wrong answer for the Monty Hall Problem you mentioned are taking the same approach to it, as you do to this one. As Fox and Levav write, they are "subjectively partitioning the sample space into n interchangeable events, editing out events that can be eliminated on the basis of conditioning information, counting remaining events, then reporting probabilities as a ratio of the number of focal to total events."

    So, for your first condition, you partition the sample space into the four interchangeable events BB, BG, GB, and GG; edit out the GG event which is the only one that is impossible if you have a boy; and report the probability as the ratio of focal event left (BB) to total events left (BB, BG, GB). Just like with Monty Hall they partition the sample space into the three interchangeable events where the prize is behind D1, D2, or D3; edit out the D3 (or whatever) event which is the only one that is impossible after you see what door the host opens; and report the probability as the ratio of focal event left (D1) to total events left (D1, D2).

    The technically-correct answer is not the ratio of the number of cases, but the ratio of the sum of the probabilities that each of the appropriate cases would produce the observed result. In the Two Child Problem, if P is the probability a parent of a boy and a girl would say "One is a boy," the answer is 1/(1+2P). In the Monty Hall Problem, if P is the probability Monty Hall would open Door #3 when he could also open Door #2, the answer is 1/(1+P). And in the Tuesday Boy problem, the answer is (1+12P)/(1+26P) for a similar P. If P=1 in any of these, you get the "wrong" answer that Fox and Levav talk about. If P=1/2, you get the right answer.

    The difference in my list of statements, is that P=1. In yours, it is really ambiguous what P is, but we can't assume anything except 1/2.
    No. It is requiring the presence of the conditional information that alters the probability. If a parent is selected at random from a group of parents who have two children and at least one boy, the answer is 1/3. If a parent is selected at random from a group of parents who have two children, and one gender is observed, the answer is 1/2 regardless of what that gender is.
    If this were true, then the answer would be 13/27 regardless of what day you mention. And if it is 13/27 regardless of what day you mention, it is also 13/27 for the first question. This is called the Law of Total Probability. If {X1, X2, ..., XN} are independent events that represent all possibilities, then for any event Y, P(Y)=P(Y|X1)*P(X1)+P(Y|X2)*P(X1)+...+P(Y|XN)*P(XN). If all the conditional probabilities are equal to the same P, then P(Y)=P because P(X1)+P(X2)+...+P(XN)=1.
     
  20. Feb 9, 2011 #19
    What is special about information which is passed on but already known?

    Your friend asks the following question

    I have two children.
    One is a boy.
    What is the probability I have two boys?

    After coming up with an answer you might then realise that the boy referred to must have been born on a certain day of the week.Your friend didn't mention that because it is a fact that is obvious.Armed with this new information,which was inherent in the question all along,how,if at all,does your answer now change?
     
    Last edited: Feb 9, 2011
  21. Feb 9, 2011 #20
    EDIT: I just read this post by davee and it makes sense now.
     
    Last edited: Feb 9, 2011
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