When intuition and math probably look wrong

I know Gary, the person who presented this, and I was there when it happened. He fully intended this to ignite the argument it has.<p>Firstly, as presented it is clearly ambiguous. It is intended to be ambiguous, but in such a way that people who are familiar with the original version will get suckered into believing that it's well formed.<p>Secondly, if presented precisely, the answer usually given is either 13/27 or 1/2, depending on which version.<p>Finally, this is like the Monty Hall problem all over again. There are people arguing vehemently and without listening at all, demonstrating clearly that they are excellent at missing the point.<p>In case you're wondering, here's one statement and answer.<p>Suppose on knock on people's doors and ask - Do you have exactly two children? If they answer no, I move on. If they answer yes I then ask - Is at least one of them a boy born on a Tuesday? If they say no, I move on.<p>If they look surprised and say "Yes," what is the probability that they have two boys?<p>Answer: 13/27.<p>Yes, it really is.<p>If you replace the second question with "Is at least one of them a boy with red hair, left-handed, plays piano, was born on Tuesday, and has a cracked left upper incisor" then if the answer is "Yes" then the probability of both children being boys is almost exactly 50%.<p>If, instead, you replace the second question with "Is at least one a boy" then the probability of two boys is 1/3.<p>Finally, suppose you see a parent that you know has two children in the park with a boy. Now the probability of two boys is 50%, because, assuming uniform probabilities, having two boys makes it more likely you see them with a boy.<p>tl;dr: It's hard, and depends precisely on the assumptions you make.

Another riddle where the math is easy, but translating the riddle into math is hard. Which is to say, it's not really a math riddle at all. In fact, the riddle is complicated by the fact that it's posed as a riddle -- if you had to answer the question "in real life," you would probably know how you came about the information, so the problem would be straightforward.

I had a math teacher that got mad when I used the word "intuitively". Probably because most people don't understand that when your intuition doesn't match the mathematical result you have to hit your intuition with a hammer until it does.

Ok, I was about to rage about yet another article going on about the Two Children problem and getting it wrong by leaving out a whole host of children (i.e. children in families of more or less than two children). Then it surprised me by not only acknowledging it, but acknowledging that the the 50% answer is correct when we are selecting from an arbitrary family (as the original problem is usually presented).<p>It then goes on to acknowledge that the 33% answer would be correct if we specifically choose a two child family that fits the parameters of the problem beforehand.<p>It's all about why the information was selected.

isn't this the same kind of "intuitive" reasoning that fails at simpson's paradox? <a href="http://en.wikipedia.org/wiki/Simpsons_paradox" rel="nofollow">http://en.wikipedia.org/wiki/Simpsons_paradox</a><p>conditioning upon more events can lead to a higher probability. p(boy = 2 | boy &#62;= 1) &#60; p(boy = 2 | boy &#62;= 1, tuesday) (or more precisely, conditioning upon more events can yield a distribution with less entropy)

This is why I'm glad I was a mathematics major :)

The answer given is not even wrong.<p>The statement "I have two children, one of whom is a son born on a Tuesday" is semantically ambiguous. It can mean (1) "I have two children, and the quantity of them who are males born on a Tuesday is exactly one", (2) "I have two children, at least one of whom is a male born on a Tuesday", or even (3) "I have two children, and the maximum quantity of males born on the same Tuesday is one".