OtherDave, it's not just about birth order, the same difference in probability distributions would apply if we indexed the children according to height (e.g. 'is your tallest child a daughter') or according to foot-size or according to academic achievements or according to paycheck-size or whatever. As long as there's indexing involved, "child A is daughter" gives us 1/2 possibility both are daughters, and "at least one child is daughter" gives us 1/3 possibility both are daughters. And more bizarre differentiations (see below) give us different results.
In the first case, "I have at least one daughter", why can't I calculate the probability of two daughters as "I know one child is female, so I'm being asked for the probability that the other child is female, which is one-half"?
Because there's no "other child" when the information he provides is "at least one". There may be a single "other one", or there may be no single "other one" if both of his children are girls.
But it may be better if you try to think of it in the sense of Bayesian evidence.
Please consider the even more bizarre scenario below:
A. - I have two children
B - Is at least one of them a daughter born on a Tuesday?
A - Yes
Now the possibility he has two daughters is higher than 1/3 but less than 1/2.
Why? Because a person having two daughters has a greater chance of having a daughter born on a Tuesday, than a person that has only one daughter does. This means that having a daughter born on a Tuesday correlates MORE with people having two daughters than with people having only one daughter.
Likewise:
The more daughters someone has, the more likely that at least one of them is class-president. This means that it increases the probability he has many daughters. But if the question is
Now (to return to the simpler problem):
I'm avoiding the math, but this mere fact ought suffice to show you that the worth of the respective evidence "at least one child is daughter " and "eldest child is daughter" correlates different between the two hypotheses, so they support each hypothesis differently.
This thought-experiment has been on my mind for a couple of days, and no doubt it's a special case of a more general problem identified somewhere by some philosopher that I haven't heard of yet. It goes like this:
You are blindfolded, and then scanned, and ninety-nine atom-for-atom copies of you are made, each blindfolded, meaning a hundred in all. To each one is explained (and for the sake of the thought experiment, you can take this explanation as true (p is approx. 1)) that earlier, a fair-coin was flipped. If it came down heads, ninety-nine out of a hundred small rooms were painted red, and the remaining one was painted blue. If it came down tails, ninety-nine out of a hundred small rooms were painted blue, and the remaining one was painted red. Now, put yourself in the shoes of just one of these copies. When asked what the probability is that the coin came down tails, you of course answer “.5”. It is now explained to you that each of the hundred copies is to be inserted into one of the hundred rooms, and will then be allowed to remove their blindfolds. You feel yourself being moved, and then hear a voice telling you you can take your blindfold off. The room you are in is blue. The voice then asks you for your revised probability estimate that the coin came down tails.
It seems at first (or maybe at second, depending on how your mind works) that the answer ought to be .99 – ninety-nine out of the hundred copies will, if they follow the rule “if red, then heads, if blue then tails”, get the answer right.
However, it also seems like the answer ought to be .5, because you have no new information to update on. You already knew that at least one copy of you would, at this time, remove their blindfold and find themselves in a blue room. What have you discovered that should allow you to revise your probability of .5 to .99?
And the answer, of course, cannot be both .5 and .99. Something has to give.
Is there something basically quite obvious that I'm missing that will resolve this problem, or is it really the mean sonofabitch it appears to be? As it goes, I'm inclined to say the probability is .5 – I'm just not quite sure why. Thoughts?