Probability chestnuts make my teeth ache, and I frequently fall for them, but… this seems bogus to me.
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”?
Introducing birth order to this reasoning seems entirely unjustified. Conversely, if it is justified, I don’t understand why “the options are (GG, BGX, BGY, GBX, GBY), therefore the probability of two girls is one-fifth” (where X and Y stand for any other factor I choose to take into consideration… for example, “taller than the median height for a member of that sex” and “shorter than...”, respectively) isn’t equally justified.
Put differently: if I’m going to consider birth order, it’s not clear why I should expect GG, BG, and GB to have equal likelihood.
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:
Do you have at least one daughter who’s class-president?
Yes
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
“Is your eldest daughter a class-president”?
Yes
This doesn’t correlate with number of daughters, because we just care about one specific them, and the rest of them may equally well be zero or a dozen.
Now (to return to the simpler problem):
person having one daughter and one son—has 50% chance of having “eldest child be a daughter”, has 100% chance of having “at least one child be a daughter”
person having two daughters—has 100% chance of having “eldest child be a daughter”, has 100% chance of having “at least one child be a daughter”
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.
Is this a joke?
Probability chestnuts make my teeth ache, and I frequently fall for them, but… this seems bogus to me.
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”?
Introducing birth order to this reasoning seems entirely unjustified. Conversely, if it is justified, I don’t understand why “the options are (GG, BGX, BGY, GBX, GBY), therefore the probability of two girls is one-fifth” (where X and Y stand for any other factor I choose to take into consideration… for example, “taller than the median height for a member of that sex” and “shorter than...”, respectively) isn’t equally justified.
Put differently: if I’m going to consider birth order, it’s not clear why I should expect GG, BG, and GB to have equal likelihood.
Nope. You’ve fallen for this one.
You might also like the scenario where the additional piece of information is that the one child was born on Tuesday.
Never mind, I lost track of who was being responded to.
I wasn’t responding to you. I agree with your analysis.
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.
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:
Do you have at least one daughter who’s class-president?
Yes
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
“Is your eldest daughter a class-president”?
Yes This doesn’t correlate with number of daughters, because we just care about one specific them, and the rest of them may equally well be zero or a dozen.
Now (to return to the simpler problem):
person having one daughter and one son—has 50% chance of having “eldest child be a daughter”, has 100% chance of having “at least one child be a daughter”
person having two daughters—has 100% chance of having “eldest child be a daughter”, has 100% chance of having “at least one child be a daughter”
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.
Not a joke at all. But it is a source of flamewars as virulent as Monty Hall and 0.999 repeating. :)
ArisKatsaris explained it well, giving the fully correct setup, so I won’t add anything more.