However, it is relevant if families that first had a boy cannot name their first girl Mary Ann because she was not the first born child. That would imply that non-Girl,Girl families with at least one girl are half as likely to have a girl named Mary Ann as Girl,Girl families. And since there are twice as many non-Girl,Girl families, half of the first-born Mary Ann’s are with them (in the Girl, Boy families) and half are in the Girl, Girl families.

]]>Eliezer, you are correct given how you interpreted the problem. This is why the postscript was added about the wording being crucial and Bayes’ rule forcing you to make your assumptions transparent.

Karim (and I) interpreted the phrase “first-born girl” to mean the first girl born to a couple, even if it is the second child to be born. Thus we interpreted the dictator’s law in the first problem as saying, “Mary Ann shall be the name of the first girl a family has, even if they have a boy first, and if they have a girl first, the second girl shall be given another name,” whereas you interpreted the law as saying, “Mary Ann shall be the name of a girl only when it is the first child of a couple.”

Your interpretation means that only half of all non-GG families with girls will have a girl named Mary Ann. Under our interpretation, the first problem is just asking you to figure out the probability a family is GG given the fact they have a girl, since all families with at least one girl have a girl named Mary Ann. And that probability is 1/3. As you said, GG families are twice as likely to have a girl named Mary Ann in the second problem, so the probability is 1/2.

One problem with interpreting the “first-born girl” naming rule as you did is that you are forced to assume all BG families in the first problem never name their second children Mary Ann, although we are not given any information about a rule against naming second-born children Mary Ann if they are the first girl born to a family. I think the second problem also must imply our interpretation, since the second-born girl’s list is shortened by the selection of the first-born girl, (“for each second-born girl in the family a name must be randomly chosen from the remaining 9 names,”) which would not be possible in BG families.

]]>In the second case, a family with 2 girls is again twice as likely to have a girl named Mary Ann as a family with 1 girl (there are 2 girls instead of 1 who might randomly be assigned that name) so prior odds of 1:2 times a likelihood ratio of 2:1 again equals posterior odds of 1:1 that both children are girls.

Am I misunderstanding the problem? This looks almost exactly like a practice problem that I designed for one of my own sessions on Bayesianism so I might be misinterpreting it to say the same thing my practice problem did.

]]>One remaining question, is Henk’s mother-in-law actually named Mary Ann??

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