{"id":1090,"date":"2009-01-03T23:23:19","date_gmt":"2009-01-03T14:23:19","guid":{"rendered":"https:\/\/regex.info\/blog\/2009-01-03\/1090"},"modified":"2009-01-03T23:23:19","modified_gmt":"2009-01-03T14:23:19","slug":"common-sense-blinded-by-math","status":"publish","type":"post","link":"https:\/\/regex.info\/blog\/2009-01-03\/1090","title":{"rendered":"Common Sense: Blinded by Math"},"content":{"rendered":"\n\n\n
\nNOTE<\/b>: Images with an icon next to them have been artificially shrunk to better fit your screen; click the icon to restore them, in place, to their regular size.\n<\/div>\n\n\n

It's amazing how many people can talk themselves into believing something that's obviously wrong.<\/p>\n\n

The other day the following question was posted<\/a>\non a widely<\/span>-read programming blog by Jeff Atwood:<\/p>\n\n

Let's say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl.<\/span> What are the odds that person has a boy<\/span> and a girl<\/span>?<\/div>\n\n

It quickly got almost 1,000 comments with people arguing about what the correct\nanswer is.<\/p>\n\n

Common sense tells us that gender of an unknown person is a 50<\/span>\/50\nproposition. You strike<\/span> up a conversation<\/span> with someone who says “yeah, I've\ngot two kids....” and you know that there's a 50<\/span>\/50 chance as to the gender\nof each kid. The person<\/span> continuing “... and that's my daughter on the\nswing” has not revealed anything about the gender of the other kid. Thus,\nthere's a 50<\/span>\/50 chance that the other kid is a boy,<\/span> and hence a 50<\/span>\/50\nchance that “the person has a boy<\/span> and a girl.<\/span>” This seems straightforward\nand common-sensical. 50% either way.<\/p>\n\n

Yet, there were a lot of comments arguing that the answer is 67%.\nConsider all four ways someone could end up with two kids:<\/p>\n

  • girl then boy<\/li>\n
  • girl then girl<\/li>\n
  • boy then girl<\/li>\n
  • boy then boy<\/li>\n<\/ul>\n\n

    ...and eliminate the “boy then boy” possibility because the question\ntells us that at least one of the kids is a girl,<\/span> you end up with three\nremaining pairings that are possible in this situation. Two of the<\/span> three\ninvolve both genders, so the odds are 2\/3... about 67%... that the person\n“has a boy<\/span> and a girl.<\/span>”<\/p>\n\n

    Well, that seems pretty solid, and we all know that math can sometimes\nbe counter-intuitive, so.... it must be true?<\/p>\n\n

    It seems that a lot of people are willing to let themselves be swayed by\nwhat they feel is a mathematical<\/span> explanation, even when it flies in the\nface of the most simple, basic common sense. Common sense tells us that\nrevealing the gender of one kid does not indicate anything about the other,\nso where is the flaw in the logic that leads to the 67% answer that\notherwise (except for common sense) seems solid?<\/p>\n\n

    The key point here<\/b> is whether the revelation about the girl gives\nus information intrinsically about a single<\/span> child, or intrinsically about\nthe pair:<\/p>\n\n

      \n
    • If the information is about one single child, it tells us everything about one kid and nothing about the other kid.<\/li>\n
    • If the information is about the pair, we know nothing specific about either kid, only one new datapoint about the pair.<\/li>\n<\/ul>\n\n

      The difference manifests itself in what we include and exclude when\ncalculating the odds based on the new information.<\/b><\/p>\n\n

      Let's look at the “information about the pair” situation first....<\/p>\n\n

      After finding out that they have two kids, if you specifically ask “is\none of your kids a girl<\/span>?” and get a yes<\/span> answer, you know, every time, that\nat least one is a girl<\/span> and that it's impossible for both to be boys.\nLikewise, if you get a “no” answer, you know every time that they have two\nboys. When the answer is yes and you move on to calculate the odds that the\nperson “has a boy<\/span> and a girl<\/span>”, you specifically include every two-kid\npermutation that includes a girl<\/span>:<\/p>\n\n