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Math Help - Logic puzzle: Child gender compounded with irrelevant data

  1. #1
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    Logic puzzle: Child gender compounded with irrelevant data

    Select all people who have two children, at least one of which is a boy born on Tuesday. Given one of those individuals, what is the probability that their other child is a boy?
    Common sense suggests there are only two possible outcomes: TuesdayBoy+Boy, and TuesdayBoy+girl, each with a probability of 1/2.

    But some hard-charging pundit for the other team has gone on to suggest otherwise by creating an outcome space like so:


    Mon-Boy Tue-Boy Wed-Boy Thu-Boy Fri-Boy Sat-Boy Sun-Boy Mon-Girl Tue-Girl Wed-Girl Thu-Girl Fri-Girl Sat-Girl Sun-Girl
    Mon-Boy
    XXXXXX











    Tue-Boy XXXXXX
    XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX
    Wed-Boy
    XXXXXX











    Thu-Boy
    XXXXXX











    Fri-Boy
    XXXXXX











    Sat-Boy
    XXXXXX











    Sun-Boy
    XXXXXX











    Mon-Girl
    XXXXXX











    Tue-Girl
    XXXXXX











    Wed-Girl
    XXXXXX











    Thu-Girl
    XXXXXX











    Fri-Girl
    XXXXXX











    Sat-Girl
    XXXXXX











    Sun-Girl
    XXXXXX












    From this table of outcomes, 27/196 possible two-child families have at least one Tuesday-Boy. Furthermore, 13/196 outcomes are TuesdayBoy+Boy, while 14/196 outcomes are TuesdayBoy+girl. And I'm willing to believe that drawing a random entry from this outcome space will result in TuesdayBoy+Boy with a probability of 13/196, and TuesdayBoy+Girl with a probability of 14/196.

    But I'm pretty sure the assessment that this represents the quoted question accurately is in error. And I think the mechanism of selecting the subset of qualified results is the source of that error.

    Any help?
    Last edited by AngleWyrm; February 8th 2013 at 03:05 PM.
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  2. #2
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    Re: Logic puzzle: Child gender compounded with irrelevant data

    The table is as good a way as any to show that the probablity in question is 13/27.

    The 'hard-charging pundit' (loving that!) just forgot the two indications ('Select all people who' and 'Given one of those') that the set of 27 is the bag out of which one of the 13 is to be drawn.
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  3. #3
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    Re: Logic puzzle: Child gender compounded with irrelevant data

    I will agree that 13/196 possibilities in the example outcomes space are boy-boy.

    But this seems like a misuse of a mathematical construct to me. Substitute any irrelevant fact into the table, and we get an equally arbitrary answer:

    I flipped a fair coin twice, and I can tell you about one of the results: One outcome was a head; also, I didn't own any cats at the time. What are the odds the other result was also a head? [P(OwnCat) declared as 1/2 my lifetime]
    Last edited by AngleWyrm; February 10th 2013 at 04:11 AM.
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